Constant-Duration Bond Portfolios: Continuous Time Models

December 31, 2024

Constant-Duration Bond Portfolios: Continuous Time Models

Abstract

Modern portfolio management frequently employs duration-targeting strategies to optimize the trade-off between risk and return. These strategies maintain a stable portfolio duration through periodic rebalancing, preserving a consistent exposure to interest rate fluctuations. Practical implementations include index-tracking and ladder portfolios. As demonstrated by Martin L. Leibowitz and his co-authors [1–4], a key feature of duration targeting is its ability to concentrate return distributions around a minimum-variance point, which occurs when the portfolio's duration is approximately half the investment horizon. In this study, we analyze stochastic interest rate models in the context of constant-duration portfolios. The analysis explores cases where the yield curve is both flat and of an arbitrary shape. A related discussion in Russian has been published on the Zen platform.

Holding Period Return Calculation for a Zero Coupon Bond Portfolio

Throughout this section and the following analysis, we assume a flat yield curve and no default risk.

The simplest constant-duration bond portfolio consists of zero-coupon bonds with a common maturity date. To maintain the target duration, the portfolio undergoes continuous rebalancing.

We demonstrate that the realized return of such a portfolio over a short time interval consists of two components: accrued interest (accruals) and price change. The value of a constant-duration portfolio at time $t$ is given by

$$V_t = N_t e^{-y_t D}, \tag{1}$$

where $N_t$ is the number of bonds, $y_t$ is the yield to maturity, and $D$ is the duration.

Suppose that after a time interval Δ$t$, the yield $y_t$ changes instantly by Δ$y_t$ . The new portfolio value becomes:

$$ V_{t + \Delta t} = N_t e^{-[y_t + \Delta y_t] [D - \Delta t]}, \tag{2}$$

To account for rebalancing, expression (2) can be rewritten as:

$$V_{t + \Delta t} = N_t e^{[y_t + \Delta y_t] \Delta t} \cdot e^{-[y_t + \Delta y_t] D}, \tag{2*}$$

This implies that the total number of bonds increases to $N_{t + \Delta t} = N_t e^{[y_t + \Delta y_t] \Delta t}$ while the price of a new bond with a unit face value is $e^{-[y_t + \Delta y_t] D}$.

With continuous compounding, the annualized portfolio return over small time interval Δ$t$ is:

$$h_t(\Delta t) = \frac{1}{\Delta t} \ln \left[ \frac{ V_{t + \Delta t}}{V_t}\right ], \tag{3}$$

Substituting (1) and (2), we derive:

$$h_t(\Delta t) \Delta t = y_t \Delta t + \Delta y_t \Delta t - D \Delta y_t, \tag{4}$$ Since Δ$y_t$ Δ$t$ is a second-order small term, it can be neglected for sufficiently small Δ$t$ The instantaneous realized return is then expressed in differential form:

$$h_t \, dt = y_t \, dt - D \, dy_t \tag{5}$$

This decomposition makes it evident that the immediate change in portfolio value arises from:

accrued interest over an infinitesimal time interval, $y_t \, dt$
price change due to yield fluctuations, $−D \, dy$

In particular, this implies that the instantaneous realized return $h_t$ coincides with the yield to maturity $y_t$ only in the absence of rate fluctuations.

The total holding period return (HPR) follows from integrating (5) and normalizing by the holding period (in years):

$$HPR_{\tau} =\frac{1}{\tau}\int_{0}^{\tau}y_s \, d s - \frac{D}{\tau}\left ( y_{\tau} - y_0 \right ) \tag{6}$$

Simple Trendline Model

Assume that the yield to maturity evolves according to a linear trend over time: $$ y_t = y_0 + k t \tag{7}$$ Substituting this into (6), we obtain: $$ HPR_{\tau} = y_0 + \frac{k}{2}(\tau - 2D) \tag {8}$$ Rewriting equation (8), we get: $$ \Delta R = - \Delta y \left ( \frac{D}{\tau} - \frac{1}{2} \right) \tag {9}$$ where $\Delta R = HPR_{\tau} - y_0$ is the deviation of the HPR from the initial yield, and $\Delta y = y_{\tau} - y_0 = k \tau$ is the total yield change over the holding period $\tau$. The term $D_{TL} = \frac{D}{\tau} - \frac{1}{2}$ represents the sensitivity (a type of duration) of the realized return to interest rate changes over the horizon $\tau$. At $\tau = 2D$ this sensitivity vanishes, making the HPR dependent solely on the initial yield $y_0$

and independent of the yield trend slope $k$.

The chart below illustrates how the yield to maturity, accumulated portfolio value, and HPR evolve for a portfolio targeting a duration of 3 years. Initially, the portfolio consists of a single 3-year zero-coupon bond. The initial yield is $y_0 = 15\%$, with yield changes $\Delta y = -2 \% , 0\% , +2 \% $ per year.

For comparison, consider the returns from a portfolio of three-year bonds, where matured bonds are reinvested into new three-year bonds at prevailing yields. Assuming the same interest rate dynamics and an initial yield of 15%, the annualized HPR for this portfolio after six years would be about 18% in a rising rate environment and around 12% in a declining rate scenario.

Some Stochastic Interest Rate Models

Linear Paths: Normally Distributed Future Yields

In a linear model framework, investors analyze different interest rate scenarios by assigning probabilities to each, creating a probability distribution of future yields over the investment horizon. The expected yield is then computed as the weighted average of these projected values. When yield changes follow linear paths, the resulting distribution can be effectively modeled using a fractional Brownian motion (fBm) framework. In this model, the Hurst index (H) is set to 1, and the yield dynamics is described by the following equation: $$ y_t = y_0 + \mu t + \sigma t z, \,\, \,\,\,\, t > 0, \tag {10} $$ where $\mu \in \mathbb R$, $\sigma > 0$ are constants, and $z \sim N(0, 1)$ is a standard normal variable.

Example of linear yield paths with $\mu = 0$, illustrating a scenario where the expected yield equals the initial yield.

Using equation (6), the holding period return (HPR) is given by: $$HPR_{\tau} = y_0 + \left( \frac {\tau}{2} - D \right) (\mu + \sigma z) \tag {11}$$The mean and variance of the HPR follow as:$$ \mathbb E [HPR_{\tau}] = y_0 + \mu \left( \frac {\tau}{2} - D \right) \tag {12}$$ $$ \mathrm {Var} [HPR_{\tau}] = \sigma^2 \left( \frac {\tau}{2} - D \right) ^2 \tag {13} $$ It follows that as the horizon approaches twice the duration, the expected return converges to the portfolio's initial yield, while return volatility declines toward zero.

Brownian Motion with Drift

Leibowitz et al. studied a random walk model without a drift [2], which corresponds to a continuous-time stochastic process where yield changes are driven by a Wiener process

$W_t$: $$ d y_t = \sigma d W_t. \tag{14}$$ Here, $\sigma$ represents constant volatility. We extend this model by introducing a constant drift term $\mu$:

\[ dy_t = \mu \, d t + \sigma \, d W_t, \,\, \,\,\,\, t > 0, \tag {14'} \] with the initial condition $y_{t=0} = y_0$ and constant parameters $\mu \in \mathbb R$, $\sigma > 0$.

