In this project we will cover the following parts:
Build a clean monthly zero-rate curve from Treasury yield data and use the fixed-income tools from Project 1.
Estimate and diagnose short-rate models: Vasicek, CIR, and rolling term-premium proxies.
Move from short-rate models to curve-level models: HW1F, G2-style two-factor dynamics, and PCA level/slope/curvature diagnostics.
Convert model-implied curve views into a dynamic duration ladder strategy using KRD exposure, carry, active risk, and transition costs.
Explain fixed-for-floating swaps and build a synthetic swap overlay that adjusts duration alongside the cash bond ladder.
Repeat the full framework on Japan JGB curves, where the economics are very different from the US rate market.
In this project, we will model the term structure of interest rates and use the resulting curve views to make duration decisions.
The yield curve connects interest rates to maturity. The 3-month rate can move as the central bank changes policy, while the 10-year rate also reflects growth expectations, inflation uncertainty, and the term premium. At the 30-year point, investors’ required compensation for long-horizon duration risk can have a large effect. These rates don’t always move together: the curve can shift, steepen, flatten, or change shape around its middle. A portfolio holding bonds at several maturities therefore needs a view of the whole curve, together with the sensitivity of its holdings to changes at different points.
We’ll use the bond mechanics from the earlier fixed-income project: discounting, duration, convexity, key-rate duration, ladder construction, and scenario PnL. Here, those tools become inputs to rate models and portfolio decisions. We’ll estimate short-rate dynamics, compare curve factors, and turn a one-month view of yields into a choice between lower, neutral, and higher duration. We can express the resulting view by changing the cash bond ladder or by adding a swap overlay to a fixed ladder.
We use the curve models to estimate both expected rate changes and the uncertainty around them, then connect those estimates to the duration exposure we want to hold.
The models will also give us a way to compare a curve view with the uncertainty and cost of implementing it across different bond maturities.
1) Setup and reused fixed-income data
We’ll start by importing the numerical and plotting tools together with the fixed-income functions used to prepare the curve and run the bond ladders. Our work follows these connected steps:
Curve data gives us the observed Treasury term structure.
Bootstrapping turns par yields into continuous zero rates.
Short-rate and curve-factor models estimate rate dynamics.
Bond ladder backtests turn curve views into portfolio returns.
Swap analytics add synthetic duration exposure on top of a cash ladder.
We use the same US Treasury yield data family that was already used in Project 1. For reproducibility, check the corresponding folder inside the repository data directory, where the README and script explain how the file is created: US Treasury yields data folder.
We’ll also use the ACM term-premium series, SOFR, and Japan MOF yields. Their data folders describe the sources and preparation steps, so we can trace each input back to the corresponding dataset.
We keep the rate-model calculations in the notebook so each state, loading, and portfolio decision can be followed as it is introduced.
The latest rows of the daily Treasury par-yield panel show the curve around January 2026. The front end is near 3.6–3.8%, while the 10-year is around 4.26% and the 30-year is close to 4.85%. So the latest observed curve is upward sloping from the 1-year area to the long end, but the very front end is still relatively high compared with the 2-year and 3-year region.
The low point at the front-to-intermediate maturities and the higher long end give us a useful starting shape. As we estimate rate dynamics, we’ll compare how changes at these different maturities affect the value and risk of a portfolio holding bonds across the curve.
The rates at each maturity are par yields: each one summarizes the coupon needed for a bond priced at par at that maturity, rather than the yield on a single zero-coupon payment.
2) Monthly curve construction and zero rates
We resample the daily curves to month-end because the strategy rebalances monthly. The selected curve starts in February 2006 and ends in January 2026, giving us 240 monthly observations. That is a reasonable sample for rolling curve estimation: long enough to include the pre-GFC tightening cycle, the zero-rate period, the 2020 shock, and the 2022–2025 rate-normalization regime.
The selected tenors are:
very short rates: 1M, 3M, 6M,
policy-sensitive rates: 1Y, 2Y, 3Y,
intermediate rates: 5Y, 7Y, 10Y,
long rates: 20Y, 30Y.
The zero curve gives us discount factors for individual cash flows at different maturities. Those discount factors connect term-structure parameters to bond prices, while the corresponding zero rates give us a consistent panel for estimating maturity-dependent curve movements. We use the bootstrapping tools introduced with the earlier bond-pricing work.
Keeping a consistent monthly panel also puts each maturity on the same observation dates before we estimate changes in the curve.
The selected monthly block runs from 2006-02-28 to 2026-01-31. Rolling estimates and strategy decisions begin after their required history has accumulated, so each decision uses a window of rates that was available at that time.
The block also starts after enough of the long-end Treasury data is available. The 20-year point has a later first valid date than most of the curve, and the selected window avoids a lot of early missing-value problems.
The resulting observations share the same set of maturities, which makes each month-to-month curve comparison directly aligned.
For model estimation we bootstrap zero rates from the par-yield curve. A par yield is the coupon rate that makes a bond price equal to par. A zero rate is the continuously compounded rate for a single maturity cash flow. If \(P(0,T)\) is the discount factor for maturity \(T\), the continuously compounded zero rate is:
\[
R(0,T) = -\frac{\log P(0,T)}{T}
\]
and the discount factor is:
\[
P(0,T) = e^{-R(0,T)T}
\]
Affine term-structure models usually start with zero-coupon bond prices. Coupon bonds and par swap rates can then be valued using the same discount curve. The resulting zero rates connect:
short-rate model equations,
yield-curve factor loadings,
bond ladder valuation,
swap fixed-leg annuities,
and scenario PnL.
We use PCHIP interpolation when bootstrapping. Its shape-preserving construction lets us interpolate between observed maturities without the large oscillations that high-order splines can produce around humps, inversions, or sparse long-end tenors. The same curve supplies the intermediate discount factors needed for cash-flow valuation.
The latest zero-rate table has the same economic shape as the par curve: the front end is around 3.6–3.8%, the 10-year zero is around 4.27%, and the 20Y/30Y area is near 4.96%. The curve is not flat. It has a low point around the 1Y–2Y region and then rises into the long end.
The low point around 1Y–2Y and the rise beyond it give us a clear example of why maturity matters. A rate at one point describes only one part of the curve, while a bond with several cash flows needs discount factors at each payment date. We’ll retain the full set of maturities for the analysis that follows.
The zero rate at each maturity summarizes the discount factor for a single payment at that particular maturity date.
We have 9023 daily observations, 240 monthly curve dates, and selected tenors from 1M to 30Y. Most Treasury tenors have a long history, while the 1M and 20Y points start later. The selected monthly block uses dates where all of these chosen maturities are available.
This tells us two things before modeling:
the monthly US sample is long enough for rolling 60-month model fits,
but the curve panel is still a constructed object, not a perfectly homogeneous raw market quote series.
The second point matters because model diagnostics later shouldn’t be read as pure theory failures. Sometimes a model looks imperfect because the observed data is a stitched historical yield curve with changing market structure, Fed regimes, and post-crisis liquidity conditions.
The observations reflect a historical curve assembled across changing market and policy environments.
3) Curve snapshots, short rates, and slopes
Before fitting the models, we’ll look at several curve snapshots. Their shapes show how the relationship between short, intermediate, and long rates changes across rate regimes, and they give us a visual reference for the dynamics we want the models to describe.
The main curve shapes we care about are:
high and inverted front end, usually connected to tight monetary policy,
steep upward curve, often connected to recession recovery or very low short rates,
flat curve, when short and long rates are similar,
humped curve, when intermediate maturities trade above both front and long end.
A one-factor short-rate model is good at describing a parallel-ish shift driven by the front end. It is weaker when the curve twists. A PCA or two-factor model is designed to handle twist and curvature more naturally.
Comparing several dates also makes it easier to see which parts of the curve have moved together and which have moved differently.
3.1 Curve shape and maturity exposure
The 10-year yield is a familiar reference rate, while a bond ladder has exposures at several maturity points. A 2Y/5Y/10Y/30Y ladder can gain or lose for different reasons depending on which part of the curve moves. Looking at the maturities together helps us connect each change to the relevant part of the portfolio.
A parallel shift moves all elements by the same amount and in the same direction. A steepener has the long end rising relative to the front end. A flattener has the front end rising relative to the long end. A curvature move changes the belly differently from the wings.
We can match each element of the change vector with a portfolio’s exposure at that maturity. That connection lets us describe how a particular movement in the whole curve affects the value of the bonds held in the ladder.
For example, a rise of 10 basis points is 0.001 in decimal yield units. We keep those units consistent when combining a change at each maturity with the portfolio’s rate sensitivity.
Show code
curve_years = np.array([1/12, 0.25, 0.5, 1, 2, 3, 5, 7, 10, 20, 30], dtype=float)sample_locs = [0, len(curve_dates) //4, len(curve_dates) //2, 3*len(curve_dates) //4, len(curve_dates) -1]sample_dates = [curve_dates[i] for i in sample_locs]fig, ax = plt.subplots()for date in sample_dates: ax.plot(curve_years, par_yields.loc[date, selected_tenors].to_numpy(float) *100, marker="o", label=date.strftime("%Y-%m"))ax.set_title("treasury par curve snapshots")ax.set_xlabel("maturity in years")ax.set_ylabel("par yield percent")ax.legend(ncol=2)plt.show()
The curve snapshots show several very different US rate regimes. The 2006 curve is high and fairly flat/inverted, which matches the late tightening-cycle environment before the global financial crisis. The 2011 curve is extremely steep: short rates were pinned close to zero, while the long end still had meaningful yield. The 2016 and 2021 curves are lower, but with different slope patterns. The 2026 curve is much higher across the curve, especially compared with the 2010s.
These snapshots cover high, low, flat, and steep curves within the same sample. A single parameter set will average over those different environments, whereas rolling estimation lets us estimate the dynamics again as the rate environment changes.
The observed shape at any one date is only one part of the information: the model also uses how rates have evolved across previous months.
We now isolate a few simple curve summaries:
a short-rate proxy from the front-end yield,
selected zero rates such as 2Y, 5Y, 10Y, and 30Y,
slope measures like \(10Y - 2Y\) and \(30Y - 5Y\).
We use these slopes as diagnostic summaries of the curve environment and the relative level of yields at different maturities. A positive \(10Y - 2Y\) means the 10-year yield is above the 2-year yield. A negative value means inversion. In strategy terms, slope helps explain whether longer bonds are compensating us more than shorter bonds, but it doesn’t fully determine whether extending duration is attractive because expected rate changes and carry also matter.
Both slopes are measured in yield differences, so multiplying a decimal difference by 10,000 expresses it in basis points.
The latest table shows a short-rate proxy near 3.76%, a 2-year zero near 3.53%, a 10-year near 4.27%, and a 30-year near 4.96%. The \(2s10s\) slope is about 75 bps, and the \(5s30s\) slope is about 116 bps.
So by the final date, the curve is no longer deeply inverted. The front end has moved down from the tightest policy levels, and the long end remains elevated. That is the kind of environment where a duration model can easily disagree with a simple “rates are high, buy duration” rule. If the long end has a persistent term premium component, long duration can still be risky even after the front end falls.
Across the sample, short rates and curve slopes move through several distinct monetary-policy and market regimes. We can follow those changes over time alongside the snapshots above.
The most important regimes are:
2006–2007: high front-end rates and a flat/inverted curve.
2008–2015: the front end collapses close to zero, while longer rates move around more gradually.
2020–2021: another zero-rate period after the pandemic shock.
2022–2024: the front end reprices upward very aggressively as the Fed tightens.
