Can you tell from the following information which of the following three bonds will have the greatest price volatility, assuming that each is trading to offer the same yield to maturity?

Question

What will be an ideal response?

Answer 1

Bond Coupon Rate (%) Maturity (years)
X 8 9
Y 10 11
Z 11 12

The price of a bond will change over time as a result of a change in the perceived credit risk of the issuer. Thus, if one of the three bonds undergoes greater change in credit risk then that bond might be expected to experience more volatility unless other factors dominate. Below we describe these factors: term to maturity, coupon rate, and yield to maturity.

For a given term to maturity and initial yield, the price volatility of a bond will increase as the coupon rate becomes smaller. Thus, ceteris paribus, we would expect bond X to have greater price volatility that bond Y, and bond Y to have greater price volatility than bond Z. However, the differences in coupon rates for bonds Y and Z are not that great. Thus, if just looking at these two bonds, the differences in price volatilities may not be that recognizable.

For a given coupon rate and initial yield, longer terms to maturity will produce greater price volatility. Thus, ceteris paribus, bond Z will have more price volatility than bond Y and bond Y will have more price volatility than bond X. However, the differences in maturities for bonds Y and Z are not that great. Thus, if just comparing these two bonds, the differences in price volatilities may not be that identifiable.

We should note that the expectations of price volatility based upon coupon rates and maturities are the reverse for bonds X and Z. For example, we expect bond X to have the greatest price volatility based upon coupon rate but the lowest based upon maturity. For bond Z, we expect it to have lowest price volatility based upon coupon rate but the greatest based upon maturity. If the coupon rate and the maturity factors or characteristics balance out then it is possible all three bonds will experience price volatilities that are very similar.

[NOTE. An implication of the maturity factor is that investors who want to increase a portfolio's price volatility because they expect interest rates to fall, all other factors being constant, should hold bonds with long maturities in the portfolio. To reduce a portfolio's price volatility in anticipation of a rise in interest rates, bonds with shorter-term maturities should be held in the portfolio.]

Although the yield to maturity is held constant for bonds X, Y, and Z, the yield to maturity can also play a role as a factor impacting a bond's price volatility. Ceteris paribus, the higher the yield to maturity at which a bond trades, then the lower the price volatility should be.

In addition to the above factors, we have to keep in mind four important properties concerning the price volatility of an option-free bond that result from the convex shape of the price-yield relationship.

First, although the prices of all option-free bonds move in the opposite direction from the change in yield required, the percentage price change (e.g., price volatility) depends on the convexity relationship between price and yield for each of the three bonds. Thus, for bonds X, Y, and Z, we will not expect the same price volatility due to likely differences in convexity. Whether the market yield rises or falls, the bond with the greatest convexity will achieve a higher price. That is, if the required yield rises, the capital loss for this bond will be less while a fall in the required yield will generate greater price appreciation.

Second, for very small changes in the yield required, the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases. Thus, for bonds X, Y, and Z if the percentage price change is very small, we will not likely detect which bond has the greatest price volatility.

Third, for large changes in the required yield, the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield. Thus, whichever bond or bonds change, the price volatility will depend on the direction of the change.

Fourth, for a given large change in basis points, the percentage price increase is greater than the percentage price decrease. Thus, whichever bond or bonds change, the price volatility will be relatively greater if there is a percentage price increase as opposed to a decrease. The implication of this fourth property is that if an investor owns a bond, the price appreciation that will be realized (if the required yield decreases) is greater than the capital loss that will be realized if the required yield rises by the same number of basis points. For an investor who is "short" a bond, the reverse is true: the potential capital loss is greater than the potential capital gain if the required yield changes by a given number of basis points.