The deviation of the yield from its initial value is normally distributed with mean $\mu t$ and variance $\sigma ^ 2 t $: $$y_t - y_0 \sim N(\mu t, \sigma^2 t),$$ which allows for the possibility of negative yield values [5–6]. The explicit solution for the yield process is:

$$y_t = y_0 + \mu t + \sigma W_t, \tag {15} $$ with expectation and variance: $$\mathbb E [y_t] = y_0 + \mu t, \,\, \,\,\,\, \forall t \geq 0 , \tag {16}$$

$$\mathrm{Var} [y_t] = \sigma^2 t, \,\, \,\,\,\, \forall t > 0 \tag {17} $$

The HPR, along with its mean and variance, are derived in Appendix A. The HPR itself is given by $$HPR_{\tau} = y_0\, +\, \mu\cdot \left ( \frac{\tau}{2}\, -\, D \right )\, + \, \frac{\sigma}{\tau} \left ( \int_{0}^{\tau} W_{s}\, d s \, - \, D\cdot W_{\tau} \right ), \tag {18} $$ with expectation: $$ \mathbb E [HPR_{\tau}] = y_0\, + \, \mu\cdot \left ( \frac{\tau}{2}\, -\, D \right ) \tag {19} $$ and variance: $$ \mathrm {Var} [ HPR_{\tau}] = \sigma^2 \tau \left( \frac{1}{3} + \left( \frac{D}{\tau}\right)^2 - \frac{D}{\tau} \right) \tag {20}$$ The minimum variance of the HPR is achieved at the horizon $ \tau = \sqrt{3} D$: $$\underset{ \tau \, > \, 0}{\mathrm{argmin}}\, \mathrm {var} [HPR_{\tau}] = \sqrt{3}\cdot D \tag {21}$$

This analysis of HPR dynamics reveals several key insights:

The behavior of $\mathbb E[HPR_\tau]$ aligns with the linear model framework. Regardless of the drift parameter $\mu$, the expected return at a horizon equal to twice the duration always converges to the portfolio's initial yield $y_0$.
The difference in expected returns between two portfolios with target durations $D_1$ and $D_2$ remains constant across all investment horizons $\tau$, given by $D_2 - D_1$, assuming all other parameters are identical. A strategy with a shorter target duration consistently yields a higher expected return.
The investment horizon that minimizes HPR variance is always $\sqrt{3}D$, independent of model parameters.
At short horizons, duration-induced price fluctuations lead to a wide range of possible HPR values. Over time, accrual effects dominate, stabilizing return volatility across a broad range of horizons.
Eventually, return volatility falls below the yield volatility. At very long horizons, the ratio of return volatility to yield volatility approaches $\frac{1}{\sqrt{3}}$.
Our findings for total return volatility are consistent with those of Leibowitz et al. in the limit as the interest accrual period approaches zero.

Geometric Brownian Motion

    The previous model allowed for the possibility of negative interest rates. However, this issue does not arise if the yield follows a geometric Brownian motion (GBM). This can be formulated as the stochastic differential equation (SDE): $$ d y_t = \mu \, y_t \, d t \, + \, \sigma \, y_t \, d W_t, \,\, \,\,\,\, \,\, t > 0, \tag {22}$$ where $y_{t=0} = y_0 > 0 $ and $\mu \in \mathbb R$, $\sigma > 0$ are constants. In this model, the yield follows a lognormal process, similar to stock prices in the Black-Scholes model. The well-known solution to equation (22) is [5-6]: $$y_t = y_0 e^{\left( \mu \,\, - \, \frac{\sigma^2}{2} \right) t \, \, + \, \sigma \, W_t}, \,\, \,\,\,\, \,\, t \geq 0. \tag {23}$$ The expected value and variance of $y_t$ are given by $$ \mathbb E [y_t] = y_0 e^{\mu t} \tag {24}$$ $$ \text {Var} [y_t] = y_0^2 \, e^{2 \mu t} \left( e^{\sigma^2 t} \, - \, 1\right) \tag {25}$$
Using these results, we can derive expressions for the holding period return (HPR), its expectation, and variance (see Appendix B for details):
$$HPR_{\tau} = \frac{1}{\tau} \left [ \left( \frac {1}{\mu}- D\right) (y_{\tau} - y_0) - \frac{\sigma}{\mu } \int_{0}^{\tau} \, y_s \, d W_s \right] \tag {26}$$
$$\mathbb E [HPR_{\tau}] = \frac {y_0}{\tau} \left( e^{\mu \tau} - 1\right) \left( \frac {1}{\mu} - D \right) \tag {27}$$
$$ \text {Var} \left[ HPR_{\tau}\right] = \frac {y_0^2}{\tau^2} \left[ \left( \frac{1}{\mu} -D\right) ^2 e^{2 \mu \tau} \left( e^{\sigma^2 \tau} -1\right) + \frac{\sigma^2}{\mu^2 (2 \mu + \sigma^2)} \left( e^{(2 \mu + \sigma^2) \tau} -1 \right) \right] -$$$$- \frac {y_0^2}{\tau^2} \left[\frac{2 \sigma^2}{\mu (\mu + \sigma^2)} \left( \frac {1}{\mu} - D\right) e^{\mu \tau} \left( e^{(\mu + \sigma^2) \tau} -1 \right) \right] \tag {28} $$


The HPR dynamics in this case exhibit similarities to the standard Brownian motion model (see the first diagram). However, there are also notable differences. From equation (27), we observe the following properties of the expected return:
As the drift parameter approaches zero, the expected return tends to the initial yield: $$ \mathbb E [HPR_{\tau}] \rightarrow y_0 \,\,\,\, \text{ as } \,\,\, \mu \rightarrow 0 $$
The expected return is zero when $\mu D = 1$. Since $D \geq 0$, meaning that a consistently rising yield is required to fully offset price effects with accruals.
If the drift is negative, the expected return converges to $y_0$ more rapidly for larger absolute values of $\mu D$, whereas for positive $\mu$, convergence is slower.
Although the variance expression in (28) is complex, it reaches a minimum near  $\tau /D = 2$
The volatility of HPR remains low across a wide range of investment horizons and increases with duration, similar to the Brownian motion model.
This confirms that, despite its differences from the standard model, the GBM framework preserves key relationships between duration, horizon, and return behavior.

Mean-reversion Yield Processes

1. Vasicek Yield Process

Consider a stochastic model where the yield follows a Vasicek process: $$ d y_t = \theta (\mu - y_t) \, d t + \sigma \, d W_t, \,\, \,\,\,\, \,\, t > 0, \tag {29}$$ with the initial condition $y_{t=0} = y_0 \in \mathbb R $ and constant parameters $\mu \in \mathbb R$, $\theta >0$, $\sigma > 0$. In this model, the yield fluctuates around its long-term mean, $\mu$, with the possibility of taking negative values. The parameter $\theta$ controls the speed of mean reversion, while $\sigma$ represents volatility.

This behavior aligns with basic economic intuition:

High interest rates dampen economic activity by making borrowing expensive, leading to reduced demand and lower rates.
Conversely, low interest rates stimulate borrowing and economic expansion, increasing demand and driving rates higher.

As a result, interest rates tend to revert to the mean, remaining within a bounded range over time.

The yield dynamics, along with its mean and variance, are well-established in the literature [5-6]:

$$y_t = y_0\cdot e^{-\theta t}\, \, +\, \, \mu\cdot \left (1- e^{- \theta t} \right )\, + \, \, \sigma \cdot \int_{0}^{t} e^{-\theta\cdot (t-s)} dW_s \tag {30}$$ $$ \mathbb E [y_t] = \mu \, + \, (y_0 - \mu) e^{- \theta t} \tag {31}$$ $$\text {Var} [y_t] = \frac{\sigma^2}{2 \theta} \left( 1 - e ^{-2 \theta t}\right) \tag {32}$$

Within this framework, we can derive analytical expressions for the mean and variance of the holding period return (HPR) (see Appendix C for details):

$$ HPR_{\tau} = \mu \, + \, \frac{1}{\tau } (y_0 \, - \, \mu) \left ( 1+ \theta D \right ) \left( \frac{1-e^{-\theta \tau}}{\theta} \right ) \, + \, \frac{\sigma} {\tau}\int_{0}^{\tau}\frac{1- (1+ \theta D)e^{-\theta (\tau-u)} }{\theta} dW_u \tag {33}$$

$$ \mathbb E [HPR_{\tau}] = \mu \, +\, \frac{1}{\tau } (y_0 \, - \, \mu) \left ( 1+ \theta D \right ) \left( \frac{1-e^{-\theta \tau}}{\theta} \right ) \tag {34}$$

$$ \mathrm {Var} [HPR_{\tau}] = \frac{\sigma^2}{\theta^2 \tau^2} \left [ \tau + (1 + \theta D) \left[ (1+ \theta D) \frac{ 1-e^{- 2 \theta \tau}}{2 \theta} - 2\cdot \frac{1- e^{-\theta \tau}}{\theta} \right ] \right ] \tag {35}$$