2025–2026: the front end begins to come down, while the long end stays relatively firm.
The same 10-year yield can mean different things depending on whether the front end is at zero, the Fed is cutting, inflation is rising, or term premium is repricing. Comparing the whole curve and its recent history gives the duration decision more context than the yield level on one date.
Show code
fig, axes = plt.subplots(1, 2, figsize=(13, 4), constrained_layout=True)axes[0].plot(short_rate.index, short_rate *100, label=f"{short_tenor.lower()} proxy")for col in ["2y", "10y", "30y"]: axes[0].plot(selected_yields.index, selected_yields[col] *100, label=col)axes[0].set_title("short rate and selected yields")axes[0].set_ylabel("percent")axes[0].legend()for col in slope_series.columns: axes[1].plot(slope_series.index, slope_series[col] *100, label=col)axes[1].axhline(0, color="black", linewidth=0.8)axes[1].set_title("curve slopes")axes[1].set_ylabel("percentage points")axes[1].legend()plt.show()
The short rate follows the policy cycle closely: it collapses rapidly in 2008, stays near zero for years, spikes after 2022, and then starts moving lower. The front end therefore reacts closely to changes in policy rates. The 10Y and 30Y rates move more gradually and don’t always follow the front end one-for-one.
The slope panel shows the large steepening after the GFC, the long flattening/inversion before and around the recent hiking cycle, and the later re-steepening. A duration strategy also chooses where on the curve to hold risk. Different slope environments can reward different maturity exposures, since changes at the front end and long end affect the positions in each part of the ladder differently.
4) ACM term premium and SOFR cash return
A Treasury yield can be decomposed roughly into two components:
where \(y_t^{(n)}\) is the \(n\)-year yield, the average expected short-rate term is the market’s expected path of policy rates, and \(TP_t^{(n)}\) is the term premium. The term premium is the extra compensation investors demand for holding a long-duration bond instead of rolling short bills.
The ACM term premium is a model-based estimate of that second component. We’ll use it as an external benchmark for the term-premium proxies from Vasicek and CIR, comparing their levels and movements with an estimate obtained from a richer term-structure model.
ACM gives us a common reference against which to compare both the direction and the level of these model-implied series.
Show code
def load_acm_10y(csv_path): raw = pd.read_csv(csv_path) clean = raw.copy() clean.columns = [str(c).strip().lower() for c in clean.columns] date_col =next((c for c in clean.columns if c in {"date", "dates"} or"date"in c), clean.columns[0]) tp_col =next((c for c in clean.columns if c.replace("_", "") in {"acmtp10", "tp10"}), None)if tp_col isNone: tp_col =next(c for c in clean.columns if"tp"in c and"10"in c) out = clean[[date_col, tp_col]].rename(columns={date_col: "date", tp_col: "acm 10y term premium"}) out["date"] = pd.to_datetime(out["date"], errors="coerce") out["acm 10y term premium"] = pd.to_numeric(out["acm 10y term premium"], errors="coerce") out = out.dropna().drop_duplicates("date").set_index("date").sort_index()if out["acm 10y term premium"].abs().median() >1: out["acm 10y term premium"] = out["acm 10y term premium"] /100return outacm_term_premium = load_acm_10y("../data/acm_term_premium.csv")acm_term_premium.tail()
acm 10y term premium
date
2025-12-31
0.007612
2026-01-30
0.007742
2026-02-27
0.006072
2026-03-31
0.006626
2026-04-30
0.007269
The ACM tail shows the 10-year term premium around 60–77 bps during late 2025 and early 2026. That is a very different world from the deeply negative term-premium estimates of some earlier post-GFC years.
A positive term premium means the estimated long yield contains compensation above the expected path of future short rates. Investors can demand that compensation for bearing inflation uncertainty, fiscal supply risk, duration volatility, or other long-horizon risks. For a duration strategy, extending exposure is more attractive when term premium is high and expected rate changes are favorable. A high premium can also compensate investors for risks that keep yields elevated, so the premium and the expected rate path both enter the investment decision.
A negative estimate can arise when long bonds provide valuable hedging benefits or when the model attributes more of the yield to the expected short-rate path.
We align ACM to the monthly curve dates by resampling to month-end and forward-filling onto the curve index. We’ll use the aligned series as a diagnostic benchmark for the model-implied proxies, rather than as an input to the trading signal.
The summary confirms no missing aligned values over the selected curve period. This is useful because the later correlation comparison between ACM and model-implied proxies won’t be distorted by a small overlapping sample.
The aligned observations share the curve dates, so correlations are computed using matching monthly observations.
The aligned ACM series ends at 0.774% in January 2026 after rising from about 0.57–0.76% in the previous months. The positive estimate indicates a meaningful model-implied risk-premium component at the long end of the US Treasury curve at the final date, in addition to the expected path of short rates.
This also gives us an interpretation checkpoint for the later models: if a simple short-rate model says the 10-year yield is far above the expected average short rate, that spread can be interpreted as a model-implied term-premium proxy. It won’t be a full no-arbitrage term-premium decomposition like ACM, but it is economically related.
We also load SOFR because the modern performance section evaluates bond returns relative to a cash benchmark. SOFR is the Secured Overnight Financing Rate. For a fixed-income strategy, raw return alone can be misleading when cash yields are high. A 3% annual return is impressive when cash is zero, but weak when cash earns 5%.
The monthly SOFR cash return is constructed from the SOFR index. If \(I_t\) is the index level at month-end, the monthly cash return is:
\[
R_{cash,t} = \frac{I_t}{I_{t-1}} - 1
\]
This gives a realistic cash baseline for the post-2020 sample. We don’t use SOFR for the full 2006–2026 backtest because the published SOFR index has a shorter history. For reproducibility, check: SOFR data folder.
Show code
sofr_start_date = pd.Timestamp("2020-03-02")sofr_raw = pd.read_csv("../data/sofr.csv")sofr_clean = sofr_raw.copy()sofr_clean.columns = [str(c).strip().lower().replace(" ", "_") for c in sofr_clean.columns]sofr_date_col =next(c for c in sofr_clean.columns if"date"in c)sofr_index_col =next(c for c in sofr_clean.columns if"sofr"in c and"index"in c)sofr_clean = sofr_clean[[sofr_date_col, sofr_index_col]].rename(columns={sofr_date_col: "date", sofr_index_col: "sofr_index"})sofr_clean["date"] = pd.to_datetime(sofr_clean["date"], errors="coerce")sofr_clean["sofr_index"] = pd.to_numeric(sofr_clean["sofr_index"], errors="coerce")sofr_daily = sofr_clean.dropna().drop_duplicates("date").set_index("date").sort_index()sofr_daily = sofr_daily.loc[sofr_daily.index >= sofr_start_date]sofr_monthly_index = sofr_daily["sofr_index"].resample("ME").last()sofr_cash_return = sofr_monthly_index.pct_change()first_month = sofr_monthly_index.index.min()sofr_cash_return.loc[first_month] = sofr_monthly_index.loc[first_month] / sofr_daily["sofr_index"].iloc[0] -1sofr_cash_return = sofr_cash_return.reindex(curve_dates).dropna().rename("sofr cash return")sofr_summary = pd.DataFrame({"item": ["source", "first sofr date", "first aligned cash return", "last aligned cash return", "monthly returns"],"value": ["data/sofr.csv", sofr_daily.index.min().date(), sofr_cash_return.index.min().date(), sofr_cash_return.index.max().date(), int(sofr_cash_return.notna().sum())],})display(sofr_summary)sofr_cash_return.tail()
The SOFR-aligned sample starts in March 2020 and runs through January 2026, with 71 monthly returns. The latest monthly cash returns are around 0.31–0.38%, which annualizes to a high cash-rate environment.
A cash return of that size gives us a meaningful benchmark for comparing a bond portfolio’s return over this period. Subtracting monthly cash returns will let us see how much return remains after accounting for the opportunity cost of holding overnight cash during the tightening and high-rate years.
The index ratio compounds the overnight return over the corresponding month before we align it to the curve dates.
5) Vasicek model: one Gaussian short-rate state
The first formal term-structure model we estimate is the Vasicek short-rate model. It assumes the instantaneous short rate \(r_t\) follows an Ornstein-Uhlenbeck mean-reverting process:
\[
dr_t = \kappa(\theta - r_t)dt + \sigma dW_t
\]
Each parameter has a specific economic meaning:
\(r_t\) is the short rate state.
\(\kappa\) is the speed of mean reversion. Larger \(\kappa\) means the short rate is pulled back to its long-run level faster.
\(\theta\) is the long-run mean level.
\(\sigma\) is the volatility of rate shocks.
\(W_t\) is a Brownian motion shock.
The drift term \(\kappa(\theta-r_t)\) is the model’s force of gravity. If \(r_t\) is above \(\theta\), the drift is negative and the model expects rates to fall. If \(r_t\) is below \(\theta\), the drift is positive and the model expects rates to rise.
The exact discrete transition over a time step \(\Delta\) is:
This transition is useful because we observe monthly data. We can estimate \(\kappa\), \(\theta\), and \(\sigma\) by maximizing the likelihood of observed short-rate transitions.
Vasicek has one major weakness: rates are Gaussian, so negative rates are possible. That is not automatically fatal. US short rates rarely went deeply negative, and many rate models used in practice allow negative rates for tractability. But it means Vasicek is better viewed as a tractable benchmark, not as a perfect description of policy rates.
5.0 Physical and risk-neutral rate dynamics
Term-structure models can describe rates under the physical measure or the risk-neutral measure, depending on the calculation we want to make. Under the physical measure, we try to describe how rates actually move through time. Under the risk-neutral measure, we price bonds and derivatives by discounting expected cash flows using adjusted dynamics. The market price of risk changes the drift used under the pricing measure, so the interpretation of a parameter depends on the measure under which it is estimated.
In symbolic form, a short-rate model under the physical measure could be written as:
\[
dr_t = \mu_P(r_t,t)dt + \sigma(r_t,t)dW_t^P
\]
while the risk-neutral version is:
\[
dr_t = \mu_Q(r_t,t)dt + \sigma(r_t,t)dW_t^Q
\]
The volatility term is usually the same, but the drift changes because investors demand compensation for bearing rate risk. The difference between \(\mu_P\) and \(\mu_Q\) is tied to the market price of risk.
We estimate models from historical curve movements and use their implied expected changes and covariances to form model-driven curve views for duration decisions. These estimates have a physical-measure forecasting interpretation. We also use affine bond-pricing formulas to construct yield loadings and term-premium proxies; we don’t calibrate a full risk-neutral derivative-pricing model to caps, floors, or swaptions in this exercise.
The two measures answer different questions about the same rate process. Historical estimation gives us information about the path and variation of observed rates, while a pricing calculation requires the drift appropriate to discounted cash flows. Keeping the measure attached to each calculation helps us interpret the model parameters and any resulting bond-price formula.
Under the pricing measure, the bond-price expectation discounts its payoff along the future short-rate path. The measure therefore affects both the expected rate dynamics and the value assigned to a future payment.
5.1 Affine bond pricing and yield loadings
Vasicek is an affine term-structure model. Zero-coupon bond prices take the form:
This equation is important because it turns the hidden short rate into a whole curve. The term \(B(\tau)/\tau\) is the yield loading on the short rate. For short maturities, the loading is close to one. For long maturities, it is smaller because a one-month change in the short rate doesn’t mechanically move the 30-year average rate one-for-one.
For example, if the 1-month short rate jumps by 100 bps, a very short zero rate should react strongly. A 30-year zero rate may react much less because it reflects the average of future short rates over a much longer horizon. This is exactly what the loading function captures.
