ORBS7220/MATH4837 Risk and Portfolio Management
Assignment 2
Instructions: Layout the intermediate steps systematically. Give exact answers or round them to 4 decimals unless specified otherwise. For percentages (e.g. return and volatility), show 4 decimals after conversion to percentages, e.g. ^σ = 12.3456%, not 0.1235.
1. [20 marks] A three-year bond with a yield of 6% p.a. (continuously compounded) pays
a 4% coupon at the end of each year. The face value is $100. (a) What is the bond's yield duration?
(b) Use the duration to estimate the bond's price if the yield decreases 0.1%.
(c) Recalculate the bond's price on the basis of a 5.9% per annum yield (continuously compounded). What is the error of the prediction in part (b)?
(d) Suppose that a second bond with a market price of $105 and a duration of 2.5 is used to hedge against interest rate risk. How much face value of the second bond should one short for each $100 face value of the first bond?
2. [30 marks] The values of a stock index over four days are given in the table below.
|
Day
|
Index
|
|
0
|
20436
|
|
1
|
20794
|
|
2
|
21059
|
|
3
|
20751
|
The volatility on Day 2 is estimated with ^(σ)2 = ju1 j .
(a) Estimate the daily volatility ^σ3 on Day 3 using the EWMA model with λ = 0.95.
(b) Estimate the daily volatility ^σ3 on Day 3 using the GARCH(1,1) model with α = 0.01, β = 0.95, γ = 0.04, and σL = 0.01.
(c) Calculate the log-likelihood for part (b).
(d) Estimate the daily volatility ^σ3 on Day 3 using the GARCH(1,1) model with α = 0.02, β = 0.96, γ = 0.02, and σL = 0.02.
(e) Calculate the log-likelihood for part (d).
(f) Which set of parameters, part (b) or (d), is better?
3. [20 marks] A bank has two $10m one-year loans
|
Outcome
|
Probability
|
|
Neither loan defaults
|
95%
|
|
Loan 1 defaults, Loan 2 does not default
|
2.5%
|
|
Loan 2 defaults, Loan 1 does not default
|
2.5%
|
|
Both loans default
|
0%
|
If Loan 1 defaults, the loss will be either $5m or $10m, with equal probability. If Loan 2 defaults, the loss will be either $2m, $4m, $6m, $8m or $10m, with equal probability. For each loan, a profit of $1m will be made in case of no default.
(a) Calculate the one-year 97% VaR for the portfolio of two loans.
(b) Calculate the one-year 97% ES for the portfolio of two loans.
4. [30 marks] Now is the end of Day 3. The stock price, daily return, and daily volatility over 4 days are depicted in the table. An investor has purchased 1000 shares.
|
Day
|
Stock price
|
daily return
|
daily volatility
|
|
0
|
5.0
|
—
|
—
|
|
1
|
4.8
|
-4.0000%
|
0.1%
|
|
2
|
4.6
|
-4.1667%
|
0.4%
|
|
3
|
5.1
|
10.8698%
|
0.2%
|
|
4
|
|
|
0.2%
|
(a) Use historical simulation to estimate the volatility-adjusted daily losses (under Scenario 1 to 3).
(b) Using the results in part (a), calculate the one-day 50% VaR on Day 4.
(c) Using the results in part (a), calculate the one-day 50% ES on Day 4.