Although closed-form solutions for the horizon at which HPR variance reaches its minimum or for the expected time to recover the initial yield are not available, several important observations can be made:

The horizon at which the expected HPR equals the portfolio’s initial yield depends on the mean-reversion speed ($\theta$ ) and the targeted duration ($D$), but not on the long-term mean ($\mu$). When $\theta > 0$, this horizon is less than $2D$
A higher mean-reversion rate (larger $\theta$ ) shortens the expected time to recover the initial yield. When $\theta = 0$, the HPR characteristics match those of the previously discussed Brownian motion model but without a drift term.
In this model, the difference in expected returns between two portfolios with target durations $D_1$ and $D_2$ is given by $$ \frac{y_0 - \mu}{\tau} \left( 1 - e^{-\theta \tau}\right)(D_1 - D_2)$$ If the long-term mean $\mu$ is not too large, specifically when $\mu < y_0$, this difference monotonically decreases as the investment horizon $\tau$ increases. In this case, a strategy with a longer target duration consistently yields a higher expected return. This contrasts with the Brownian motion with drift case, where the expected return difference remains constant over time, and a shorter target duration results in higher returns.
The existence of a local minimum in HPR variance depends on the relationship between $D$ and $\theta$. If $\theta D$ exceeds a certain threshold, no minimum exists, and HPR volatility decreases monotonically.
Regardless of parameter values, for sufficiently long horizons, HPR volatility diminishes as $\frac{\sigma^2}{\tau \, \theta^2 }$, approaching zero in the limit.

2. Cox–Ingersoll–Ross Yield Process

Another prominent mean-reverting model is the Cox–Ingersoll–Ross (CIR) process. It describes the yield dynamics via the following SDE:

$$ d y_t = \theta(\mu - y_t) \, d t + \sigma \sqrt{y_t}\, d W_t, \,\,\,\,\,\, \,\,\,\,\, t > 0, \tag{36}$$ with initial condition $y_{t=0} = y_0 \in \mathbb R$ and constant parameters $\mu \in \mathbb R$, $\theta >0$, $\sigma > 0$.

Like the Vasicek yield model, this process features a mean-reverting drift term. However, its volatility scales with the square root of the current yield, making it state-dependent. As a result, higher yields are associated with greater volatility, while small yields exhibit lower fluctuations. Notably, if the Feller condition $$\sigma^2 \leq 2 \theta \mu$$ is satisfied, and the initial yield is positive ($y_0 > 0$), the CIR yield process remains strictly positive at all times.

The integral representation of this process is given by [6]:

$$ y_t = y_0 e^{-\theta t} + \mu \left( 1 - e^{-\theta t}\right) + \sigma \int_0^t e^{-\theta (t-s)} \sqrt{y_s} \, d W_s \tag {37}$$ Although equation (37) is implicit, it allows us to derive the expected value and variance of the yield [6]: $$\mathbb E [y_t] = y_0 e^{-\theta t} + \mu \left( 1 - e^{-\theta t}\right) \tag {38}$$ $$\text {Var} [y_t] = \frac {\sigma^2}{\theta} \left( 1 - e^{-\theta t}\right) \left( y_0 e^{-\theta t} + \frac{\mu}{2} \left( 1- e^{-\theta t}\right) \right) \tag {39}$$

A similar approach to that used for the Vasicek yield model can be applied to derive the mean and variance of holding period returns (HPR). In particular, the expected return follows the same formula as in equation (34), while the variance is given by: $$\text{Var} [HPR^{CIR}_{\tau}] = \mu \, \text{Var}[HPR^{Vas}_{\tau}] \,\,\,+$$$$+\,\,\, (y_0 - \mu)\frac{\sigma^2}{\theta^3 \ \tau ^2} \left[ \left(1+ e^{-\theta \tau}\left( 1+ \theta D\right)^2 \right) \left( 1-e^{-\theta \tau}\right) - 2 \theta \tau (1+ \theta D) e^{-\theta \tau} \right], \tag {40}$$ where $\text{Var}[HPR^{Vas}_{\tau}]$ is the HPR variance under the Vasicek yield process, given by equation (35).

The figure below compares HPR volatility in the Vasicek and CIR yield models under identical parameter values, illustrating a key distinction: the CIR yield model exhibits lower volatility due to its strictly positive yields, which reduce the range of potential outcomes.

Finally, we emphasize the importance of using mean-reverting processes in stochastic interest rate modeling. These processes reflect the natural tendency of interest rates to stabilize over time, making them a more economically justified choice, particularly for evaluating long-term holding period returns. A key implication is that long-term HPR volatility remains substantially lower than short-term yield volatility—an effect consistently observed in empirical data.

Arbitrary Yield Curve Shape: One-factor Short Rate Models

When modeling interest rate dynamics, we initially assumed a flat yield curve. However, this assumption has theoretical drawbacks. A flat yield curve that allows only parallel shifts violates the local expectations theory and creates arbitrage opportunities, making it an inadequate stochastic model for bond prices in general.

Moreover, in practice, the yield curve is rarely flat, and changes in short-term and long-term rates are not perfectly correlated. A more sophisticated approach is needed to capture the complexity of long-term interest rate movements.

The standard approach to term-structure modeling starts with an equivalent martingale measure (EMM), $ \mathbb Q$, under which all discounted security prices follow a martingale process [7-8]. In this framework, all bonds yield the same expected return over an infinitesimal time interval. This return is given by the short rate, $r_t$, which represents the instantaneous risk-free interest rate. The dynamics of $r_t$ is typically modeled as $$dr_t = \alpha (t, r_t) \, dt + \beta (t, r_t) \, dw_t, \tag {41}$$ where $w_t$ is $ \mathbb Q$-Brownian motion. Here, $\alpha (t, r_t)$ represents the drift term, which captures the deterministic trend of $r_t$, while $\beta (t, r_t)$ is the volatility term, determining the magnitude of stochastic fluctuations.

Such models are known as one-factor models, as they incorporate only one source of uncertainty. They assume all rates move in the same direction over short intervals, though not necessarily by the same magnitude.

The price of a zero-coupon bond at time $t$, denoted as $Z_{t, \,T}$, with a unit face value maturing at time $T$, is calculated under the martingale approach as $$Z_{t, \,T} = \mathbb E^Q_t \left[ e^{- \int_t^T \, r_s ds}\right] \tag{42}$$

The annualized yield between times $t$ and $T$, derived from the bond price $Z_{t, \,T}$, is defined as $$y_{t, \,T} = - \frac{\text{ln} \, Z_{t, \,T}}{T-t} \tag {43}$$

We fix a positive relative maturity $D$ and define the yield at time $t$ on the zero-coupon bond with relative maturity $D$ (i.e., the bond maturing at date $t + D$) as the long-term rate $y_t^{(D)}$: $$y_t^{(D)} = - \frac{\text{ln} \, Z_{t, \, t+D}}{D} \tag {44}$$ Under the risk-neutral measure, a model for the short rate $r_t$ determines the price of a zero-coupon bond maturing at $(t + D)$ and thereby fully specifies the long-term rate $y_t^{(D)}.$

We define $f_{t, \, T_1, \, T_2}$ as the forward rate for the period between time $T_1$ and $T_2$, observed at time $t:$ $$f_{t, \, T_1, \, T_2} = - \frac {\text{ln} Z_{t, \, T_2} - \text{ln} Z_{t, \, T_1}}{T_2 - T_1} \tag {45}$$ As $T_1$ approaches $T_2 = T$, the forward rate $f_{t, \, T_1, \, T_2}$ converges to the instantaneous forward rate $f_{t, \, T}$.

For a zero-coupon bond with price $Z_{t, \,T}$ maturing at time $T$, the instantaneous forward rate $f_{t, \,T}$ at time $t$ is given by $$f_{t, \, T} = - \frac{\partial \, \text{ln} Z_{t, \,T}}{\partial T} \tag {46}$$ Using equation (44), the relationship between the forward rate and the long-term yield is expressed as: $$f_{t, \, t+D} \equiv f_t^{(D)} = y_t^{(D)} + D \frac{\partial y_t^{(D)}}{\partial D} \tag {47}$$ This formula shows that the forward rate incorporates both the current yield and the slope of the yield curve over the horizon $D$.