For \(\kappa>0\), \(B(\tau)/\tau\) approaches 1 as \(\tau\) approaches zero and approaches 0 as maturity grows. The intercept \(-A(\tau)/\tau\) collects the remaining maturity-dependent contribution, including the effects of the long-run level and volatility. Together, the intercept and loading map one value of \(r_t\) into a yield at each maturity.
A useful way to understand the affine structure is to read the zero yield as a regression-like equation where maturity controls the intercept and loading:
The loading \(\beta(\tau)\) is usually between 0 and 1. Short maturities have \(\beta(\tau)\) close to 1 because a short bond mostly reflects the current short rate. Long maturities have smaller \(\beta(\tau)\) because the current short rate is only one point in a long future average.
This is the mathematical reason a one-factor short-rate model creates a very specific curve shape. If \(r_t\) increases, every maturity changes in the same direction, but short maturities change more than long maturities. The model can create level movement with a short-end tilt, but it cannot freely create a steepener, flattener, or belly-only movement. That limitation will show up later when PCA says slope and curvature are important.
For example, if the 3-month yield rises because the central bank hikes, Vasicek can represent that through a higher \(r_t\). But if the 30-year yield rises while the 2-year yield falls, one short-rate state can’t explain that move well. It would need to move \(r_t\) up and down at the same time. A model with additional independent factors can represent moves where the short and long ends respond differently, including changes in the slope of the curve.
5.2 Filtering a hidden short rate from observed yields
Even though we have a front-end yield proxy, the model doesn’t have to treat it as the exact true short rate. The observed curve contains noise, interpolation error, liquidity effects, and maturity-specific premia. So we use a Kalman filter to infer a latent short-rate state from several observed zero rates.
Here \(\mathbf{y}_t\) is the vector of observed zero rates at selected maturities, \(\mathbf{a}\) contains intercepts, \(\mathbf{b}\) contains the loadings \(B(\tau)/\tau\), and \(\boldsymbol{\epsilon}_t\) is observation noise. The affine formula gives the default intercepts \(-A(\tau)/\tau\). When we supply a short-rate hint \(h_t\), as in these fits, we instead estimate each maturity’s intercept by averaging \(y_{t,j}-b_j h_t\) across the supplied history. That aligns the observed yields with the hinted rate before filtering.
The Kalman filter has two jobs:
Prediction: use last month’s filtered short rate to predict the current month’s hidden short rate.
Correction: compare the current month’s predicted curve to its observed curve and update the hidden state.
The notation \(t|t-1\) means our estimate for time \(t\) before observing the curve at time \(t\). In the variance prediction, \(\phi^2\) scales the previous estimation uncertainty and \(q\) adds uncertainty from the new state shock. Mean reversion can reduce the inherited part, so the total predicted variance needn’t always be larger than the preceding filtered variance.
It also helps to compare the Kalman filter with ordinary regression. In ordinary regression, we estimate one static coefficient vector. In the Kalman filter, we estimate a state that changes through time. The hidden state has its own dynamics, and observations only give noisy information about it.
In this one-state Vasicek case, the hidden state is \(r_t\). The curve observations are several yields at time \(t\). The filter combines two sources of information:
the model transition, which says where \(r_t\) probably moved from last month,
the cross-section of observed yields, which says what value of \(r_t\) best explains today’s curve.
If the model transition says the short rate should be 3.7%, but the observed 3M/2Y/5Y/10Y curve implies something closer to 3.4%, the filter doesn’t simply choose one. It weights the difference by the Kalman gain. The gain is high when state uncertainty is large or observation noise is small. It is low when the state is already estimated tightly or the observed curve is noisy.
The innovation covariance \(S_t\) describes uncertainty in the predicted yield vector. The diagonal entries of \(S_t\) describe uncertainty at individual maturities, while the off-diagonal entries describe how their forecast errors move together. A 10 bp forecast error has a different weight when prediction uncertainty is small than when the same maturity is noisy or the hidden state is uncertain.
The update combines the predicted state with the new curve according to those uncertainties. It gives us a filtered estimate of the short rate at each date, which we can then compare with the observed short-rate proxy.
The correction step starts by asking what curve the predicted state would imply:
If observed yields are higher than predicted yields, \(\mathbf{v}_t\) is positive. If they are lower, it is negative. The model then asks how surprising that innovation is. The innovation covariance is:
\[
S_t = P_{t|t-1}\mathbf{b}\mathbf{b}^\top + R
\]
where \(R\) is the observation-noise covariance. The first term says uncertainty in the hidden short rate creates uncertainty in every observed yield through the loading vector \(\mathbf{b}\). The second term says each market quote also has its own noise.
The Kalman gain is:
\[
K_t = P_{t|t-1}\mathbf{b}^\top S_t^{-1}
\]
This is the weight placed on the innovation. A high Kalman gain means the filter trusts the observed curve a lot and moves the hidden short rate strongly. A low gain means the observed curve is noisy or the state is already estimated precisely, so the filter updates only slightly.
Financially, this is a disciplined way to smooth the short-rate state. If one yield point jumps for a noisy reason, the filter doesn’t automatically treat it as a full policy-rate shock. If the whole curve moves in a way consistent with a short-rate change, the filter reacts more.
If we observe \(m\) maturities, \(\mathbf{b}\) and \(\mathbf{v}_t\) are \(m\times1\) vectors, \(S_t\) is an \(m\times m\) matrix, and \(K_t\) is a \(1\times m\) row vector. Multiplying \(K_t\mathbf{v}_t\) therefore gives a scalar correction to the single short-rate state. In our implementation, observation noise has the same variance at each selected maturity and no off-diagonal covariance.
def fit_vasicek_ar(short_history): x = pd.Series(short_history).dropna().astype(float)iflen(x) <12:return {"kappa": 0.35, "theta": float(x.mean()) iflen(x) else0.03, "sigma": 0.01, "obs_sd": 0.001,"method": "ar fallback", "optimizer success": False} y = x.iloc[1:].to_numpy() lag = x.iloc[:-1].to_numpy() slope, intercept = np.polyfit(lag, y, 1) phi =float(np.clip(slope, 0.02, 0.995)) dt =1/12 kappa =-np.log(phi) / dt theta = intercept /max(1- phi, 1e-6) resid = y - (intercept + slope * lag) sigma =float(np.std(resid, ddof=1) * np.sqrt(2* kappa /max(1- phi**2, 1e-8)))return {"kappa": kappa, "theta": theta, "sigma": max(sigma, 1e-5), "obs_sd": 0.0015,"method": "ar fallback", "optimizer success": False}def vasicek_kalman(yields, maturities, params, state_hint=None): data = np.asarray(yields, dtype=float) maturities = np.asarray(maturities, dtype=float) affine_level, b = vasicek_loadings(params, maturities)if state_hint isnotNone: hint = np.asarray(state_hint, dtype=float)iflen(hint) ==len(data): a = np.nanmean(data - hint[:, None] * b[None, :], axis=0)else: a = affine_levelelse: a = np.asarray(params.get("obs_alpha", affine_level), dtype=float) kappa =max(float(params["kappa"]), 1e-8) theta =float(params["theta"]) sigma =max(float(params["sigma"]), 1e-8) obs_sd =max(float(params.get("obs_sd", 0.001)), 1e-8) dt =1/12 phi = np.exp(-kappa * dt) q = sigma**2* (1- phi**2) / (2* kappa) mean =float(state_hint[0]) if state_hint isnotNoneandlen(state_hint) else theta var = sigma**2/ (2* kappa) filtered = [] nll =0.0if np.isfinite(data).all(): eye = np.eye(data.shape[1]) outer_b = np.outer(b, b) log_const = data.shape[1] * np.log(2* np.pi)for row in data: mean_pred = theta + phi * (mean - theta) var_pred = phi**2* var + q innov = row - (a + b * mean_pred) s = var_pred * outer_b + obs_sd**2* eye sign, logdet = np.linalg.slogdet(s) inv_s = np.linalg.pinv(s) nll +=0.5* (logdet + innov @ inv_s @ innov + log_const) if sign >0else1e6 gain = var_pred * b @ inv_s mean = mean_pred + gain @ innov var =max(float((1- gain @ b) * var_pred), 1e-10) filtered.append(mean)returnfloat(nll), np.asarray(filtered)for row in data: mean_pred = theta + phi * (mean - theta) var_pred = phi**2* var + q mask = np.isfinite(row)if mask.any(): obs = row[mask] aa = a[mask] bb = b[mask] innov = obs - (aa + bb * mean_pred) s = var_pred * np.outer(bb, bb) + obs_sd**2* np.eye(mask.sum()) sign, logdet = np.linalg.slogdet(s) inv_s = np.linalg.pinv(s) nll +=0.5* (logdet + innov @ inv_s @ innov + mask.sum() * np.log(2* np.pi)) if sign >0else1e6 gain = var_pred * bb @ inv_s mean = mean_pred + gain @ innov var =max(float((1- gain @ bb) * var_pred), 1e-10)else: mean, var = mean_pred, var_pred filtered.append(mean)returnfloat(nll), np.asarray(filtered)
5.3 Vasicek estimation results
We estimate the Vasicek parameters from the short-rate transition likelihood, then use those parameters in the Kalman filter for the curve observation equation. The resulting estimates and diagnostics let us examine both the fitted dynamics and the numerical fit.
The output reports:
\(\kappa\), the mean-reversion speed,
\(\theta\), the long-run mean short rate,
\(\sigma\), the short-rate volatility,
observation noise,
optimizer success,
Hessian condition number,
and residual autocorrelation.
The optimizer status and Hessian condition number help us read the numerical fit, while residual autocorrelation indicates whether serial structure remains after estimating the one-factor transition.
The likelihood used for the short-rate transition comes directly from the discrete Gaussian transition. If we observe \(r_t\) and \(r_{t+\Delta}\), the conditional density is:
The squared residual divided by \(v\) becomes large when the predicted variance is tiny and the observed transition differs from its conditional mean. The \(\log(v)\) term moves in the opposite direction: it decreases as variance becomes smaller, so the likelihood balances a narrow distribution against the size of the observed errors. Minimizing both terms together estimates the rate dynamics and their conditional uncertainty.
The Hessian condition number then tells us how sharply the likelihood identifies the parameters. A very high condition number means the objective is flat or stretched in some direction. That usually means parameters are hard to estimate separately. In rate models, this often happens because \(\kappa\), \(\theta\), and \(\sigma\) can compensate for each other over short samples.
The Vasicek fit succeeds, but the parameter estimates are telling. The estimated \(\kappa\) is about 0.099, which is slow mean reversion. The long-run level \(\theta\) is about 1.19%, far below the final short-rate environment. The monthly short-rate volatility parameter is around 0.73% in annualized model units.
The low \(\theta\) reflects the long zero-rate and low-rate sample from 2008 through 2021. Because the full sample includes many years near zero, the model’s long-run gravity sits much lower than the 2022–2026 rate regime. A rolling estimate lets the mean respond to a more recent window rather than giving equal weight to all of those earlier regimes.
The residual autocorrelation is around 0.27, which is not ideal. A good transition model would leave less serial dependence in residuals. This tells us the short-rate process has structure the simple OU transition doesn’t fully capture, especially across regime changes and policy cycles.
