Making mathematical sense of the expression for realized bond return

Question

I came across the following statement regarding the realized 10-year maturity bond's return over a year:

The realized bond return (H) over a year has two components: the yield income earned over time and the capital gain or loss due to yield changes: $$H_{10} \approx Y_{10}-\text{Duration}_{10} \times \Delta Y_{10}.$$

I am a complete economics rookie and I'm trying to understand what's going on here from the mathematical point of view, which I'm gonna present here, but my calculations don't seem to add up.

If we denote the bond's coupon with $C$, and the bond's time $t=0$ yield to maturity with $y_0$, then the bond's value at time $t=0$ equals: $$V_0=\frac{C}{1+y_0}+\frac{C}{(1+y_0)^2}+\dots+\frac{C}{(1+y_0)^9}+\frac{F+C}{(1+y_0)^{10}}.$$

At time $t=1$, we can express the bond's value as the function of the time $t=1$ yield to maturity $y$, so we have $$V_1(y)=\frac{C}{1+y}+\frac{C}{(1+y)^2}+\dots+\frac{C}{(1+y)^8}+\frac{F+C}{(1+y)^{9}}.$$ Derivative of $V_1$ with respect to $y$ is equal to: $$\frac{dV_1}{dy}=-1\cdot\frac{C}{(1+y)^2}-2\cdot\frac{C}{(1+y)^3}-\dots-8 \cdot \frac{C}{(1+y)^9}-9 \cdot\frac{F+C}{(1+y)^{10}} .$$ Now, we can apply some basic calculus here and state that for $\Delta y$ small "enough", we have that $$ V_1(y_0+\Delta y)\approx V_1(y_0)+\frac{d V_1}{dy}(y_0)\cdot \Delta y. $$ So now, if we consider the absolute return on our position (buying this bond at time $t=0$, selling it at $t=1$) from the time $t=0$ perspective, under the assumption that the time $t=1$ bond's yield to maturity is $y_1=y_0+\Delta y$, we have that: $$\text{AbsReturn} \approx-V_0+\frac{C}{1+y_0}+\frac{V_1(y_0)+\frac{d V_1}{dy}(y_0)\cdot \Delta y}{1+y_0}.$$ That is - we buy the bond for $V_0$, at the end of the first year we are paid the coupon which discounted value is $\frac{C}{1+y_0}$, and the approximation of the time $t=1$ bond's value taking into the account the YTM change is $V_1(y_0)+\frac{d V_1}{dy}(y_0)\cdot \Delta y$ and we also discount it to time $t=0$.

Now, we can simplify the expression for AbsReturn since $-V_0+\frac{C}{1+y_0}+\frac{V_1(y_0)}{1+y_0}=0$ and we get: $$ \text{AbsReturn}= \frac{\frac{d V_1}{dy}(y_0)\cdot \Delta y}{1+y_0} ,$$ which I guess we can also divide with our initial investment of $V_0$ to get the rate of return so we get: $$ \text{RateOfReturn}= \frac{\frac{d V_1}{dy}(y_0)\cdot \Delta y}{V_0(1+y_0)} ,$$ aaand this is where I completely lose it. I can't seem to understand the connection between the original expression and the thing I end up with. What does the term $\text{Duration}_{10}$ in the original formula even stand for - I guess it is the derivation of bond's value with respect to yield - but bond's value at what time: $t=0$ or $t=1$? Does it even make any difference? If it is at time $t=0$, how can we be using linear approximation of that function for approximating bond's value change at time $t=1$? I'm completely puzzled over this. Am I doing something completely wrong in this derivation? I appreciate any insights on this. Thanks!

Answer 1

Let's go back to basics. In terms of its yield $y$, the price of a bond maturing in $n$ years is

$$ P_n(y) = \sum_{i=1}^n\frac{c}{(1+y)^i} + \frac{100}{(1+y)^n} $$

One year later, the yield is now $y^*$ and the bond now matures in $(n-1)$ years, and its price is

$$ P_{n-1}(y^*) = \sum_{i=1}^{n-1}\frac{c}{(1+y^*)^i} + \frac{100}{(1+y^*)^{n-1}} $$

We can write $y^* = y + \Delta y$ and expand in powers of $\Delta y$ to get

$$ P_{n-1}(y^*) \approx P_{n-1}(y) + \frac{\partial P_{n-1}}{\partial y}\Delta y + O(\Delta y^2) $$

The duration of this bond is defined to be

$$ -D_{n-1}(y) = \frac{1}{P_{n-1}(y)}\frac{\partial P_{n-1}}{\partial y} $$

The total return of the bond, including the cashflow of $c$ received, is therefore

$$ R = \frac{c + P_{n-1}(y) - D_{n-1}(y)P_{n-1}(y)\Delta y - P_n(y)}{P_n(y)} + O(\Delta y^2) $$

Noting that

$$ c + P_{n-1}(y) = (1+y)P_{n}(y) $$

we can write the total return as

$$ R = y - \frac{P_{n-1}(y)}{P_n(y)}D_{n-1}(y) \Delta y + O(\Delta y)^2 $$

Note that the only approximation that has been made is in approximating the price of the bond $P_{n-1}(y + \Delta y)$ as a linear function of $\Delta y$. Other than that, we have used exact formulas throughout. The expression that you often see, and that you quoted in your question,

$$ R = y - D_n(y) \Delta y + O(\Delta y^2) $$

is incorrect when we consider returns over finite time periods - it is only valid for an instantaneous change in yields. In particular, as you found out, it obscures the choice of which duration to use - the correct answer is neither the duration $D_n(y)$ nor the forward duration $D_{n-1}(y)$, but instead the forward duration adjusted by the ratio of bond prices assuming an unchanged yield.

Answer 2

I don't agree with your expression for AbsReturn. It contains two terms which have been divided by (1+y0). Why have these been discounted? The AbsReturn is just value(t=1) - value(t=0) with no discounting. I believe that correction explains your dilemma.