Holding Period Return Calculation for an Arbitrary Yield Curve Shape

In this section, we derive the formula for the instantaneous return of a constant-duration portfolio under a non-flat yield curve. We consider a portfolio consisting of zero-coupon bonds with a common maturity date, continuously rebalanced to maintain the target duration $D$. The portfolio value at time $t$ is given by: $$V_t = N_t e^{-y_t^{(D)}D}. \tag {48}$$ Here, $N_t$ represents the number of bonds, $y_t^{(D)}$ denotes the long-term interest rate, and $D$ is the portfolio duration. To examine the impact of yield curve shifts, we analyze how the portfolio value evolves over a small time interval $\Delta t$. First, the yield curve experiences an instantaneously shift at time $t$. After a short time interval $\Delta t$, the updated portfolio value is given by: $$V_{t+\Delta t} = N_t e^{-\tilde {y}_{t+ \Delta t} (D - \Delta t)}, \tag {49}$$ where $\tilde {y}_{t+ \Delta t}$ is the yield at time $t+\Delta t$ if the shifted yield curve remains unchanged. Expanding this yield in a Taylor series with respect to $D$ gives:

$$ \tilde {y}_{t+ \Delta t} = \tilde{y}_{t}^{(D - \Delta t)} \approx \tilde{y}_t^{(D)} - \frac{\partial \tilde{y}_t^{(D)}}{\partial D} \Delta t \tag {50}$$ where $\tilde {y_t}^{(D)} = y_t^{(D)} + \Delta y_t^{(D)}, $ $\Delta y_t^{(D)}$ is the instantaneous change of $y_t^{(D)}$ at time $t$. Using the definition of instantaneous portfolio return from equation (3) and applying algebraic transformations, we obtain: $$ h_t^{(D)} \Delta t \approx -D \Delta y_t^{(D)} + D \frac{\partial}{\partial D}\left[y_t^{(D)} + \Delta y_t^{(D)} \right] \Delta t + y_t^{(D)} \Delta t \,\, + \tag {51}$$ $$ \,\,\, +\, \, \Delta y_t^{(D)} \Delta t - \frac{\partial}{\partial D} \left[y_t^{(D)} + \Delta y_t^{(D)}\right] \Delta t^2$$ Since $\Delta t$ is small, we neglect high-order terms involving $\Delta y_t^{(D)} \Delta t$ and $\Delta t^2$. Then, the instantaneous holding period return can be represented in differential form, considering the expression for the instantaneous forward rate from equation (46): $$h^{(D)}_t dt = f_t^{(D)} dt - D \, d y_t^{(D)} \tag {52}$$ The total holding period return over horizon $\tau$ is given by integral expression: $$HPR_{\tau}^{(D)} = \frac{1}{\tau}\int_0^{\tau}{h^{(D)}_u du} \tag{53}$$

Next, we analyze basic short-rate models, where the expected return and variance can be easily derived.

The Merton Model

In Merton's model (1973), the risk-neutral process for the short rate is described by the following SDE: $$ d r_t = \mu \, d t + \sigma \, d w_t, \,\,\,\,\,\,\, t>0,\tag {54}$$ where $\mu \in \mathbb R$ and $\sigma > 0$ are constants, and $w_t$ is a standard Brownian motion under the EMM. In this approach, the short rate follows an arithmetic Brownian motion. Integrating equation (54) from $t$ to $u$ ($0 \leq t < u $), we obtain:

$$r_u = r_t + \mu \, (u-t) + \sigma \int_t^u d w_s. \tag {55}$$ The distribution of $r_u$ is normal, with conditional expectation and variance given by $$\mathbb E [r_u|r_t] = r_t + \mu (u-t) \tag {56}$$ $$\text {Var}[r_u|r_t] = \sigma^ 2 (u-t) \tag {57}$$ In this model, the zero-coupon bond price takes the form [10]: $$Z_{t, \,T} = e^{-r_t(T-t) - \mu \frac{(T-t)^2}{2}+\sigma^2\frac{(T-t)^3}{6}} \tag {58}$$ Therefore, using equation (44), the long-term rate is $$ y_{t}^{(D)} = r_t + \frac {\mu D}{2} - \frac {\sigma^2 D^2}{6}\tag {59}$$ Note that since the Merton model allows short rates to be negative, the bond price $Z_{t, T}$ in this model goes to infinity as $T \rightarrow \infty$

From equation (59), it follows that the yield curve (the dependence of the long-term rate on duration) can take a normal, inverted, or humped shape. Furthermore, in this model, only parallel shifts of the yield curve are possible. The figure below illustrates an example of such a yield curve shift, showing the curves at $t=0$ and $t=1$

The expression for the instantaneous forward rate in Merton's model is given by $$f_t^{(D)} = r_t + \mu D - \frac{\sigma^2 D^2}{2} \tag {60}$$ Thus, in Merton's model, for any fixed duration $D$, the instantaneous forward rate deviates from the short rate by a constant $\alpha = \mu D - \frac{\sigma^2 D^2}{2}$. Using the results from the section 'Brownian Motion with Drift,' we can derive the mean and variance of the holding period return (HPR): $$\mathbb E \left[HPR_{\tau}^{(D)}\right] = r_0 + \alpha + \,\mu \left( \frac {\tau}{2} - D\right) = \, f_0^{(D)} + \mu \left( \frac {\tau}{2} - D\right) \tag {61}$$ $$\text {Var} \left[ HPR_{\tau}^{(D)}\right] = \sigma^2 \tau \left[ \frac{1}{3} + \left( \frac{D}{\tau}\right)^2 - \frac {D}{\tau} \right] \tag {62}$$

Consequently, the findings from 'Brownian Motion with Drift' also apply to Merton's model, with the key distinction that the expected return over a horizon of twice the duration converges to the portfolio’s initial instantaneous forward rate.

Similarly to the 'Brownian Motion with Drift' model, the difference in expected returns for two fixed duration values $D_1$ and $D_2$ remains constant, regardless of the investment horizon $\tau$: $$\mathbb E \left[HPR_{\tau}^{(D_1)}\right] -\mathbb E \left[HPR_{\tau}^{(D_2)}\right] = \frac{\sigma^2}{2} \left( D_2^2 - D_1^2\right) \tag {63}$$ A strategy with a lower target duration has a higher expected return.

The figure below shows the expected return graphs for two duration values, $D = 1$ and $D = 3$. The excess return is calculated as the difference between the expected return and the initial instantaneous forward rate.

The Ho-Lee model

The drawback of Merton's model is that it does not automatically fit the current term structure of interest rates. In contrast, no-arbitrage models are specifically designed to be consistent with the observed yield curve.

The Ho-Lee model (1986) assumes that the short rate satisfies SDE $$dr_t = \theta_t \, dt + \sigma \, dw_t, \,\,\,\,\,\,\,\, t >0, \tag{64}$$ with a constant parameter $\sigma >0$ and $w_t$ represents a standard Brownian motion under the equivalent martingale measure (EMM). This model is also frequently referred to as the Extended Merton's Model. The function $\theta_t$ can be determined analytically and calibrated using the initial term structure of interest rates [9]:

$$\theta_t = \frac{\partial f_{0, \, t}}{\partial t} + \sigma^2 t, \tag{65}$$ where $\frac{\partial f_{0, \, t}}{\partial t}$ is the partial derivative of the instantaneous forward rate with respect to $t$, as observed today. The variable $\theta_t$ determines the average direction in which $r_t$ moves at time $t$, regardless of its level.

Of course, for the Ho-Lee model to be well-defined, the initial yield curve should be sufficiently smooth, meaning it must be at least twice differentiable. This requirement ensures that the forward rate function is well-behaved and that $\theta_t$ can be computed reliably. In practice, yield curve modeling often relies on appropriate approximation techniques. One such method is the Nelson-Siegel (1987) model, which provides a parsimonious representation of the yield curve [10]. Later, Svensson (1994) extended this approach by introducing additional parameters to improve flexibility and better capture complex term structure dynamics [11].