Show code
fig, ax = plt.subplots()ax.plot(short_rate.index, short_rate *100, label="observed short proxy", alpha=0.8)ax.plot(vasicek_filtered.index, vasicek_filtered *100, label="filtered vasicek short rate", linewidth=2)ax.set_title("vasicek filtered short rate")ax.set_ylabel("percent")ax.legend()plt.show()
The filtered short-rate plot compares the observed short proxy with the latent Vasicek short rate. The filter tracks the big regimes well: the pre-GFC high-rate period, the post-GFC near-zero period, the COVID zero-rate period, and the sharp hiking cycle after 2022.
The filtered line is smoother than the observed proxy. The model treats the short rate as a hidden state that evolves according to the transition equation, while the observed curve can contain quote noise and maturity-specific distortions. The gain determines how much of each new curve observation enters the state estimate.
The largest gap appears around rapid policy transitions. When the Fed cuts or hikes quickly, a slow mean-reverting one-factor model reacts with delay. That delay is economically important because it can cause the model to underestimate how fast the front end has repriced.
6) CIR model: square-root volatility and rate positivity
The Cox-Ingersoll-Ross model modifies Vasicek by making volatility proportional to the square root of the short rate:
The drift has the same interpretation as Vasicek: \(\kappa\) pulls the rate toward \(\theta\). The difference is the diffusion term. When \(r_t\) is low, \(\sqrt{r_t}\) is small, so rate volatility is lower. When \(r_t\) is high, volatility increases. This is often more realistic for interest rates because the level of rates and the volatility of rate changes tend to be connected.
CIR is also famous for its positivity condition. A sufficient condition for the process to stay strictly positive is the Feller condition:
\[
2\kappa\theta \geq \sigma^2
\]
The quantity
\[
2\kappa\theta - \sigma^2
\]
is the Feller margin. For a positive starting rate, a nonnegative margin satisfies the condition above and the continuous-time process stays strictly positive. With a negative margin, zero can be reached, although the CIR process remains nonnegative under its standard parameter assumptions.
Because US rates touched the zero lower bound for years, we use a small shift in the implementation. The model is fitted to shifted rates \(r_t + s\) so the square-root term is well-defined. We report the shift alongside the fitted parameters because it is a numerical device used to apply the square-root structure to near-zero historical rates. After the shifted process is evaluated, the shift is removed to return to the original rate scale.
6.1 CIR estimation in this project
The exact CIR transition has a noncentral chi-square distribution. We’ll estimate a more transparent quasi-likelihood using the Euler approximation. Over a short step \(\Delta\), the mean-reverting drift and the state-dependent shock give us:
The variance depends on \(r_t\), so observations at high rates receive different statistical treatment than observations near zero. Near zero, the model expects smaller absolute rate changes. At higher rates, it allows larger moves.
The implementation also penalizes violations of the Feller condition. The optimizer can still end near the boundary, especially when the data includes near-zero rates, but the penalty discourages parameter combinations that make the process numerically unstable.
The Feller penalty and parameter bounds are part of the estimation, so we’ll read the fitted parameters alongside optimizer status and residual autocorrelation. Comparing those diagnostics with Vasicek helps us see how the square-root specification represents the observed short-rate transitions.
Since the conditional variance depends on the starting rate, the same absolute transition error has a different standardized size at a low rate than at a high rate. The likelihood uses that state-dependent scale when comparing the transitions.
The CIR fit succeeds, but it looks very different from the Vasicek fit. The estimated \(\kappa\) is about 0.130, slightly faster than the full-sample Vasicek estimate. The estimated long-run shifted mean \(\theta\) is around 1.55%, and the volatility parameter \(\sigma\) is much larger at about 6.35% because CIR volatility is multiplied by \(\sqrt{r_t}\).
The shift is 0.20%, which reflects the near-zero periods in the data. The Feller margin is slightly negative, very close to zero. So the fitted process is almost at the positivity boundary, but not strongly inside the strict Feller region. This is realistic for a dataset that includes years at or near zero.
The residual autocorrelation is much lower than Vasicek, around 0.04. That is a meaningful improvement in this transition-residual diagnostic. The state-dependent volatility gives CIR a different way to represent short-rate changes at high and low rate levels. The Hessian condition number is very high, though, which warns us that the likelihood surface is numerically ill-conditioned. In simple terms, multiple parameter combinations can fit the data similarly well.
7) Short-rate simulation fan charts
After estimating Vasicek and CIR, we simulate future short-rate paths from the latest observed rate. The resulting distributions let us compare the range, shape, and mean reversion implied by each model’s fitted parameters.
and then floors the shifted rate at a small positive value. This is less elegant than the exact noncentral chi-square transition, but it is stable and transparent for simulation diagnostics.
The fan charts summarize uncertainty by showing quantile bands rather than every simulated path. The median tells us the central model path. The 5–95% and 25–75% bands show the range of plausible outcomes.
Simulated paths make the parameter estimates easier to interpret. With \(\kappa=0.13\) and \(\sigma=0.063\), for example, we can see how the fitted speed of mean reversion and the size of shocks translate into the distribution of future rates.
Mean reversion controls the median path. If the current short rate is above the estimated long-run mean, the median path drifts lower. Volatility controls the width of the fan. In CIR, the starting rate also controls the width because volatility is scaled by \(\sqrt{r_t}\).
Each fan describes a distribution conditional on the fitted model and the latest starting rate. A sudden policy pivot, fiscal shock, or inflation surprise can produce moves that differ from those implied by the fitted dynamics. We’ll compare the bands to understand the assumptions about mean reversion and rate uncertainty in the two models.
The Vasicek and CIR fan charts both slope downward from the latest high-rate environment. That follows directly from their estimated long-run means being below the current short rate. The models read the latest short rate as above equilibrium, so the drift points lower.
CIR produces a wider and more asymmetric fan because volatility depends on the rate level. At higher starting rates, the square-root diffusion allows larger absolute moves. Vasicek produces a more symmetric Gaussian band. The comparison plot makes this clearer: the CIR lower tail reaches lower levels, while the upper band remains broad.
Economically, this says the fitted short-rate models believe the front end should mean-revert lower, but they disagree about uncertainty. The CIR model says uncertainty is more state-dependent. That becomes important when we later use models to decide whether extending duration is worth the risk.
8) Rolling short-rate models and model-implied term premium
A single full-sample fit is useful for teaching, but it is not enough for trading. Interest-rate dynamics change across regimes. A model fitted on 2006–2026 data averages together the GFC, the zero-rate decade, the pandemic shock, and the inflation hiking cycle.
So we estimate Vasicek and CIR on a rolling 60-month window. The rolling window creates time-varying parameters:
\[
\kappa_t, \theta_t, \sigma_t
\]
These parameters are then used to construct a simple model-implied 10-year term-premium proxy:
where \(\bar{r}^{model}_{10,t}\) is the model’s expected average short rate over the next 10 years. For Vasicek, the expected average short rate has a closed form:
This expression is easy to interpret. If the current short rate \(r_0\) is above \(\theta\), the expected average short rate is above \(\theta\) but below \(r_0\), because mean reversion pulls it down over the horizon. The difference between the observed 10-year yield and this expected average short-rate path is the proxy for term premium.
To obtain the average, first take the expected Vasicek path \(E[r_u\mid r_0]=\theta+(r_0-\theta)e^{-\kappa u}\) and integrate it from 0 to \(T\). Dividing that integral by \(T\) gives the expression above. The ratio \((1-e^{-\kappa T})/(\kappa T)\) is the average weight placed on today’s deviation from the long-run mean over the horizon.
8.1 Interpreting rolling parameter estimates
Rolling short-rate parameters are useful, but they can look strange because the model is deliberately local. A 60-month window doesn’t know the full history of the rate market. It only sees the last five years. If those five years contain a hiking cycle, the estimated long-run mean can move upward. If they contain a zero-rate period, the long-run mean can collapse.
Using recent data gives the strategy an estimate that responds to the current regime. At the same time, a five-year window can make a temporary period of high or low rates look like a persistent equilibrium, so the estimated long-run mean should be read in the context of that window.
For example, after several years of high short rates, a rolling Vasicek fit may estimate a high \(\theta_t\). Then the model expects the average future short rate to remain high, so the 10-year yield can look low relative to that expected short-rate path. That creates a negative term-premium proxy. If the high short-rate regime is actually temporary and policy cuts are coming, that proxy may be misleading.
Comparing the model-implied proxy with ACM will help us see how much the estimated expected short-rate path influences the resulting premium. Both estimates use the same general idea of separating long yields from expected short rates, while their model specifications differ.
The latest rolling parameter table shows a very different picture from the full-sample estimates. By late 2025 and early 2026, Vasicek \(\kappa\) rises toward 0.34, and \(\theta\) rises toward 5.6%. CIR \(\theta\) is even higher, around 8.9% on the shifted scale, while CIR \(\kappa\) remains slow.
The recent window is dominated by the post-2022 higher-rate regime. The model’s local long-run mean rises sharply from the full-sample estimate near 1% because the observations in the five-year window now include those higher rates. But the rolling residual diagnostics are still imperfect. Vasicek residual autocorrelation is around 0.64 in the latest window, while CIR residual autocorrelation is closer to 0.29. CIR is again better on that diagnostic, but neither model is fully capturing the policy-cycle structure.
The latest model-implied term-premium proxies are negative, around -82 bps for Vasicek and -90 bps for CIR. That contrasts with the positive ACM estimate. A high expected path of short rates leaves less of the observed 10-year yield in the residual term-premium proxy, especially when the recent window produces a high local mean. ACM uses a richer no-arbitrage macro-finance decomposition, so its estimate need not match the simple proxy.
The reported parameter estimates use the most recent 60 months, and the residual autocorrelations are calculated from the fitted one-step errors in that same history.
Show code
term_premium_plot = term_premium_comparison.dropna()term_premium_correlation = term_premium_plot.corr().loc[["acm 10y term premium"], ["vasicek term premium proxy", "cir term premium proxy"]]fig, ax = plt.subplots()for col in term_premium_plot.columns: ax.plot(term_premium_plot.index, term_premium_plot[col] *100, label=col)ax.set_title("acm benchmark and model-implied term-premium proxies")ax.set_ylabel("percentage points")ax.legend()plt.show()display(term_premium_correlation)
vasicek term premium proxy
cir term premium proxy
acm 10y term premium
0.541233
0.528921
The simple Vasicek/CIR proxies and ACM move in the same broad direction during some periods, while their levels and timing also differ across the sample. The reported correlations with ACM are about 0.54 for Vasicek and 0.53 for CIR. That is meaningful but not strong enough to treat these proxies as clean replicas.
The model proxies fall sharply negative around 2022–2024, while ACM behaves differently. This divergence is economically interpretable. The rolling short-rate models see the high recent short-rate environment and infer a high expected short-rate path. If the expected average short rate is high, then the observed 10-year yield can look low relative to that path, producing a negative term-premium proxy. ACM, by contrast, uses a richer term-structure model and estimates risk premia separately.
The short-rate models give us a transparent decomposition: the fitted expected average path is subtracted from the observed yield. The differences from ACM show how the estimated path and the choice of model affect the resulting premium, even when the broad series share some common movement.
Correlation measures co-movement across the aligned monthly observations; agreement in direction does not require the two series to have the same level or the same size of movement.
9) HW1F: one-factor curve scenarios with mean reversion
The Hull-White one-factor model is often written as:
The time-dependent drift \(\theta(t)\) distinguishes the model from constant-parameter Vasicek: a time-dependent drift chosen so the model can fit the current initial yield curve. That makes HW1F a no-arbitrage extension of Gaussian short-rate modeling. Instead of forcing a constant long-run mean to explain the whole curve, the deterministic drift term fits today’s curve exactly, while the stochastic factor describes future curve movements.
