Suzanne Evans
Well-Known Member
AIMs: Calculate the convexity effect using Jensen’s inequality. Calculate the price and return of a zero coupon bond incorporating a risk premium
Questions:
301.1. The interest rate tree below shows the true process for a one-year interest rate. The current one-year spot rate is 7.0%. Next year, investors expect the one-year rate to either increase to 10.0% or drop to 4.0%, with equal probability of an increase or drop. In the subsequent year (Year 2), investors similarly expect the future one-year rate to again either increase or decrease by +/- 3%, with equal likelihood. Graphically, as follows:
Before the inclusion of a risk premium, assuming the above interest rate tree the true and known process, the price of a $1,000 par two-year zero-coupon bond would be $874.13; i.e., this price is the expected discounted value under annual compounding. However, let us modify this and instead assume that investors are risk-averse: they would prefer a certain 7.0% return to an expected 7.0% return with volatility. Consequently, let us assume investors require (charge) a risk premium of 90 basis points. As compensation for the volatility, the require an expected return of 7.90%.
Compared to the risk-neutral price of $874.13, what is the change in price due to the introduction of the risk premium?
a. Increase bond price by $14.58
b. Increase bond price by $7.30
c. Decrease bond price by $7.30
d. Decrease bond price by $14.58
301.2. Assume the current (spot) one-year interest rate is 10.0% and a true binomial interest rate tree is known (understood) to investors. In each of the next two years, the rate will either increase or decrease by +/- 4.0% with equal probabilities. Next year, the rate will increase to 14.0% (with 50% probability) or decrease to 6.0% (with 50% probability). Assume investors are risk-neutral: they require no risk premium due to volatility. Under these assumptions and assuming annual compounding, both the current (spot) one-year rate and the expected one-year rate in one year are 10.0%. However, the convexity effect due to Jensen's Inequality implies a different two-year spot rate. If 10.0% is the yield in the absence of uncertainty, what is the yield change implied by Jensen's Inequality due to the presence of volatility (but assuming no risk aversion)?
a. An increase of ~ 25 basis points to ~ 10.25%
b. None
c. A decrease of ~ 7 basis point to 9.93%
d. A decrease ~ 25 basis points to 9.75%
301.3. Robert's manager requests that he graph a five-year term structure of interest rates based on a binomial interest rate tree. The tree recombines with up- and down-jumps that are equal in magnitude (plus or minus X basis points) and likelihood (50%) such that the expected one-year rate on each of years two, three, four and five is today's one-year spot rate. However, the interest rate assumptions in the tree's nodes reflect an assumption of significant rate volatility. Further, a key input into the model is a risk premium which reflects an assumption that investors are risk-averse. Which of the following is the most plausible predication about the produced graph of the term structure?
a. The term structure is upward-sloping throughout
b. In the shorter-end the yield increases but in the longer-end the yield decreases
c. In the shorter-end the yield decreases but in the longer-end the yield increases
d. The term structure is downward-sloping throughout
Answers:
Questions:
301.1. The interest rate tree below shows the true process for a one-year interest rate. The current one-year spot rate is 7.0%. Next year, investors expect the one-year rate to either increase to 10.0% or drop to 4.0%, with equal probability of an increase or drop. In the subsequent year (Year 2), investors similarly expect the future one-year rate to again either increase or decrease by +/- 3%, with equal likelihood. Graphically, as follows:
Before the inclusion of a risk premium, assuming the above interest rate tree the true and known process, the price of a $1,000 par two-year zero-coupon bond would be $874.13; i.e., this price is the expected discounted value under annual compounding. However, let us modify this and instead assume that investors are risk-averse: they would prefer a certain 7.0% return to an expected 7.0% return with volatility. Consequently, let us assume investors require (charge) a risk premium of 90 basis points. As compensation for the volatility, the require an expected return of 7.90%.
Compared to the risk-neutral price of $874.13, what is the change in price due to the introduction of the risk premium?
a. Increase bond price by $14.58
b. Increase bond price by $7.30
c. Decrease bond price by $7.30
d. Decrease bond price by $14.58
301.2. Assume the current (spot) one-year interest rate is 10.0% and a true binomial interest rate tree is known (understood) to investors. In each of the next two years, the rate will either increase or decrease by +/- 4.0% with equal probabilities. Next year, the rate will increase to 14.0% (with 50% probability) or decrease to 6.0% (with 50% probability). Assume investors are risk-neutral: they require no risk premium due to volatility. Under these assumptions and assuming annual compounding, both the current (spot) one-year rate and the expected one-year rate in one year are 10.0%. However, the convexity effect due to Jensen's Inequality implies a different two-year spot rate. If 10.0% is the yield in the absence of uncertainty, what is the yield change implied by Jensen's Inequality due to the presence of volatility (but assuming no risk aversion)?
a. An increase of ~ 25 basis points to ~ 10.25%
b. None
c. A decrease of ~ 7 basis point to 9.93%
d. A decrease ~ 25 basis points to 9.75%
301.3. Robert's manager requests that he graph a five-year term structure of interest rates based on a binomial interest rate tree. The tree recombines with up- and down-jumps that are equal in magnitude (plus or minus X basis points) and likelihood (50%) such that the expected one-year rate on each of years two, three, four and five is today's one-year spot rate. However, the interest rate assumptions in the tree's nodes reflect an assumption of significant rate volatility. Further, a key input into the model is a risk premium which reflects an assumption that investors are risk-averse. Which of the following is the most plausible predication about the produced graph of the term structure?
a. The term structure is upward-sloping throughout
b. In the shorter-end the yield increases but in the longer-end the yield decreases
c. In the shorter-end the yield decreases but in the longer-end the yield increases
d. The term structure is downward-sloping throughout
Answers:
