Learning objectives: Explain the role of interest rate expectations in determining the shape of the term structure. Apply a risk-neutral interest rate tree to assess the effect of volatility on the shape of the term structure. Estimate the convexity effect using Jensen’s inequality. Evaluate the impact of changes in maturity, yield, and volatility on the convexity of a security. Calculate the price and return of a zero-coupon bond incorporating a risk premium.
Questions:
23.3.1. Below (in the left-hand panel) is a non-granular, dramatic interest rate tree: it is the true process of the one-year spot rate that starts at 9.0% and jumps up/down by 300 basis points over the next two years. It assumes that next year the one-year spot rate will be either 6.0% or 12.0%. The right-hand panel is the associated price tree (aka discount factors) implied by the interest rate tree; for example, 0.89260 = 1/(1+12%) under an assumption of annual compound frequency.
For the two-year spot rate, which of the following is nearest to the value of convexity?
a. Negative 167.0 basis points
b. Zero
c. 4.13 basis points
d. 300.0 basis points
23.3.2. Below (in the left-hand panel) is an interest rate tree for the one-year rate where the initial rate of 8.0% jumps 300 basis points to either 11.0% or 5.0% next year. In two years, the one-year rate will be either 2.0%, 8.0%, or 14.0%.
Next, we will assume a three-year zero-coupon bond. Without a risk premium, for example, the date 0, state 2 price would be 0.877193 = 1/(1+14%). However, we will instead assume a risk premium. Specifically, investors require compensation of 50 basis points for each year of duration risk. The right-hand panel displays the associated risk-neutral (i.e., , assumes the risk premium) price tree for this three-year zero-coupon bond
In regard to the three-year zero-coupon bond, which of the following is true?
a. The expected return over the next year is 7.50%
b. The expected return over the next year is 9.00%
c. The implied 3-year spot rate term structure is downward sloping (aka, inverted)
d. Jensen's inequality implies that expected excess return over two years is less than 50 basis points
23.3.3. The shape of the term structure is summarized by the determinants of the forward rate (Tuckman 8.28, Third Edition):(†)
If we accept this as a valid model of the forward rate term structure, then each of the following statements is true EXCEPT which is false?
a. The forward rate is composed of the instantaneous rate plus the expected change in the rate multiplied by the duration of the zero-coupon bond corresponding to the term of the forward rate
b. The forward rate increases with the risk premium in proportion to the corresponding duration
c. The forward rate is reduced by greater interest rate volatility and the corresponding zero-coupon volatility
d. If lambda, λ, equals zero, the implied future scenario is Unchanged Term Structure; but if E[dr/dt] equals zero, then the future scenario is Realized Forwards
Answers here:
(†) Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today’s Markets, 3rd Edition (New York: Wiley, 2011)
Questions:
23.3.1. Below (in the left-hand panel) is a non-granular, dramatic interest rate tree: it is the true process of the one-year spot rate that starts at 9.0% and jumps up/down by 300 basis points over the next two years. It assumes that next year the one-year spot rate will be either 6.0% or 12.0%. The right-hand panel is the associated price tree (aka discount factors) implied by the interest rate tree; for example, 0.89260 = 1/(1+12%) under an assumption of annual compound frequency.
For the two-year spot rate, which of the following is nearest to the value of convexity?
a. Negative 167.0 basis points
b. Zero
c. 4.13 basis points
d. 300.0 basis points
23.3.2. Below (in the left-hand panel) is an interest rate tree for the one-year rate where the initial rate of 8.0% jumps 300 basis points to either 11.0% or 5.0% next year. In two years, the one-year rate will be either 2.0%, 8.0%, or 14.0%.
Next, we will assume a three-year zero-coupon bond. Without a risk premium, for example, the date 0, state 2 price would be 0.877193 = 1/(1+14%). However, we will instead assume a risk premium. Specifically, investors require compensation of 50 basis points for each year of duration risk. The right-hand panel displays the associated risk-neutral (i.e., , assumes the risk premium) price tree for this three-year zero-coupon bond
In regard to the three-year zero-coupon bond, which of the following is true?
a. The expected return over the next year is 7.50%
b. The expected return over the next year is 9.00%
c. The implied 3-year spot rate term structure is downward sloping (aka, inverted)
d. Jensen's inequality implies that expected excess return over two years is less than 50 basis points
23.3.3. The shape of the term structure is summarized by the determinants of the forward rate (Tuckman 8.28, Third Edition):(†)
If we accept this as a valid model of the forward rate term structure, then each of the following statements is true EXCEPT which is false?
a. The forward rate is composed of the instantaneous rate plus the expected change in the rate multiplied by the duration of the zero-coupon bond corresponding to the term of the forward rate
b. The forward rate increases with the risk premium in proportion to the corresponding duration
c. The forward rate is reduced by greater interest rate volatility and the corresponding zero-coupon volatility
d. If lambda, λ, equals zero, the implied future scenario is Unchanged Term Structure; but if E[dr/dt] equals zero, then the future scenario is Realized Forwards
Answers here:
(†) Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today’s Markets, 3rd Edition (New York: Wiley, 2011)