For yield changes, the one-factor loading has the familiar mean-reverting shape:
\[
L(T) = \frac{1-e^{-aT}}{aT}
\]
This loading is high at the short end and lower at the long end. If the factor shock is positive, short maturities move more than long maturities. That means HW1F is mainly a level/front-end dominated curve model. It can create parallel-ish shifts with stronger short-end response, but it can’t independently create slope and curvature shocks.
In this implementation, we estimate a practical HW1F-style model from recent curve changes. We estimate \(a\) from short-rate mean reversion, estimate the monthly factor volatility from historical yield changes, and compute how much of total curve-change variance the one factor explains.
9.1 Fitting the initial curve with HW1F
Plain Vasicek with constant \(\theta\) has a problem: one set of parameters has to explain both the current curve shape and future dynamics. Hull-White fixes this by allowing the drift to be time-dependent. The function \(\theta(t)\) is chosen so the model fits today’s observed discount curve.
We use the observed current curve as the starting curve, and the stochastic factor describes movements from that starting point. The deterministic drift supplies the initial fit, while the factor loading determines the relative change at each maturity.
In a full HW1F no-arbitrage setup, the drift function is tied to the initial instantaneous forward curve \(f(0,t)\) and the volatility structure. The exact formula can get heavy, but the intuition is simple: \(\theta(t)\) bends the model’s deterministic path so that model-implied zero-coupon prices match market zero-coupon prices at time zero.
This is why HW1F is widely used for rate derivatives as a first model. It has only one stochastic factor, so it is simple, but it can fit the initial curve. Its weakness is not curve fitting at time zero. Its weakness is future curve movement: one stochastic factor can’t produce enough independent slope and curvature behavior.
The HW1F diagnostic table reports mean reversion around 0.342, monthly factor volatility around 30.6 bps, annualized factor volatility around 106 bps, and a curve-change variance share of about 54%.
One factor explains a little more than half of recent curve-change variance. The remainder includes movements in curve shape, including slope and curvature, as well as variation that the single loading doesn’t capture. If the strategy only used HW1F, it would tend to interpret curve risk as mostly one common rate shock. That can work when the whole curve is moving together, but it will miss environments where the front end and long end move in opposite directions.
In a one-factor scenario, the random yield change at every maturity is proportional to the same shock. The loading fixes the relative size of those changes within each simulated curve.
The one-month HW1F scenario summary shows a large short-end distribution and a much smaller long-end distribution. The 10% to 90% range is about -29 to +45 bps at the 3-month maturity, but only about -3 to +5 bps at the 30-year maturity.
The loading is strongest near the front end and decays with maturity, so the same simulated factor produces a wider yield-change distribution at shorter maturities. Economically, the model is saying recent curve dynamics are dominated by short-rate/policy movement rather than long-end term-premium shocks.
This is useful for strategy design, but it has a clear limitation: if the next month’s move is a long-end selloff caused by fiscal supply or term premium repricing, HW1F will likely underestimate that risk.
10) G2-style two-factor curve model
A two-factor Gaussian term-structure model gives the curve more flexibility. A common G2-style structure is:
\[
r_t = x_t + y_t + \varphi(t)
\]
with two mean-reverting factors:
\[
dx_t = -a x_t dt + \sigma dW_t^{(1)}
\]
\[
dy_t = -b y_t dt + \eta dW_t^{(2)}
\]
and correlated shocks:
\[
dW_t^{(1)}dW_t^{(2)} = \rho dt
\]
The first factor and second factor can have different speeds of mean reversion. A faster factor tends to load more heavily on short/intermediate maturities. A slower factor can influence the longer end more. Together they can produce richer curve shapes than HW1F.
We use a practical G2-style curve-factor model, estimated from observed yield changes. We construct two maturity loadings from HW-style exponential decay shapes, orthogonalize the second loading relative to the first, estimate the two factor returns by least squares, and use their covariance matrix for scenario generation. A full market-calibrated swaption model would require additional market prices and calibration steps.
The useful object is the curve-change covariance implied by the factors:
\[
\Sigma_{\Delta y} = L\Sigma_f L^\top
\]
where \(L\) is the matrix of maturity loadings and \(\Sigma_f\) is the factor covariance matrix. This covariance is what the duration strategy later uses to estimate active risk.
With \(m\) maturities, \(L\) has \(m\) rows and two columns, while \(\Sigma_f\) is a \(2\times2\) covariance matrix of monthly factor changes. Their product produces an \(m\times m\) matrix: each entry describes how yield changes at a pair of maturities move together under the fitted factors.
10.1 Two factors and active curve risk
A duration strategy’s active risk depends on the covariance of yield changes across the entire curve, together with its exposure at each maturity. Suppose a candidate ladder is long 30Y KRD and short 5Y KRD relative to neutral. If the curve steepens, that candidate can lose even if the average rate level barely moves. A one-factor model may understate this risk because it treats most moves as a single loading pattern.
Each term has a role in the active-risk calculation:
\(\Delta KRD\) says what maturities the strategy is exposed to.
\(L\) maps factor shocks into yield changes at those maturities.
\(\Sigma_f\) says how volatile and correlated the factors are.
The outer quadratic form turns curve-factor covariance into active portfolio risk.
The active variance enters the denominator of the strategy score. A candidate can therefore receive a different score when the covariance includes an additional factor and its exposure to a curve twist carries more estimated risk.
The vector and covariance must use the same maturity ordering when the active risk is calculated.
The G2-style diagnostic table shows first-factor mean reversion around 0.342 and second-factor mean reversion around 0.085. The first monthly factor volatility is about 30.6 bps, and the second is about 24.5 bps. The factor correlation is about 0.65, so the two factors are not independent, but the model is not degenerate.
The slower second factor adds a different maturity-loading pattern to the first factor. After orthogonalization, the pair can represent curve changes that a single decaying loading cannot capture. The positive correlation says level and slope-like moves have tended to occur together in the latest 60-month window.
HW1F has one smooth decaying loading. G2 has a first loading similar to HW1F and a second loading that changes sign across maturities. The sign change allows a factor shock to move different parts of the curve in opposite directions, giving the model a slope/twist component.
The scenario fan also shows the difference. HW1F produces a curve-change distribution dominated by the short end. G2 allows a richer distribution across the curve because two correlated factors can move the short and long end in different proportions.
When we compare a lower-duration ladder with a higher-duration ladder, the active KRD vector has exposure to different key rates. The two models attach different covariances to those exposures, so the estimated risk of the same ladder can change with the chosen factor structure.
For a ladder tilted toward the long end, the second loading lets a long-end move contribute separately from the common front-end-heavy shock when we estimate its active risk.
11) PCA level, slope, and curvature diagnostics
PCA is not a no-arbitrage model. It is an empirical decomposition of yield-curve changes. But it is one of the most useful diagnostics in fixed income because it tells us how many curve factors we actually need.
Start with a matrix of demeaned monthly yield changes:
\[
X_t = \Delta y_t - \bar{\Delta y}
\]
The sample covariance matrix is:
\[
\Sigma_X = \frac{1}{T-1}X^\top X
\]
PCA solves the eigenvalue problem:
\[
\Sigma_X v_j = \lambda_j v_j
\]
where \(v_j\) is the \(j\)th loading vector and \(\lambda_j\) is the variance explained by that component. The score for date \(t\) is:
\[
f_{t,j} = X_t v_j
\]
In yield-curve data, the first three PCs usually have intuitive shapes:
PC1 level: most maturities move in the same direction.
PC2 slope: short and long maturities move in opposite directions.
PC3 curvature: the belly moves differently from the wings.
This tells us how much of total curve-change variance each factor captures. A good two-factor model should be judged partly against PCA: if PCA says slope is important, but a model only has one level factor, that model is structurally incomplete.
Each row of \(X\) is one month’s demeaned changes across the selected maturities, and each column corresponds to a maturity. The eigenvectors give directions in that maturity space; the eigenvalues give the sample variance along those directions. Multiplying a row by \(v_j\) gives that month’s score, while the explained-variance ratio compares the component’s variance with the total across all maturities.
11.1 PCA sign conventions
PCA loadings are only identified up to sign. If \(v_j\) is an eigenvector, then \(-v_j\) is also an eigenvector with the same variance explained. That means a raw PCA routine can randomly flip signs across windows. For interpretation, we orient the components consistently:
PC1 is oriented so its average loading is positive.
PC2 is oriented so it has the expected slope direction.
PC3 is oriented so the belly curvature has a consistent sign.
We flip a loading and its corresponding score together. Their product is unchanged, so the reconstructed curve change and explained variance stay the same. Consistent orientation makes it easier to compare loading shapes and rolling diagnostics across windows.
PCA also helps diagnose HW1F and G2. If PC1 explains 95% of curve moves, a one-factor model may be enough. If PC2 and PC3 explain meaningful variance, a one-factor model will miss important risk. In this sample, PC2 and PC3 together are too large to ignore.
After orientation, a positive PC1 score points in the direction of the plotted component with positive average loading. That gives the sign of the score a consistent interpretation across windows.
In the latest window, the first three PCA components have the familiar level, slope, and curvature patterns seen in yield-curve changes. PC1 explains about 76.8% of variance, PC2 about 15.0%, and PC3 about 6.8%. Together the first three PCs explain more than 98% of recent curve-change variance.
The loading plot shows the classic structure: PC1 is level-like, PC2 is slope-like, and PC3 is curvature-like. The rolling explained-variance panel shows that PC1 is not constant through time. Some regimes are more level-dominated, while others need more slope and curvature.
The AR(1) rho values also have interpretation. PC2 has the highest clipped persistence at 0.35, suggesting slope dynamics have been more persistent than level or curvature in the latest window. That helps explain why a model that can represent slope, like G2 or PCA itself, can matter for strategy decisions.
12) Duration ladder candidates and KRD exposure
We now connect the curve models to an implementable bond strategy. The cash-bond core is a ladder across 2Y, 5Y, 10Y, and 30Y buckets. We compare three duration targets:
lower duration: target around 3 years,
neutral duration: target around 5 years,
higher duration: target around 7 years.
We already built the mechanics of duration and KRD earlier, so we don’t repeat the full derivation. The only formula we need for the strategy is the first-order curve PnL approximation:
We use this active KRD vector with each term-structure model’s expected curve change and covariance. If the model expects rates to fall, positive active duration is attractive. If the model expects rates to rise, lower duration is safer. If the model expects a curve twist, the distribution of KRD across maturities determines how the different parts of the ladder respond.
The candidate ladders have distinct realized duration exposures, giving the model a choice between different amounts and locations of rate risk. The fixed/neutral ladder has average duration close to 5.0. The lower-duration ladder has average duration around 4.29, and the higher-duration ladder around 6.98.
The higher-duration ladder has much higher convexity, around 88.4 compared with 42.2 for neutral. That is expected because adding long-maturity exposure increases both duration and convexity. The KRD profile also shifts strongly toward the 30-year bucket for the higher-duration ladder, with latest 30Y KRD around 2.38. The lower-duration ladder has zero latest 30Y KRD and concentrates more in the 2Y/5Y/10Y area.
Each choice corresponds to an actual ladder with its own KRD vector, convexity, turnover, and trading costs. Those exposures are the inputs we need when evaluating a model’s curve view against the candidate portfolios.
13) Turning model curve views into duration decisions
Each model produces two objects for the next month:
\[
E_t[\Delta y]
\]
and
\[
\Sigma_t^{\Delta y}
\]
The first is the expected curve change. The second is the covariance matrix of curve changes. For a candidate ladder, the active expected curve return relative to neutral is approximately:
This score is deliberately practical. It doesn’t pick the biggest expected return blindly. It compares expected active return to active risk and subtracts a risk penalty. If no candidate has positive expected active return, we stay neutral.