Integrating equation (64) from $0$ to $t$, we obtain the expression for the short rate: $$r_t = f_{0, \, t} + \frac{\sigma^2 t^2}{2} - \sigma w_t \tag {66}$$ The distribution of $r_t$ is Gaussian, with expectation and variance given by $$ \mathbb E [r_t] = f_{0, \, t} + \frac{\sigma^2 t^2}{2} \tag {67}$$ $$\text{Var} [r_t ] = \sigma^2 t \tag {68}$$

Under the Ho-Lee model, the price of a zero-coupon bond, $Z_{t, \, T}$, is given by [9] $$Z_{t, \, T} = e^{ \, A_{t, \,T} \, - \, r_t (T-t)}, \tag {69}$$ where $$A_{t, \, T} = \text {ln} \frac{Z_{0, \, T}}{Z_{0, \, t}} + f_{0, \, t} (T-t) - \frac{1}{2}\sigma^2 t (T-t)^2 \tag {70}$$ Considering equations (44) and (47), we obtain the expressions for the long-term rate, $y_t^{(D)}$ and the instantaneous forward rate, $f_t^{(D)}$: $$y_t^{(D)} = r_t + f_{0, \, t, \, t+D} - f_{0, \, t} + \frac{1}{2}D \sigma^2 t \tag {71} $$ $$f_t^{(D)} = r_t + f_{0, \, t+D} - f_{0, \, t}+ D \sigma^2 t, \tag {72}$$ where the forward rate for the period between time $t$ and $t+D$ is denoted $f_{0, \, t, \, t+D}$, and the instantaneous forward rates for maturities $t$ and $t+D$ are denoted $f_{0, \, t}$ and $f_{0, \, t +D}$ respectively. All of them are observed today. Although these forward rates depend on time $t$, they are determined based on the information contained in the current yield curve $y_0^{(D)}$.

Note that both the long-term rate and the forward rate are expressed as the sum of the short rate and deterministic functions. The expected value of a deterministic function is the function itself, and its contribution to the variance of a random variable is zero. Therefore, we can use the results from Section 'Brownian Motion with Drift' to obtain expressions for the mean and variance of the HPR in the Ho-Lee model. $$\mathbb E \left[HPR_{\tau}^{(D)}\right] = y_0^{(\tau)} + \frac{\sigma^2}{6} \left( \tau^2 - 3 D^2 \right), \tag {73}$$ where $y_0^{(\tau)}= - \frac{\text{ln }Z_{0, \, \tau}}{\tau}$ is obtained from the initial yield curve as the yield corresponding to the maturity $\tau$ $$\text {Var} \left[HPR_{\tau}^{(D)}\right] = \sigma^2 \tau \left[ \frac{1}{3} + \left( \frac{D}{\tau}\right)^2 - \frac {D}{\tau} \right] \tag {74}$$

The figure below shows the expected return and standard deviation of HPR in the Ho-Lee model for two duration values, $D=1$ and $D=3$. The shape of the current yield curve is given by the function $$y_0^{(D)} = L + (S - L) e^{-\lambda D},$$ where $L = 0.2$ is long-term rate level, $S = 0.15$ is short-term yield at $D=0$, $\lambda = 0.5$ is the decay rate (which controls how quickly the curve approaches the long-term level), and $D$ represents the duration (time to maturity) The excess return is calculated as the difference between the expected return and the initial yield for the horizon $\tau$ .

Surprisingly, in the Ho-Lee model, the expected return does not converge to the initial instantaneous forward rate corresponding to the horizon $D$, as one might intuitively expect. Instead, it approaches the initial yield for the horizon $\tau$ (see details in Appendix E). Notably, this initial yield is exactly attained at the point of the lowest HPR variance, when $\tau =\sqrt{3} D$.

The Vasicek model

In Vasicek’s model (1977), the risk-neutral process for the short rate is $$dr_t = a(b-r_t)dt + \sigma dw_t \tag {75}$$ where $a$, $b$ and $\sigma$ are nonnegative constants, and $w_t$ is a standard Brownian motion under EMM. The short rate is pulled to the long-term mean level $b$ at 'reversion rate' $a$. A normally distributed stochastic term $\sigma dw_t$ is superimposed on this pull.

Integrating equation (75) from $t$ to $u$ ($0 \leq t < u$) we obtain $$r_t = b + (r_s - b) e^{-a(t-s)}+\sigma \int_s^t e^{-a(t-u)} dw_u \tag {76}$$ Thus, the distribution of $r_t$ is normal with mean and variance given by $$\mathbb E \left[ r_t|r_s \right] = b + (r_s - b) e^{-a(t-s)} \tag {77}$$ $$Var \left[ r_t| r_s\right] = \frac{\sigma^2}{2a} \left(1-e^{-2a(t-s)} \right) \tag {78}$$ The price of a zero-coupon bond with unit face value can be shown to be [9] $$Z_{t, \,T} = e^{A_{t, \, T} \, \,- \,r_t \, B_{t, \, T}}\tag {79}$$ where $A_{t, T}$ and $B_{t, T}$ for $a> 0$ are defined as $$B_{t, \, T} = \frac{1-e^{-a(T-t)}}{a} \tag {80}$$ $$ A_{t, \, T} = \frac{\left( B_{t, \, T} - (T-t)\right) \left( a^2b - \frac{\sigma^2}{2}\right)}{a^2} - \frac{\sigma^2 B_{t, \, T}^2}{4a} \tag {81}$$ The long-term rate in this model is given by $$y_t^{(D)} = \frac{1}{D} \left[ r_t \, B_D- \left( B_D - D \right) \left( b - \frac{\sigma^2}{2 a^2}\right) + \frac{\sigma^2}{4a} {B^2_D}\right] \tag {82}$$ where $$B_D = \frac{1-e^{-aD}}{a} \tag {83}$$

As in the Merton model, the yield curve in the Vasicek model can take a normal, humped, or inverted shape. When $a>0$, the curve tends to decline with maturity. The rate of mean reversion, $a$, controls the shape of the curve. A higher value of $a$ causes the influence of short-term rate fluctuations on long-term rates to diminish more quickly.

In contrast to the Merton model, shifts in the yield curve under the Vasicek model are not strictly parallel due to the mean-reversion mechanism of the short rate. As a result, changes in the short rate have a more pronounced effect on short-term yields compared to long-term yields. This leads to non-parallel shifts in the yield curve. Nevertheless, all long-term yields move in the same direction. The steepness of the yield curve depends on both the magnitude and persistence of changes in the short rate.

The expression for the instantaneous forward rate in Vasicek's model is given by [13] $$f_t^{(D)}= r_t \, e^{-aD} + ab B_D - \frac{\sigma^2}{2} B_D^2 \tag {84}$$

Note that in this model, both the long-term and instantaneous forward rates depend linearly on the short rate. Following the approach outlined in the 'Mean Reversion Yield Process' section, we derive the mean and variance of the holding period return (HPR) The mean of HPR is given by (see Appendix F for details): $$\mathbb E\left[ HPR{\tau}^{(D)} \right] = f_0^{(D)} +(r_0 - b) \left( \frac{B_{\tau}}{\tau} - e^{-a D}\right) \tag {85}$$ where $$B_{\tau} = \frac{1-e^{-a \tau}}{a} \tag {86}$$ and $$f_0^{(D)} = r_0 e^{-a D} + ab B_D -\frac{\sigma^2}{2}B_D^2 \tag{87}$$ is the initial instantaneous forward rate.

The variance of HPR is given by: $$\text{Var} \left[ HPR_{\tau}^{(D)}\right] =\frac{\sigma ^2}{a^2 \, \tau^2} \left[ \tau e^{-2aD} + B_{\tau}\left( \frac{1 + e^{-a \tau}}{2} - 2 e^{-aD}\right) \right] \tag {88}$$

The Vasicek yield process (where a flat yield is assumed) and the Vasicek short rate model share several common features. In the Vasicek short rate model, if $aD$ exceeds a certain threshold, the volatility of the holding period return (HPR) decreases monotonically with an increasing investment horizon, without a local minimum. In both models, HPR volatility diminishes over time, approaching zero for sufficiently long horizons, although the rate of this decay differs between the two processes.

