# Making mathematical sense of the expression for realized bond return

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!

• The expression you try to understand is very vaguely and badly expressed. For example, the "yield income" is nothing but the coupon, because the holder of the bond gets the coupon and nothing else at the end of the year. And there is no change in the coupon, as the $\Delta Y_{10}$ symbol appears to suggest. – Alecos Papadopoulos Mar 28 '17 at 0:15