The expected yield-change vector and covariance use decimal yield units. For example, a 10 bp change is \(0.001\); using basis points directly in the dot product would overstate the return. The quadratic form maps covariance across the four key maturities into the variance of the active KRD exposure, and its square root gives active risk in return units.
13.1 A concrete KRD scoring example
Imagine the neutral ladder has KRD vector:
\[
KRD^{neutral} = [0.5,\;2.1,\;1.5,\;0.7]
\]
at the 2Y, 5Y, 10Y, and 30Y key rates. A higher-duration ladder might have:
\[
KRD^{high} = [0.3,\;2.2,\;2.1,\;2.4]
\]
Then the active KRD is:
\[
\Delta KRD = [-0.2,\;0.1,\;0.6,\;1.7]
\]
This candidate is mostly adding long-end exposure. If the model expects the 10Y and 30Y yields to fall, the curve-return term is positive:
\[
-\Delta KRD^\top E[\Delta y] > 0
\]
But if the model covariance says long-end shocks are very volatile, active risk rises too. So the strategy doesn’t choose higher duration only because expected return is positive. It chooses it only if the expected active return is high enough relative to active risk and transition cost.
The portfolio decision combines the expected curve move with its covariance and the candidate’s active KRD profile. We can therefore compare the return offered by this long-end exposure with the risk and transition cost needed to hold it.
13.2 Carry, roll, and curve return are different pieces
We separate carry/roll from the return caused by the expected change in the curve. Each component describes a different part of the candidate ladder’s expected active return.
Carry is the income earned from holding the bonds over the period, including coupons and financing/cash effects. Roll is the price effect from the bond aging down the curve if the curve shape stays the same. A steep curve can make roll attractive because a bond moves toward a lower-yield maturity as time passes. Curve return is the mark-to-market effect from the model’s expected yield changes.
A simple approximate active return decomposition is:
We compute carry and roll together through the reference ladders by holding the positions over the period using the starting curve. A higher-duration ladder may have a carry/roll advantage in one environment while also having a different sensitivity to future rate changes.
This is why a strategy can choose higher duration even if the model doesn’t strongly expect rates to fall. If the carry/roll advantage is positive and active risk is controlled, higher duration can still have a better score. Conversely, if expected curve return is favorable but transition costs and active risk are too high, the strategy can stay neutral.
The latest target table shows clear model disagreement. In the last few months, CIR stays mostly neutral at 5 years, HW1F wants higher duration at 7 years, and G2 wants lower duration at 3 years after August 2025.
The models use different descriptions of curve dynamics, so their preferred duration targets can differ on the same date. CIR is driven by a short-rate mean-reversion view. HW1F is driven by a one-factor front-end-heavy scenario. G2 can see a richer curve twist and may decide that long-end exposure is less attractive even if the one-factor model likes duration.
We’ll compare the recent performance of each model’s decisions in a rolling validation window, then use the selected model’s target for the following decision date.
Each model’s target history is evaluated against the same neutral ladder, so the validation comparison reflects the results of its own duration decisions under a common benchmark.
14) Model selection by rolling validation
Instead of permanently choosing CIR, HW1F, or G2, we use a rolling validation rule. Each model has its own candidate target series. We backtest those model-specific decisions, compare their recent active returns against the neutral ladder, and select the model with the best recent information ratio-like score.
computed over the trailing validation window. For each candidate model, the numerator is its mean monthly active return and the denominator is the standard deviation of those active returns over the same trailing window. The rule chooses the model with the highest resulting score using data available before the decision date.
The selected model then supplies the target duration for the next decision date. This structure is useful because rate regimes change. A one-factor model can work in level-shift regimes. A two-factor model can work better in twist regimes. CIR can work better when the front-end rate level dominates. Rolling validation lets the strategy adapt instead of forcing one model forever.
The model-selection table shows CIR selected for 84 months, HW1F for 46 months, and G2 for 13 months. CIR dominates the selected sample, but G2 has the highest average validation score among selected months. That suggests G2 is used less often, but when it is selected, it has recently been useful.
The target-count table shows higher duration selected most often: 100 months. Neutral duration appears 32 months, and lower duration only 11 months. So the strategy has a structural tendency to extend duration over this sample. That makes sense in a long period where falling or low rates rewarded duration, but it also creates vulnerability during rising-rate shocks.
The selected targets therefore have a bias toward higher duration in this sample. The distribution of target choices gives useful context for reading the strategy’s active returns alongside the additional rate exposure it takes.
The count measures how often each target was selected. Actual duration on a date also depends on the ladder’s realized holdings and how closely its available bonds reach the target.
Show code
fig, axes = plt.subplots(1, 2, figsize=(13, 4), constrained_layout=True)nav_series(baseline_returns.reindex(dynamic_duration_returns.index).dropna(), start_value=1).plot(ax=axes[0], label="fixed 5y ladder")nav_series(dynamic_duration_returns.dropna(), start_value=1).plot(ax=axes[0], label="dynamic duration ladder")axes[0].set_title("fixed ladder vs dynamic duration ladder")axes[0].set_ylabel("nav")axes[0].legend()dynamic_active_returns.dropna().cumsum().plot(ax=axes[1], color=colors[1])axes[1].axhline(0, color="black", linewidth=0.8)axes[1].set_title("dynamic ladder active return")axes[1].set_ylabel("cumulative active return")plt.show()
The dynamic duration ladder outperforms the fixed 5-year ladder over the evaluation period. Cumulative active return is mostly positive and trends upward, with setbacks during difficult rate periods and changes in the interest-rate environment.
This behavior matches the target-count table. Since the model often chose higher duration, it benefited during periods where duration was rewarded. But the active-return line is not smooth. It shows that model-driven duration timing is noisy. The strategy improves long-run performance, but it doesn’t avoid every drawdown.
The long-sample performance table shows the dynamic duration ladder with annualized return around 2.23%, compared with 1.63% for the fixed 5-year ladder. Volatility rises from about 3.91% to 4.68%, and the max drawdown worsens slightly from -13.94% to -14.72%.
The higher return and slightly better Sharpe come with more volatility and a somewhat deeper drawdown. The dynamic allocation therefore changes the amount of rate risk held as well as the return relative to the fixed ladder. The active performance table is cleaner: the dynamic ladder earns about 0.62% annual active return with active volatility around 1.22%, giving an active Sharpe around 0.51.
The active results give us a way to assess the model-based duration allocation separately from the return of the underlying fixed ladder. Across this evaluation sample, the improvement is moderate relative to the active volatility taken to obtain it.
15) Fixed-for-floating swaps as a duration overlay
Before building the overlay, we need to understand what a swap is. A plain-vanilla interest-rate swap is a contract where two parties exchange cash flows:
one side pays a fixed rate on a notional amount,
the other side pays a floating rate tied to a short-rate benchmark.
The notional is not exchanged. It is only used to compute cash flows. For a $100 million notional swap with a 4% annual fixed rate and semiannual fixed payments, the fixed payer owes roughly $2 million every six months on the fixed leg, while receiving floating payments that reset with the benchmark.
There are two standard positions:
Receiver swap: receive fixed, pay floating.
Payer swap: pay fixed, receive floating.
A receiver swap behaves like receiving the cash flows of a fixed-rate bond financed by floating-rate borrowing. Its value rises when swap rates fall because the fixed coupon you receive becomes more attractive. A payer swap behaves like short duration. Its value rises when swap rates rise because paying the old fixed rate becomes cheaper relative to the new market fixed rate.
For example, suppose you enter a 10-year receiver swap at a fixed rate of 4%. If one month later the market 10-year swap rate falls to 3.5%, your right to receive 4% fixed is valuable. The receiver swap gains. If the market rate rises to 4.5%, receiving 4% is unattractive and the receiver loses.
With a swap overlay, we can add or reduce rate sensitivity while keeping the cash bond ladder in place. The position gains or loses value as the market curve changes, just like any other source of duration exposure.
15.1 Receiver and payer intuition through rate moves
We can understand receiver and payer positions by following their value as market rates move after the trade begins.
For a receiver swap, we receive fixed and pay floating. If market swap rates fall after we enter, receiving the old fixed rate becomes valuable. The receiver gains. If market swap rates rise, receiving the old fixed rate becomes unattractive. The receiver loses.
For a payer swap, we pay fixed and receive floating. If market swap rates rise, paying the old fixed rate is valuable because new market participants have to pay a higher fixed rate. The payer gains. If market swap rates fall, the payer loses.
So the sign mapping is:
Position
Gains when rates
Duration sign
Portfolio use
Receiver swap
fall
positive duration
add duration
Payer swap
rise
negative duration
reduce duration
A receiver swap adds positive rate duration while the existing ladder remains in place. The position is marked to market as rates move, so an increase in the market rate after entering the receiver produces a loss on the overlay.
A par swap rate is the fixed rate that makes the swap value zero at initiation. If payment times are \(t_1,\ldots,t_n\), the corresponding accrual fractions are \(\alpha_i\), and the discount factors are \(P(0,t_i)\), we can value each fixed payment at the start of the swap. The fixed-leg annuity is:
\[
A = \sum_{i=1}^{n}\alpha_i P(0,t_i)
\]
The present value of the fixed leg for fixed rate \(K\) and notional \(N\) is:
\[
PV_{fixed} = NKA
\]
In the single-curve approximation used here, a newly reset floating leg telescopes to the difference between unit principal today and the final discounted principal. For a standard par floating leg, its value is approximately:
\[
PV_{float} = N(1-P(0,T))
\]
The par swap rate \(K^*\) sets fixed and floating leg values equal. Dividing both sides by the notional \(N\) and the annuity \(A\) gives:
The swap PV01 measures the change in swap value for a 1 bp move in rates. Receiver PV01 is positive in the convention used here because a downward rate shift increases receiver value. Payer PV01 is negative because a downward rate shift hurts the payer.
A swap overlay changes the portfolio’s interest-rate exposure by adding a receiver or payer position alongside the cash ladder. The position’s value and trading costs still enter the portfolio return.
In a receiver swap, our position receives fixed and pays floating. In a payer swap, our position pays fixed and receives floating. The same contract has two sides, so one investor’s receiver position is another investor’s payer position.
The fixed payments are set using the agreed rate, while floating payments reset against the benchmark. Once the contract starts, its value changes as the market curve and the value of the remaining fixed and floating payments change.
Both sides apply their rates to the same notional. The actual fixed and floating payments also depend on the accrual periods in the contract’s respective payment schedules.
16) Swap valuation functions and the 10-year overlay instrument
We now build the swap analytics from the zero curve. The core functions are:
interpolate zero rates at payment times,
convert zero rates to discount factors,
build a semiannual fixed-payment schedule,
compute the fixed-leg annuity,
compute the par swap rate,
compute receiver/payer mark-to-market value,
compute PV01 by bumping the curve up and down.
The discount factor for a payment time \(t\) is:
\[
P(0,t) = e^{-R(0,t)t}
\]
The annuity is:
\[
A = \sum_i \alpha_i P(0,t_i)
\]
The par rate is:
\[
K^* = \frac{1-P(0,T)}{A}
\]
We’ll use a 10-year synthetic swap overlay to add or remove duration on top of the cash ladder. The 10-year point is a liquid, widely followed maturity and gives the overlay substantial rate sensitivity relative to its notional.
At each payment date, the interpolated zero rate supplies the discount factor for that cash flow. Summing the discounted accrual fractions gives the annuity used in the par-rate calculation.