However, for the same value of the scaled horizon, HPR volatility in the Vasicek short rate model decreases as the duration increases. This reflects the fact that the influence of the short rate on long-term yields weakens with longer durations — long-term yields are inherently less volatile than the short rate itself.

Moreover, in contrast to the Vasicek yield process, a strategy with a shorter target duration consistently delivers a higher expected return, highlighting a key difference under the Vasicek short rate model.

In the Vasicek short rate model, the expected return converges more slowly to the initial instantaneous forward rate than in the Merton model, where this occurs over a horizon equal to twice the portfolio’s duration. This slower convergence becomes particularly pronounced as the parameters $a$ and $D$ increase, as illustrated in the figure below.

At the same time, the analysis shows that in this model, for certain parameter ratios, the expected return converges to the initial yield over a horizon significantly shorter than twice the duration. For example, the figure below illustrates the expected time to recover the initial yield when the current yield curve has a positive slope.

Thus, it can be concluded that the dynamics of the expected return for constant duration portfolios in models with mean-reverting interest rates can differ significantly from those in models based on random walks. Since mean-reverting models have a strong economic rationale, their further investigation remains of great interest.

Appendix

Let us first make a brief note. Given the well-behaved nature of the yield processes under consideration (where flat yield curve assumed), the expected HPR can be directly computed from equation (6) across all cases: $$\mathbb E [HPR_{\tau}] = \frac {1}{\tau} \int_0^{\tau} \mathbb E [y_s] ds - \frac{D}{\tau}(\mathbb E [y_{\tau}] - y_0).$$ Because the expected yield in the discussed models is well-known, finding the expected return is straightforward. However, for the sake of logical consistency, we will first derive expressions for the HPR in sections (A) - (C), and then determine its expected value and variance.

(A) Brownian Motion with Drift

The HPR in this case is given by $$HPR_{\tau} = y_0\, +\, \mu\cdot \left ( \frac{\tau}{2}\, -\, D \right )\, + \, \frac{\sigma}{\tau} \left ( \int_{0}^{\tau} W_{s}\, d s \, - \, D\cdot W_{\tau} \right ) $$ In this section, we calculate its expected value and variance.

1. Expected Value of HPR

We start by noting that the Wiener process $W_t$ has zero mean, i.e., $\mathbb E [W_t] =0$. Using this property we compute: $$ \mathbb{E} \left[ \frac{\sigma}{\tau} \left ( \int_{0}^{\tau} W_{s}\, d s \, - \, D\cdot W_{\tau} \right ) \right] = \frac{\sigma}{\tau} \int_0^\tau \mathbb{E}[W_s] \, ds \, - \, \frac{D \sigma}{\tau} \mathbb{E}[W_{\tau}] = 0 $$ Thus, the expected value of HPR simplifies to: $$ \mathbb E [HPR_{\tau}] = y_0\, + \, \mu\cdot \left ( \frac{\tau}{2}\, -\, D \right ) $$

2. Variance of HPR

Since the term $ y_0 + \mu\cdot \left ( \frac{\tau}{2}\, -\, D \right )$ contains only deterministic components, its contribution to the variance is zero. Therefore, the variance of HPR is determined by the stochastic terms, and can be expressed as: $$ \mathrm {Var} [ HPR_{\tau}] = \frac {\sigma ^2}{\tau ^2}\left( \mathrm {Var} \left[ \int_{0}^{\tau} W_{s}\, d s \right] + D^2 \, \mathrm {Var}\, \left[W_\tau \right] - 2D \, \mathrm{Cov}\left[\int_{0}^{\tau} W_{s}\, d s \,\, , W_\tau \right] \right) $$

To compute this, we calculate each component separately.

2.1.Variance of the Area Under Brownian Motion Path

Using the stochastic product rule $$ \int_{0}^{\tau} d (s W_{s}) = \int_{0}^{\tau} s \, d W_{s} + \int_{0}^{\tau} W_{s} \, d s $$

and noting that $$ \int_{0}^{\tau} d (s W_{s}) = \tau \, W_{\tau} - 0 \,W_{0}, $$ we can rewrite: $$ \int_{0}^{\tau} W_{s} \, d s = \int_{0}^{\tau} (\tau - s) \, d W_{s} $$ This is a stochastic integral with a deterministic integrand. By applying Ito isometry, we find: $$ \mathrm {Var} \left[\int_{0}^{\tau} W_{s}\, d s \right] = \mathrm {Var} \left[ \int_{0}^{\tau} (\tau - s) \, d W_{s} \right] = \int_{0}^{\tau} (\tau - s)^2 \, d s = \frac {\tau^3}{3}$$

2.2 Variance of $W_{\tau}$

By the properties of the Wiener process: $$\mathrm {Var}\, \left[W_\tau \right] = \tau$$

2.3. Covariance Between $ \int_0^\tau W_s \, \mathrm d s $ and $W_{\tau}$

Using the result for the area under the Brownian motion path and applying Ito isometry, we compute: \[ \text{Cov} \left( \int_0^\tau W_s \, d s, W_{\tau} \right) = \mathbb{E} \left[ \int_0^\tau W_s \, d s \cdot W_{\tau} \right] = \mathbb{E} \left[\int_{0}^{\tau} (\tau - s) \, d W_{s} \cdot \int_{0}^{\tau} \, d W_{s} \right] = \int_0^\tau (\tau - s) \, ds = \frac{\tau^2}{2} \]

Substituting all components into the variance formula, we obtain: $$ \mathrm {Var} [ HPR_{\tau}] = \sigma^2 \tau \left( \frac{1}{3} + \left( \frac{D}{\tau}\right)^2 - \frac{D}{\tau} \right)$$

(B) Geometric Brownian Motion

1. Calculation of HPR

Integration (22) give:

$$ \int_{0}^{\tau} \, d y_{s} = \int_{0}^{\tau} \, \mu \, y_s \, d s + \int_{0}^{\tau} \sigma \, y_s \, d W_s $$ which we can rewrite:

$$ y_{\tau} - y_0 = \mu \int_{0}^{\tau} \, y_s \, d s + \sigma \int_{0}^{\tau} \, y_s \, d W_s $$ Using (6), a series of algebraic rearrangements yields: $$ HPR_{\tau} = \frac{1}{\tau} \left [ \left( \frac {1}{\mu}- D\right) (y_{\tau} - y_0) - \frac{\sigma}{\mu } \int_{0}^{\tau} \, y_s \, d W_s \right] $$ where $y_t$ determined from (23)

2. Expected Value of HPR

By properties of the stochastic integral, $$ \mathbb E \left[ \int_{0}^{\tau} \, y_s \, d W_s \right] = 0$$ Therefore, using (23): $$ \mathbb E [HPR_{\tau} ] = \mathbb E \left[ \frac{1}{\tau} \left( \frac {1}{\mu}- D\right) (y_{\tau} - y_0) \right] = \frac{y_0}{\tau} \left( \frac {1}{\mu}- D\right) \mathbb E \left[ e^{\left( \mu \,\, - \, \frac{\sigma^2}{2} \right) \tau \, \, + \, \sigma \, W_{\tau}} \, -1 \right] $$ Accounting for $$\mathbb E \left[ e^{\sigma W_{\tau}}\right] = e^{\frac{\sigma^2 \tau}{2}}$$ finally we have $$\mathbb E [HPR_{\tau}] = \frac {y_0}{\tau} \left( e^{\mu \tau} - 1\right) \left( \frac {1}{\mu} - D \right)$$

3. Variance of HPR

As the expression containing the factor $y_0$ is entirely deterministic, its contribution to the variance or covariance is zero.

