PROBLEM SET 1
Investment Policy and Bond Portfolio management
Due date: Friday, September 17, 5:00 pm. No late problem sets will be accepted.
1. Assume that at retirement you have accumulated $825,000 in a variable annuity contract. The assumed investment return is 5.5% and your life expectancy is 18 years. What is the hypothetical constant benefit payment?
PV = -825,000, i = 5.5, n = 18, PMT = 73,358.93.
2. You manage a portfolio for Ms. Greenspan, who has instructed you to be sure her portfolio has a value of at least $350,000 at the end of six years. The current value of Ms. Greenspan's portfolio is $250,000. You can invest the money at a current interest rate of 8%. You have decided to use a contingent immunization strategy. - What amount would need to be invested today to achieve the goal, given the current interest rate? - Suppose that four years have passed and the interest rate is 9%. What is the trigger point for Angel's portfolio at this time? (That is, how low can the value of the portfolio be before you will be forced to immunize to be assured of achieving the minimum acceptable return?) - Illustrate the situation graphically.
- If the portfolio's value after 4 years is $291,437 what should you do?
Calculations are shown below.
- Amount needed to reach the goal = $350,000/1.086 = $220,559.37 - The trigger point = $350,000/1.092 = $294,588.00
- The graph should look like the ones in Figure 16.11 on page 542. - You should immunize the portfolio because its value is below the trigger point. If the value is $291,437 you will need to earn a rate of 9.59% over the remaining two years to achieve the goal of $350,000: $350,000 = $291,437 * (1+r)2. Solving for r yields 9.59%
3. True or False (Please explain your answer!)
Two bonds are selling at par value and each has 17 years to maturity. The first bond has a coupon rate of 6% and the second bond has a coupon rate of 13%. Thus, there is no consistent statement that can be made about the durations of the bonds.
As the price and maturities are the same, the bond with higher coupon rate will have lower duration.
4. Consider a bond selling at par with modified duration of 12 years and convexity of 265. A 1 percent decrease in yield would cause the price to increase by 12%, according to the duration rule. What would be the percentage price change according to the duration-with-convexity rule?
(P/P = -D*(y + (1/2) * Convexity * ((y)2; = -12 * -.01 + (1/2) * 265 * (.01)2 = .12 + .01325 = .13325 or (13.3%)
5. Please answer questions 12, 13 on page 544 and question 13 on page 550.
Solutions for question 12 & 13 on page 544.
12.a.The duration of the perpetuity is: 1.05/0.05 = 21 years
Call w the weight of the zero-coupon bond. Then:
(w × 5) + [(1 – w) × 21] = 10 ( w = 11/16 = 0.6875
Therefore, the portfolio weights would be as follows: 11/16 invested in the zero and 5/16 in the perpetuity.
b.Next year, the zero-coupon bond will have a duration of 4 years and the perpetuity will still have a 21-year duration. To obtain the target duration of nine years, which is now the duration of the obligation, we again solve for w:
(w × 4) + [(1 – w) × 21] = 9 ( w = 12/17 = 0.7059
So, the proportion of the portfolio invested in the zero increases to 12/17 and the proportion invested in the perpetuity falls to 5/17.
13.a.The duration of the annuity if it were to start in 1 year would be: |(1) |(2) |(3) |(4) |(5) | |Time until Payment |Cash Flow |PV of CF (Discount |Weight |Column (1) ( Column (4)| |(years) | |rate = 10%) | | | |1 |...