16.1 Swap duration versus bond duration
A swap’s PV01 acts like duration exposure, but the cash-flow structure is different from a bond. A fixed-rate bond has an upfront price and coupon/principal cash flows. A par swap starts near zero value and exchanges fixed versus floating cash flows over time.
The receiver swap’s PV01 is close to the PV01 of the fixed leg because the floating leg resets. The fixed leg has long-duration exposure. The floating leg has much shorter duration because its coupon resets to current rates. So receiving fixed and paying floating leaves us with positive duration.
A rough intuition is:
\[
PV01_{receiver} \approx N \times A \times 0.0001
\]
where \(A\) is the swap annuity. The annuity scales both the fixed-leg payments and this approximate sensitivity. For a given target PV01, a larger annuity means that a smaller notional is needed to reach the same rate exposure.
For the overlay, this is very convenient. If we know the PV01 we want to add, and we know the PV01 per unit notional, we can solve for notional directly. That makes the overlay more targeted than buying a mix of bonds and hoping the KRD exposure comes out right.
overlay_tenor =10.0overlay_slippage_bp =0.5overlay_duration_budget =1.5overlay_dv01_fraction_cap =0.40example_swap = pd.DataFrame([ {"side": side, "par rate": par_swap_rate(curve_dates[-1], overlay_tenor),"annuity": swap_annuity(curve_dates[-1], overlay_tenor),"pv01 per 1 notional": swap_pv01(curve_dates[-1], overlay_tenor, side=side)}for side in ["receiver", "payer"]])display(example_swap)
side
par rate
annuity
pv01 per 1 notional
0
receiver
0.042576
8.169869
0.000822
1
payer
0.042576
8.169869
-0.000822
The latest 10-year par rate is about 4.26% and the corresponding annuity is around 8.17, using the Treasury-implied zero curve for valuation. The receiver PV01 per unit notional is about 0.000822, while the payer PV01 is the negative of that.
The signs follow the two swap positions: a receiver adds positive duration exposure, and a payer subtracts duration exposure. Since the overlay is synthetic, we can adjust duration without physically selling 2Y/5Y/10Y/30Y ladder positions.
The magnitude also matters. A unit-notional 10-year receiver has about 8.22 bps of value change for a 1 bp yield move under this normalization. The strategy later scales notional so the overlay PV01 stays inside a duration budget and doesn’t dominate the cash ladder.
17) Swap overlay strategy
We use the dynamic duration signal to set a swap position on top of the fixed cash ladder. The position expresses the same higher, lower, or neutral duration direction through the receiver or payer instrument.
The signal is:
\[
s_t = \begin{cases} +1 & \text{if target duration is above neutral} \\ -1 & \text{if target duration is below neutral} \\ 0 & \text{if target duration is close to neutral} \end{cases}
\]
A positive signal means we want more duration, so we enter a receiver swap. A negative signal means we want less duration, so we enter a payer swap. A zero signal means flat overlay.
The overlay target PV01 is capped two ways:
\[
PV01^{target}_t = s_t \cdot \min\left(BD_t,\; c \cdot |PV01^{base}_t|\right)
\]
where \(B\) is the overlay duration budget, \(D_t\) scales by NAV, and \(c\) is the fraction cap relative to the base ladder’s DV01. The notional is then:
The duration budget limits the amount of PV01 relative to NAV, and the fraction cap limits it relative to the base ladder’s DV01. Taking the smaller amount keeps the swap exposure tied to the scale and rate sensitivity of the underlying portfolio.
17.1 Overlay PnL and implementation
At each decision date, the swap is initiated near par, so its starting value is approximately zero. Over the next month, its PnL comes from revaluing the remaining swap at the new curve. If the fixed rate locked at initiation is \(K_t\), then one month later the receiver value is approximately:
We calculate the cost term from the change in signed notional at each rebalance, so its size reflects the amount traded. If we go from receiver to payer, the signed notional changes a lot, so the trade cost is larger. If we slightly resize an existing receiver, cost is smaller.
Using a swap reduces the need to rebalance the cash ladder, while the overlay still depends on tenor, notional cap, slippage, reset frequency, and collateral or cash treatment. Our synthetic swap is priced from the Treasury-implied curve, so differences from actual swap-market pricing also affect how closely the exercise represents a traded swap.
Dividing the swap PnL and cost by starting NAV expresses both in return units. We can then add that component to the bond ladder’s return over the same monthly holding period.
The strategy is in a receiver position through late 2025 and January 2026, according to the latest overlay log. That means it is adding duration on top of the fixed 5-year ladder. The notional is around 0.184–0.186 relative to NAV, and the overlay PV01 is around 0.00015.
The overlay contributed positively in September, October, and November 2025, then lost money in December and January as rates moved against the receiver position. That pattern follows the receiver position’s positive duration: it gains when rates fall and loses when rates rise. The core ladder return and overlay return are shown separately, which is useful because we can see whether the overlay is truly adding value or simply increasing total volatility.
The modern swap overlay runs from March 2020 to January 2026, with 47 active months. Average absolute notional is around 0.11 of NAV, and average cost return is about 0.22 bps per month, or around 0.000022 in decimal return units. Multiplying the decimal figure by 10,000 converts it to basis points.
The signed notional plot shows frequent switching between receiver, payer, and flat/near-flat states. The cumulative return-pieces plot shows the overlay return is small relative to the core ladder but still meaningful. It adds a separate source of active duration timing without requiring the cash bond ladder itself to move as aggressively.
18) Raw, SOFR-excess, and active performance
We evaluate three versions of performance:
Raw returns: total portfolio returns.
SOFR-excess returns: returns minus the monthly SOFR cash return.
Active returns: returns minus the fixed 5-year ladder benchmark.
After 2020, the SOFR-excess view compares the portfolio with a cash alternative whose return changes through time. For each month, we calculate:
These answer different questions. SOFR-excess asks whether the strategy beat cash. Active return asks whether the strategy beat the fixed ladder. A duration strategy can beat the fixed ladder while still failing to beat cash in a high-cash-rate period.
18.1 Comparing returns with cash and the fixed ladder
SOFR-excess returns measure the portfolio’s return after subtracting the return on overnight secured cash. This gives us a way to compare the bond allocation with the cash alternative over the same months.
During a high-rate period, the cash benchmark compounds at a meaningful monthly rate. If the bond ladder earns coupon income but loses price value because rates rise, its raw return can be positive while its cash-excess return is negative. This is exactly what happened in many fixed-income portfolios during the recent rate-normalization period.
We can read the three performance measures alongside each other, since each one uses a different reference:
Did the strategy make money at all? Raw return answers that.
Did it beat cash? SOFR-excess return answers that.
Did the model improve on a passive duration benchmark? Active return answers that.
Active return isolates the improvement relative to the fixed ladder, while SOFR-excess return measures the opportunity cost relative to cash. Raw return keeps the total portfolio outcome visible alongside both comparisons.
The raw modern-sample results are modest because this period includes the brutal bond-market drawdown after 2021. The fixed 5-year ladder earns about 0.28% annualized with volatility around 4.49%. The dynamic ladder improves raw return to about 0.56% and the swap overlay version to about 0.40%.
All strategies are negative versus SOFR in this sample, as cash was very competitive during the high-rate period. The dynamic ladder has the least negative SOFR-excess annual return, around -2.24%, compared with -2.52% for fixed. So the active model helps, but it doesn’t overcome the cash hurdle.
Comparing active returns against the fixed ladder isolates the contribution of the duration decisions. The dynamic ladder has annual active return around 0.29% with active volatility around 1.04%, producing an active Sharpe near 0.29. The swap overlay has smaller active return around 0.12%, but also slightly lower active volatility. The dynamic cash-ladder implementation is stronger here than the synthetic overlay.
Show code
fig, axes = plt.subplots(1, 2, figsize=(13, 4), constrained_layout=True)for col in modern_returns.columns: nav_series(modern_returns[col].dropna(), start_value=1).plot(ax=axes[0], label=col)axes[0].set_title("modern sample raw nav")axes[0].legend()for col in modern_excess_returns.columns: drawdown_series(modern_excess_returns[col].dropna(), input_kind="returns").plot(ax=axes[1], label=col)axes[1].set_title("sofr-excess drawdown")axes[1].legend()plt.show()
Raw NAVs decline heavily during the 2022 rate shock and then recover gradually. The plotted NAV paths make the size and timing of that drawdown visible across the three portfolios. The dynamic strategy does somewhat better, but it doesn’t avoid the main drawdown because its duration exposure still suffers when rates reprice upward.
The SOFR-excess drawdowns show that the opportunity cost of holding duration relative to cash was persistent during this part of the sample. In a zero-rate world, bond carry looks attractive. In a high-SOFR world, the hurdle rate is high, and a bond strategy has to earn enough price return or carry advantage to beat cash. During much of this sample, that was difficult.
The dynamic ladder and swap overlay don’t materially reduce the reported tail risk relative to the fixed ladder in this sample. Historical 5% VaR is around 2.2% monthly across the strategies, and expected shortfall is around 2.6%. The Cornish-Fisher numbers are similar.
The cash ladder has average turnover around 0.33, while the dynamic target changes in about 0.23 of months. These turnover measures describe different actions, so we read each one in the context of its strategy. The swap overlay’s notional change is much smaller on average, and its average cost return is low. That is the practical advantage of a swap overlay: it can adjust duration with low transaction cost. But in this sample, cheaper implementation doesn’t automatically mean better active performance.
19) Parallel-rate scenario diagnostics
The scenario diagnostic uses the KRD approximation:
\[
\Delta NAV \approx -\sum_j KRD_j \Delta y_j
\]
For a parallel shift, every selected key rate moves by the same amount. A +100 bp shock means:
\[
\Delta y_j = 0.01 \quad \text{for all } j
\]
so the PnL is approximately minus total key-rate duration times 1%. The approximation uses current KRD exposure and omits the convexity contribution that would appear in a full repricing.
Scenario diagnostics are useful because backtests tell us what happened historically, while KRD scenarios tell us what the current portfolio would lose or gain under a simple shock. A strategy with higher recent return can still be dangerous if it has too much current duration.
The scenario table shows the dynamic duration ladder has the largest current parallel-rate sensitivity. A -100 bp shock would add about +6.99%, while a +100 bp shock would cost about -6.99%. The fixed 5-year ladder has about ±4.91%, and the synthetic swap overlay ladder sits in between at about ±6.45%.
So the dynamic model is currently taking more duration than the fixed benchmark. That explains why it benefits more if rates fall, but loses more if rates rise. The swap overlay also adds duration, but less than the full dynamic ladder.
At this final date, the dynamic strategy is positioned to benefit more from falling rates and to lose more from rising rates than the fixed ladder. The historical return comparison and this current exposure describe different aspects of the same strategy.
20) Model stability and diagnostic plots
We can now compare model estimates and fit diagnostics through time. The plots and summary cover:
rolling Vasicek and CIR mean reversion,
rolling Vasicek and CIR volatility,
HW1F and G2 scenario-factor volatilities,
G2 factor correlation,
PCA three-factor explained variance,
success rates and residual autocorrelation.
Parameter jumps, remaining residual autocorrelation, and a nearly degenerate factor covariance can all affect how we interpret a model’s signals. Looking at these series through different rate regimes helps us understand the stability of the estimates behind the strategy.