3.1 Calculating $ I_1 = \text {Var} \left[ \frac{1}{\tau} \left( \frac{1}{\mu} -D\right) y_{\tau} \right]$

Taking (25) into account we have:

$$ I_1 = \frac{1}{\tau ^2} \left( \frac{1}{\mu} -D\right)^2 \text{Var}[y_{\tau}] = \frac{y_0^2}{\tau ^2} \left( \frac{1}{\mu} - D\right)^2 e^{2 \mu \tau } \left(e^{\sigma^2 \tau} \, - 1 \right) $$

3.2 Calculating $ I_2 = \text {Var} \left[ \frac{\sigma}{\mu \tau} \int_{0}^{\tau} \, y_s \, d W_s \right] $

Using Ito isometry we have: $$ I_2 = \frac{\sigma^2}{\mu^2 \tau^2} \text{Var} \left[ \int_{0}^{\tau} \, y_s \, d W_s \right] =\frac{\sigma^2}{\mu^2 \tau^2} \int_{0}^{\tau} \mathbb E \left[ \, y_s^2 \, \right] d s $$ and taking into account that $$\mathbb E \left[y_s^2\right] = \text {Var} [y_s] + (\mathbb E [y_s])^2 = y_0^2 \, e^{(2 \mu \,+\, \sigma^2) s}$$ finally we have: $$ I_2 = \frac{y_0^2 \sigma^2}{\mu^2 \tau ^2 (2\mu + \sigma^2)} (e^{(2\mu + \sigma^2) \tau} \, - 1)$$

3.3 Calculating $ I_3 = 2 \, \text {Cov} \left[\frac{1}{\tau} \left( \frac{1}{\mu} -D\right) y_{\tau}, \, - \frac{\sigma}{\mu \tau} \int_{0}^{\tau} \, y_s \, d W_s \right] $

Note that

$$M_t = e^{\sigma W_t \, - \, \frac{\sigma^2}{2} t}$$ is a martingale that satsifes the SDE $$ dM_t = \sigma M_t \, d W_t$$ Therefore $$M_T = 1 + \sigma \int_0^T M_s \, dW_s$$ Then the calculation of original expectation is equivalent to calculating the following expression:

$$y_0^2 \, \sigma \, e^{\mu \, T} \, \mathbb E \left [\int_0^T M_s dW_s \int_0^T e^{ \left(\mu \, - \, \frac {\sigma^2}{2} \right)s} e^{\sigma W_s} dW_s \right]$$ Using Ito isometry it reduses to $$y_0^2 \, \sigma \, e^{\mu T} \int_0^T e^{(\mu -\sigma^2) s} \, \mathbb E \left[ e^{2 \sigma W_s} \right] ds$$ Using the fact $$\mathbb E \left [e^ {\alpha W_s} \right] = e^ {\frac {\alpha ^2 t}{2}}$$ finally we have $$ \mathbb E \left[ y_T \cdot \int_0^T y_s \, dW_s\right] = \sigma y_0^2 e^{\mu T} \, \cdot \frac{e^{\left( \mu + \sigma^2\right)T}-1}{\mu + \sigma^2}$$

$$ I_3 = -2 \frac{\sigma^2 y_0^2}{\mu \tau^2} \left( \frac {1}{\mu} - D\right) e^{\mu \tau} \cdot \frac{e^{(\mu + \sigma^2) \tau} -1}{\mu + \sigma^2}$$ Then variance of HPR is: $$ \mathrm {Var} [HPR_{\tau}] = \frac {y_0^2}{\tau^2} \left[ \left( \frac{1}{\mu} -D\right) ^2 e^{2 \mu \tau} \left( e^{\sigma^2 \tau} -1\right) + \frac{\sigma^2}{\mu^2 (2 \mu + \sigma^2)} \left( e^{(2 \mu + \sigma^2) \tau} -1 \right) \right] -$$ $$ - \frac {y_0^2}{\tau^2}\left[\frac{2 \sigma^2}{\mu (\mu + \sigma^2)} \left( \frac {1}{\mu} - D\right) e^{\mu \tau} \left( e^{(\mu + \sigma^2) \tau} -1 \right) \right]$$

(C) Vasicek Yield Process

1. Calculation of HPR

Substituting the expression for $y_t$ from (30) into equation (6) yields

$$HPR_{\tau} = \frac {1}{\tau} \left[\int_0^{\tau} y_0 e^{-\theta s} + \mu \left( 1 - e^{-\theta s}\right) \, ds+ \sigma \int_0^{\tau} ds \int_0^s e^{-\theta(s-u)} \, d W_u\right]-$$ $$- \frac{D}{\tau} \left[ y_0 e^{-\theta \tau} + \mu\left(1- e^{-\theta \tau}\right) + \sigma \int_0^{\tau} e^{-\theta(\tau - u)} dW_u\right]$$ Having integrated the first deterministic term, by grouping the remaining deterministic ones, we obtain: $$ HPR_{\tau} = \mu \, + \, \frac{1}{\tau } (y_0 \, - \, \mu) \left ( 1+ \theta D \right ) \left( \frac{1-e^{-\theta \tau}}{\theta} \right ) \, + \frac {\sigma}{\tau} \int_0^{\tau} ds \int_0^s e^{-\theta(s-u)} \, d W_u \, \,\,-$$ $$- \frac{D \sigma}{\tau} \int_0^{\tau} e^{-\theta (\tau - u)} d W_u $$ Using Fubini's theorem for the stochastic part and considering the Wiener process $W_u$ as the unique source of uncertainty, we obtain: $$\frac {\sigma}{\tau} \int_0^{\tau} ds \int_0^s e^{-\theta(s-u)} \, d W_u - \frac{D \sigma}{\tau} \int_0^{\tau} e^{-\theta (\tau - u)} d W_u = $$$$ = \frac {\sigma}{\tau} \int_0^{\tau} d W_u \int_u^{\tau} e^{-\theta(s-u)} \, d s - \frac{D \sigma}{\tau} \int_0^{\tau} e^{-\theta (\tau - u)} d W_u =$$$$=\frac {\sigma}{\tau} \int_0^{\tau} \frac {1- e^{-\theta(\tau - u)}}{\theta} d W_u - \frac{D \sigma}{\tau} \int_0^{\tau} e^{-\theta (\tau - u)} d W_u = \frac{\sigma} {\tau}\int_{0}^{\tau}\frac{1- (1+ \theta D)e^{-\theta (\tau-u)} }{\theta} dW_u$$ Gathering stochastic and deterministic terms, we obtain the final expression for HPR: $$ HPR_{\tau} = \mu \, + \, \frac{1}{\tau } (y_0 \, - \, \mu) \left ( 1+ \theta D \right ) \left( \frac{1-e^{-\theta \tau}}{\theta} \right ) \, + \, \frac{\sigma} {\tau}\int_{0}^{\tau}\frac{1- (1+ \theta D)e^{-\theta (\tau-u)} }{\theta} dW_u $$

2. Expected Value of HPR

By the properties of the stochastic integral, the expression $$ \mathbb E \left[ \frac{\sigma} {\tau}\int_{0}^{\tau}\frac{1- (1+ \theta D)e^{-\theta (\tau-u)} }{\theta} dW_u \right] = 0; $$ therefore the expected value is equal to the deterministic part of HPR: $$ \mathbb E [HPR_{\tau}] = \mu \, +\, \frac{1}{\tau } (y_0 \, - \, \mu) \left ( 1+ \theta D \right ) \left( \frac{1-e^{-\theta \tau}}{\theta} \right )$$

3. Variance of HPR

Since the deterministic part makes a zero contribution to the variance, we obtain $$\text{Var} [HPR_{\tau}] = \text{Var} \left[ \frac{\sigma} {\tau}\int_{0}^{\tau}\frac{1- (1+ \theta D)e^{-\theta (\tau-u)} }{\theta} dW_u \right] $$ Using Ito isometry we have $$\text{Var} \left[ \frac{\sigma} {\tau}\int_{0}^{\tau}\frac{1- (1+ \theta D)e^{-\theta (\tau-u)} }{\theta} dW_u \right] = \frac {\sigma^2}{\theta^2 \tau^2 } \int _{0}^{\tau} \left[ 1 - (1+ \theta D) e^{-\theta(\tau - u)}\right]^2 d u$$ After integrating we get the variance of HPR: $$ \mathrm {Var} [HPR_{\tau}] = \frac{\sigma^2}{\theta^2 \tau^2} \left [ \tau + (1 + \theta D) \left[ (1+ \theta D) \frac{ 1-e^{- 2 \theta \tau}}{2 \theta} - 2\cdot \frac{1- e^{-\theta \tau}}{\theta} \right ] \right ] $$

