<p>The approximation given is off at a key point - it is the duration of the bond at 9 years maturity, not the original duration.</p>

<p>The approximation works in two steps.</p>

<ol>
<li>If the yield is unchanged over one year, the total return over the year is roughly equal to the starting yield.</li>
<li>If the yield changes, we add a capital gain/loss which is equal to the change in yields times the yield sensitivity. This is the modified duration of the bond, at 9 years maturity.</li>
</ol>

<p>This approximation ignores the reinvestment of any coupon income, and the convexity effect (the return sensitivity changes slightly as yields change). Also, the bond yield convention is probably different than the convention used for total return. That said, for yield changes under 50 basis points, the approximation is typically pretty good.</p>

CC BY-SA 3.0
2017-04-15T14:26:11.940Z
2017-04-15T14:26:11.940Z
Brian Romanchuk
<p>I came across the following statement regarding the realized 10-year maturity bond's return over a year: </p>

<blockquote>
  <p>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:
  $$H_{10} \approx Y_{10}-\text{Duration}_{10} \times \Delta Y_{10}.$$</p>
</blockquote>

<p>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. </p>

<p>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:
$$V_0=\frac{C}{1+y_0}+\frac{C}{(1+y_0)^2}+\dots+\frac{C}{(1+y_0)^9}+\frac{F+C}{(1+y_0)^{10}}.$$</p>

<p>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 $t=0$.</p>

<p>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!</p>

CC BY-SA 3.0
2017-03-28T00:15:30.450Z
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
<bonds>
Making mathematical sense of the expression for realized bond return
Milan
# Making mathematical sense of the expression for realized bond return

(Asked by: username_0 on 2017-03-26)

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!

**Tags:** bonds
> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2017-04-15. Score: 1 | Accepted answer

The approximation given is off at a key point - it is the duration of the bond at 9 years maturity, not the original duration.

The approximation works in two steps.

1. If the yield is unchanged over one year, the total return over the year is roughly equal to the starting yield.
2. If the yield changes, we add a capital gain/loss which is equal to the change in yields times the yield sensitivity. This is the modified duration of the bond, at 9 years maturity.

This approximation ignores the reinvestment of any coupon income, and the convexity effect (the return sensitivity changes slightly as yields change). Also, the bond yield convention is probably different than the convention used for total return. That said, for yield changes under 50 basis points, the approximation is typically pretty good.


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