20.1 Reading residual autocorrelation correctly
Residual autocorrelation measures the serial structure left after fitting the conditional dynamics. If the model describes those dynamics well, successive one-step errors should have little remaining correlation. In a one-step transition model, the residual is roughly:
\[
e_{t+1} = r_{t+1} - E_t[r_{t+1}]
\]
The first-order autocorrelation is:
\[
\rho_1 = Corr(e_t,e_{t-1})
\]
If \(\rho_1\) is strongly positive, the model is missing persistence. If it is strongly negative, the model may be overreacting or creating reversal errors. Values close to zero are better.
In rate data, some residual autocorrelation is hard to avoid because central-bank policy is persistent. Rate changes are not independent coin flips. Hiking cycles and cutting cycles create serial dependence. So we shouldn’t expect perfect white noise. But comparing residual autocorrelation across models still tells us which model absorbs more structure.
We can compare CIR and Vasicek on this residual measure alongside the conditioning and stability of their parameter estimates. Those diagnostics describe different features of the fit: less serial dependence does not by itself establish that the parameters are precisely estimated.
The estimated autocorrelation comes from a finite rolling window, so we can compare its value across models and also watch how it changes as the historical window moves.
Show code
stability_rows = []for date in model_parameters.index[::2]: history = zero_rates.loc[:date].tail(rolling_window) hfit = estimate_hw1f(history, scenario_maturities) gfit = estimate_g2(history, scenario_maturities) pfit = estimate_pca(history, scenario_maturities, n_components=3) cov = gfit["factor covariance"] stability_rows.append({"date": date, "hw1f sigma": hfit["sigma monthly"] * np.sqrt(12),"hw1f mean reversion": hfit["a"], "g2 first vol": np.sqrt(cov[0, 0]) * np.sqrt(12),"g2 second vol": np.sqrt(cov[1, 1]) * np.sqrt(12),"g2 correlation": gfit["factor correlation"],"pca three pc total": pfit["explained"].sum()})scenario_parameters = pd.DataFrame(stability_rows).set_index("date")fit_diagnostic_summary = pd.DataFrame({"metric": ["vasicek success rate", "cir success rate", "median vasicek residual acf1","median cir residual acf1", "median pca three pc explained"],"value": [fit_diagnostics["vasicek success"].mean(), fit_diagnostics["cir success"].mean(), fit_diagnostics["vasicek residual acf1"].median(), fit_diagnostics["cir residual acf1"].median(), pca_parameters["three pc total"].median()],})fig, axes = plt.subplots(2, 2, figsize=(13, 7), constrained_layout=True)model_parameters[["vasicek kappa", "cir kappa"]].plot(ax=axes[0, 0], title="rolling short-rate mean reversion")model_parameters[["vasicek sigma", "cir sigma"]].plot(ax=axes[0, 1], title="rolling short-rate volatility")scenario_parameters[["hw1f sigma", "g2 first vol", "g2 second vol"]].plot(ax=axes[1, 0], title="scenario factor volatilities")scenario_parameters[["g2 correlation", "pca three pc total"]].plot(ax=axes[1, 1], title="factor diagnostics")plt.show()display(fit_diagnostic_summary)
metric
value
0
vasicek success rate
1.000000
1
cir success rate
1.000000
2
median vasicek residual acf1
0.143875
3
median cir residual acf1
-0.060870
4
median pca three pc explained
0.984549
The rolling short-rate parameters vary across the sample, particularly around changes in the rate regime. CIR volatility starts very high in earlier windows, falls during low-rate regimes, and rises again in the higher-rate period. Vasicek volatility is much more stable but also less flexible. Mean reversion spikes around some regime transitions and then settles.
The scenario factor volatility panel shows elevated factor risk around crisis periods and again during the recent hiking cycle. The PCA diagnostic is strong: the median three-PC explained variance is about 98.45%, which confirms that level, slope, and curvature are a compact but powerful description of curve changes.
The fit diagnostic summary reports 100% success rates for both Vasicek and CIR rolling fits. Median Vasicek residual autocorrelation is about 0.144, while CIR’s is about -0.061. CIR again looks better on residual autocorrelation, although the earlier Hessian diagnostics warn us that CIR parameters can be numerically delicate.
21) Secondary application: Japan JGB term-structure market
We’ll apply the same framework to Japanese government bond yields. Japan offers a different term-structure setting from the US, with a long history of low rates and policy measures that influenced both the level and shape of the curve.
For many years, Japan lived with near-zero or negative policy rates, quantitative easing, and yield curve control. In that environment, the meaning of duration is different. A 10-year yield moving from 0.05% to 0.50% can be a huge relative regime change even though the absolute yield level still looks low compared with the US. Long periods of suppressed yields also affect model estimation: mean reversion, volatility, and factor dynamics are all shaped by policy constraints.
We use the Japan MOF yield data through the library workflow. For reproducibility, check the corresponding repository folder: Japan MOF yields data folder.
The Japan run reuses the same broad structure:
prepare monthly JGB curves and zero rates,
fit Vasicek/CIR-style short-rate diagnostics,
estimate HW1F, G2-style, and PCA curve factors,
build fixed and dynamic duration ladders,
evaluate a synthetic JGB-implied swap overlay,
compare strategy NAV, active return, scenario exposure, and rolling active return.
The same models and portfolio construction can now be examined in a market where rates were pinned for years and long-end moves were often influenced by monetary policy. We’ll use the library to implement the full secondary application.
21.1 Japan’s rate environment
Japan’s yield curve history forces the models to work in a different environment. The US sample contains large policy cycles and a violent repricing after 2021. Japan’s sample contains long periods where rates were compressed near zero and policy frameworks constrained the curve.
This changes model interpretation in several ways:
A small move in JGB yields can be economically meaningful because the starting level is low.
Mean-reversion estimates can be dominated by policy anchoring instead of market-driven cycles.
Long-end yield changes can reflect changes in policy credibility or curve-control expectations, not just inflation risk.
Duration carry can look stable for long periods, then reprice quickly when the market questions the anchor.
Applying the framework here lets us compare its estimates and portfolio decisions across a different absolute rate level, volatility scale, and policy regime. The maturity pattern of changes is as relevant as their small absolute size.
For example, a 10 bp yield move can change the value of a duration position even when the starting yield is near zero. The portfolio effect still depends on maturity and on which part of the curve moved.
We have 13,129 daily observations and 316 monthly dates, from September 1999 to December 2025, in the Japan data summary. That gives a longer monthly sample than the US run, but the economic regime is more unusual.
The JGB curve snapshots show low-rate history and later upward movement. Unlike the US curve, where the front end and long end experienced dramatic tightening after 2022, Japan’s curve has been more constrained and gradual for much of the sample. The short-rate fan comparison reflects this: simulated paths are tighter in absolute terms than the US rate fans.
In a low-rate, policy-managed market, curve changes may be distributed differently across maturities than in the US sample. The PCA and HW/G2 loading panels help diagnose whether a one-factor model is enough or whether slope/long-end dynamics matter.
The factor loadings describe changes in absolute yield units at each maturity. Even a small numerical move can matter to a bond portfolio when it holds substantial duration exposure.
The Japan application produces a strong active result over its evaluation sample, with the model-driven ladder improving on the fixed-duration benchmark. The dynamic duration ladder earns about 0.68% annualized with volatility around 1.43%, compared with about 0.38% annualized and 1.25% volatility for the fixed 5-year ladder. The dynamic ladder’s Sharpe is about 0.48, versus 0.31 for fixed.
The active performance table is more impressive: the dynamic ladder earns about 0.30% annual active return with active volatility around 0.33%, giving an active Sharpe around 0.91. That is a strong result for a bond-duration timing overlay. It means the model-driven target changes added meaningful return relative to their active risk.
The JGB-implied swap overlay also adds value versus fixed, but it underperforms the dynamic cash-ladder implementation. Its annual active return is about 0.19% with active volatility around 0.27%, giving an active Sharpe around 0.70. So the overlay is smoother and lower risk, but it captures less of the dynamic ladder’s active return.
The model-selection counts are balanced between G2 and HW1F: G2 is selected 123 months and HW1F 108 months. That is very different from the US run, where CIR dominated selection. Japan’s secondary implementation uses HW/G2 for the target-selection layer, and the market seems to alternate between one-factor and two-factor curve behavior.
The target-count table shows only higher duration and neutral duration selected: 118 months higher and 113 months neutral. There are no lower-duration target months in the final selected log. That makes economic sense for much of Japan’s sample. In a low-rate but policy-supported bond market, the model often finds positive carry/roll or favorable expected curve movement from holding duration, but doesn’t often see enough reason to go structurally short duration.
The historical preference for higher duration is connected to Japan’s low-rate and policy-supported regime. If those dynamics change substantially, the balance between carry, expected rate changes, and duration risk can change too. The model worked in the historical sample, but its future usefulness depends on whether Japan’s rate dynamics remain partially anchored or become more like other developed markets.
The latest parallel-rate exposure is identical across the fixed, dynamic, and swap-overlay strategies in the Japan scenario table: about +5.02% for a -100 bp shock and -5.02% for a +100 bp shock. This doesn’t mean the strategies were identical through history. It means their latest effective KRD exposure ended up essentially the same under the scenario diagnostic.
In the US, the latest dynamic strategy was taking more duration than fixed. In Japan, by the final date, the strategies converge to a similar current parallel-shift risk. The historical active return therefore came from timing through the sample, not from ending with a dramatically different current duration exposure.
The conservative overlay was selected over the stronger setting in the overlay comparison. The stronger overlay has higher annual active return, but it also has higher active volatility and deeper drawdown. The Sharpe ratios are almost identical, and the conservative overlay has the smaller drawdown. The stronger setting is selected only when it has both at least as high an annual active return and at least as favorable a maximum drawdown as the conservative setting. In this result, the drawdown comparison keeps the conservative setting.
We can read the Japan results across the twelve panels, following the curves, model factors, selected targets, and portfolio outcomes:
The JGB curve snapshots show the long period of low yields and the later steepening/upward shift.
The short-rate fan charts show low absolute volatility compared with US rates.
PCA and HW/G2 panels show that curve shape still matters even in a low-rate market.
Model counts show a real alternation between HW1F and G2, not one model dominating mechanically.
Target counts confirm the strategy mostly chooses neutral or higher duration.
Strategy NAV shows dynamic duration leading fixed duration over time.
Active NAV versus fixed 5Y rises steadily, with some drawdowns but no collapse.
Scenario bars converge across strategies at the final date, which matches the equal KRD scenario table.
The rolling 12-month active return plot shows that the active edge is episodic, not constant.
Japan’s curve market rewarded controlled duration timing over this long sample, mostly through decisions between neutral and higher duration rather than aggressive duration reduction. This is a completely different regime from the modern US SOFR period, where beating cash was extremely difficult and high duration created much larger drawdown risk.
The same modeling framework can produce different outcomes across markets because the rate environment and economic constraints differ. The Japan and US applications give us two settings in which to compare the duration decisions. In the US, the post-2020 sample is dominated by a high cash hurdle and a large duration drawdown. In Japan, the long sample has lower absolute volatility and a different policy anchor, so controlled duration timing can look much more attractive on an active-risk basis.
We can compare where the active edge comes from in each market and how much risk the strategy takes to earn it. In the US, the active edge is modest and partly overwhelmed by the cash benchmark during the SOFR period. In Japan, the active edge is more visible versus the fixed ladder because the strategy times duration inside a lower-volatility curve environment.
The secondary application also gives us a view of the models in Japan’s local market regime and lets us compare the resulting duration choices. A curve model is mathematical, but its output is economic. A higher-duration signal in Japan doesn’t mean the same thing as a higher-duration signal in the US because the distribution of rate moves, policy reaction function, and long-end risk premium are different.
