Yield Volatility: Duration and Convexity as Approximated Price Change
Many models of interest rate risk assume a parallel shift in the yield curve. This is not realistic because all bonds and portfolios have unique characteristics that influence yield volatility.
An analyst would not expect the stock of two large technology firms to have the same volatility over an investment horizon. In a similar manner, you would not expect 30-year New York City general obligation bonds and two-year U.S. Treasury notes to have the same volatility.
Which bond would have the greatest price change if a parallel change in yield curves was assumed?
Correct!
Bond 1 would have the greatest price change because a parallel shift in the yield curve assumes that both bonds would have the same yield change. Bond 1 has a higher duration.
Incorrect.
Bond 2 would not have a greater price change if the yield curve had a parallel shift.
In order to analyze the risk of this bond, the investor looks at the duration of the bond and the yield volatility. Which of the bonds has the most risk based on duration alone?
Correct!
Bond 1 has a higher modified duration. Based only on modified duration, it has more risk.
Incorrect.
Bond 2 does not have the most risk based on duration.
Incorrect.
The risk of the two bonds is different.
The investor can estimate the expected change in the bond price for each bond based on duration and convexity. For bond 1, the expected percentage change in price for a 15-basis-point change in yield is equal to:
$$\displaystyle \% \Delta PV^{Full} \approx -(\text{AnnModDur} \times \Delta \text{Yield}) + \left[ \frac{1}{2} \times \text{AnnConvexity} \times (\Delta \text{Yield})^2 \right] $$
In this case:
$$\displaystyle \% \Delta PV^{Full} \approx (-13.838)(0.0015)+\frac{1}{2}(296.51)(0.0015)^2 \approx -0.02042$$
and the change in price per 100 dollars in value is equal to:
$$\displaystyle 100 \times -0.02042 = -2.042$$
For bond 2, the expected percentage change in price for a 30-basis-point change in yield is equal to:
$$\displaystyle \% \Delta PV^{Full} \approx (-10.315)(0.0030) + \frac{1}{2}(136.65)(0.003)^2 \approx -0.03033$$
and the change in price per 100 dollars in value is equal to:
$$\displaystyle 92.056 \times -0.03033 = -2.792$$
Incorrect.
A parallel shift means the yields on both bonds change by the same amount. Since they have different durations, the price changes would be different.
To summarize:
[[summary]]
For the two bonds, the risk summary is as follows:
| Bond | Maturity | Price | Duration | Convexity | Yield Volatility | Price Change |
|---|---|---|---|---|---|---|
| 1 | 30 years | $100 | 13.838 | 296.51 | 15 basis points | -2.042 |
| 2 | 15 years | $92.056 | 10.315 | 136.65 | 30 basis points | -2.792 |
Notice that bond 2 has a higher expected price change even though the modified duration is lower. That is because it has more yield volatility.
This shows how important it is for the investor to consider yield volatility when comparing bonds.
Suppose an investor has two bonds and is interested in comparing the risk of the two over the next six months. The two bonds have the following risk characteristics:
| Bond | Maturity | Price (of 100 par) | Duration | Convexity | Yield Volatility |
|---|---|---|---|---|---|
| 1 | 30 years | 100 | 13.838 | 296.51 | 15 basis points |
| 2 | 15 years | 92.056 | 10.315 | 136.65 | 30 basis points |
Bond 1
Bond 2
Both bonds would change the same
Bond 1
Bond 2
The risk is the same
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