(D) The Merton model

Integrating expression for the short-rate from $t$ to $T$, we get: $$\int_t^T r_u \, du = r_t(T-t) + \frac{\mu}{2}(T-t)^2 + \sigma \int_t^T \left( \int _t^u dw_s\right) du $$ Applying Fubini’s theorem, we swap the order of integration in the last term and compute the inner integral, obtaining: $$\int_t^T \left( \int _t^u dw_s\right) du = \int_t^T \left( \int_s^T \, du\right) dw_s = \int_t^T (T-s) \, dw_s$$ Finally, we arrive at the expression: $$\int_t^T r_u \, du = r_t(T-t) + \frac{\mu}{2}(T-t)^2 + \sigma \int_t^T (T-s) \, dw_s$$ It follows that $\int_t^T r_u \, du$ is normally distributed.

Using properties of the stochastic integral and applying Ito isometry, we compute its conditional mean and variance: $$\mathbb E \left[\int_t^T r_u \, du \, | \, r_t \right] = r_t(T-t) + \frac{\mu}{2}(T-t)^2 $$$$\text {Var} \left[\int_t^T r_u \, du \, | \, r_t \right] = \sigma^2 \int_t^T (T-s)^2 \, ds = \frac {\sigma^2 (T-t)^3}{3} $$ Using the fact that for a normally distributed random variable $X$, the following equality holds $$ \mathbb E \left[e^{\phi X}\right] = e^{\phi \mathbb E[X] + \frac{\phi^2}{2} \text{Var} [X]} $$ we substitute the conditional mean and variance of $\int_t^T r_u \, du$, obtaining $$Z_{t, T} = \mathbb E^Q_t \left[ e^{- \int_t^T \, r_s ds}\right] = e^{-r_t(T-t) \, - \, \frac{ \mu (T-t)^2} {2}\, + \, \frac{\sigma^2 (T-t)^3}{6}}$$

(E) The Ho-Lee model

Expected Value of HPR

The expected value of HPR can be derived directly from equation (53): $$\mathbb E \left[HPR_{\tau}^{(D)}\right] = \frac {1}{\tau} \int_0^{\tau} \mathbb E \left[f^{(D)}_s \right] ds - \frac{D}{\tau}\left( \mathbb E \left[y_{\tau}^{(D)}\right] - y_0^{(D)}\right).$$ Substituting equations (71) and (72) and using (67), followed by algebraic manipulations, we obtain: $$\mathbb E \left[HPR_{\tau}^{(D)} \right] = \frac{1}{\tau}\int_0^{\tau} f_{0, \, s+D} \, \, ds - \frac {D}{\tau} \left( f_{0, \, \tau, \, \tau + D} - f_{0, \, 0, \, D}\right)+\frac{\sigma^2}{6} \left( \tau^2 - 3 D^2 \right) $$ Utilizing the property of the forward rate $$f_{t, \, t', \, T} = \frac{1}{T - t'} \int_{t'}^T f_{t, \, s} \,ds,$$ we obtain: $$ \frac{1} {\tau}\int_0^{\tau} f_{0, \, s+D} \, \, ds = \frac{1}{\tau}\int_D^{\tau + D} f_{0, \, u} \, du=f_{0, \, D, \, \tau + D }$$ Next, all forward rates can be expressed in terms of zero-coupon bond prices using equation (45): $$f_{0, \, \tau, \, \tau +D} = -\frac{\text{ln} \, Z_{0, \, \tau +D} - \text{ln} \, Z_{0, \, \tau }}{D}$$$$f_{0, \, D, \, \tau +D} = -\frac{\text{ln} \, Z_{0, \, \tau +D} - \text{ln} \, Z_{0, \, D }}{\tau}$$ $$f_{0, \, 0, \, D} = -\frac{\text{ln} \, Z_{0, \, D} }{D} $$ Thus, $$f_{0, \, D, \, \tau +D}- \frac {D}{\tau} \left( f_{0, \, \tau, \, \tau + D} - f_{0, \, 0, \, D}\right) = $$$$=-\frac{\text{ln} \, Z_{0, \, \tau +D} - \text{ln} \, Z_{0, \, D }}{\tau} +\frac{\text{ln} \, Z_{0, \, \tau +D} - \text{ln} \, Z_{0, \, \tau }}{\tau} - \frac{ \text{ln} \, Z_{0, \, D }}{\tau} =$$$$ = - \frac{ \text{ln} \, Z_{0, \, \tau }}{\tau} = y_0^{(\tau)}$$ Finally, we obtain $$\mathbb E \left[HPR_{\tau}^{(D)}\right] = y_0^{(\tau)} + \frac{\sigma^2}{6} \left( \tau^2 - 3 D^2 \right) $$

(F) The Vasicek Short Rate model

1. Expected Value of HPR

The expected value of holding period return can be derived directly from equation (53): $$\mathbb E \left[HPR_{\tau}^{(D)}\right] = \frac {1}{\tau} \int_0^{\tau} \mathbb E \left[f^{(D)}_s \right] ds - \frac{D}{\tau}\left( \mathbb E \left[y_{\tau}^{(D)}\right] - y_0^{(D)}\right).$$ Substituting equations (82) and (84) and using (77), followed by algebraic manipulations, we obtain: $$\mathbb E \left[HPR_{\tau}^{(D)}\right] = be^{-aD} + \frac{1}{\tau}(r_0 - b) B_{\tau}e^{-aD} + \beta + \frac{1}{\tau} B_D(r_0 - b)(1-e^{-a \tau}) $$ Rearranging the terms, this simplifies to: $$\mathbb E \left[HPR_{\tau}^{(D)}\right]= b e^{-aD} + \beta + \frac{1}{\tau} (r_0 - b)B_{\tau},$$ where $$\beta = abB_D - \frac{\sigma ^2}{2}B_D^2, \qquad B_{\xi} = \frac{1- e^{-a \xi}}{a}$$ Finally, using equation (84) for $t=0$, we obtain: $$\mathbb E \left[HPR_{\tau}^{(D)}\right] =f_0 + (r_0 - b)\left( \frac{B_{\tau}}{\tau} - e^{-aD}\right)$$

2. Variance of HPR

Since the deterministic components in the HPR calculation have no impact on its variance, we can write, using equations (82) and (84): $$\text{Var} \left[ HPR_{\tau}^{(D)}\right] = \text{Var} \left[ \frac{1}{\tau}\int_0^{\tau} r_u e^{-aD} \, du - \frac{B_D}{\tau} r_{\tau}\right] $$ Given the expression for the short rate (76) and using the approach employed in Appendix (C), we reduce the problem to finding the variance in the form: $$ \text{Var} \left[ HPR_{\tau}^{(D)}\right] = \text{Var} \left[ \frac{\sigma}{a \tau} \int_0^{\tau} e^{-aD} - e^{-a(\tau - u)} \, dw_u \right]$$ Applying Itô’s isometry and evaluating the integral, we obtain: $$\text{Var} \left[ HPR_{\tau}^{(D)}\right] = \frac{\sigma ^2}{a^2 \, \tau^2}e^{-2 aD} \left( \tau + \frac {\left( e^{aD} - e^{a(D-\tau)} \right) \left( e^{aD} + e^{a(D-\tau )} -4\right)}{2a} \right)$$ Alternatively, it can be expressed as$$\text{Var} \left[ HPR_{\tau}^{(D)}\right] =\frac{\sigma ^2}{a^2 \, \tau^2} \left[ \tau e^{-2aD} + B_{\tau}\left( \frac{1 + e^{-a \tau}}{2} - 2 e^{-aD}\right) \right] $$

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