Exam FM/2 Review Cash Flows, portfolios, duration, & immunization

1. Exam FM/2 ReviewCash Flows, portfolios, duration, & immunization

2. Yield rates • Net Present Value method= simply compare present values • Second method= Compare internal rates of return • Dollar weighted: Calculate i, assuming simple interest • Time weighted: Multiply together growth factors, find equivalent for one year, keep careful track of deposits and withdrawals, do not include them in the factors • Also, approximation assuming all transactions occur at mid year, • where I= interest earned, A= beginning value, B= end value

3. Example Dollar/Time Weighted Find the time weighted and dollar weighted yields if the original deposit of 100,000 dropped to 90,000 at mid-year but the deposit made at that point was 10,000 and the final amount in the fund was 110,000. Ans: Time: -1% Dollar: 0%

4. Portfolio Methods • Portfolio method • Everybody receives same interest every year • Just read down the table • Investment year method • Interest rates are based on investment for a few years, then pool under the portfolio rate • Read across the row and then down the table

6. Rates • Spot rate- yield rate for zero coupon bond bought now • Forward rates- yield rate for bond bought in the future • Get comfortable deriving these rates from each other • Formula • Inflation • Consider it a negative interest rate • Just divide by (inflation rate)

7. Duration • Duration is a measure of time until cash flows, can be used to measure price sensitivity to changes in interest rates • Macaulay duration, or just duration • Weight times using PV of cashflow (current price) at those times • Also the relative change in price due to changes in force of interest • Modified duration • Simply v times the Macaulay duration • Relative change in price due to changes in i

8. Convexity • Convexity • Relative second derivative of price, with respect to interest rate • Second order approximations using convexity

9. Immunization • Immunization- protecting from changes in interest rates • Cash-flow matching/ exact matching/ dedication • Match liabilities exactly with assets, one for one • Redington immunization • PV(assets)=PV(liabilities) • ModD(assets)=ModD(liabilities), or first order derivatives equal • Convexity(assets)>Convexity(liabilities), or second order derivative greater • Protects against small changes in i • Full immunization • PV(assets)=PV(liabilities) • ModD(assets)=ModD(liabilities), or first order derivatives equal • One asset cash inflow before and after liability cash outflow

10. Problem 1 • You are given this information about the activity in two different investment accounts. During 1999, the dollar weighted return for investment account K equals the time weighted return for investment account L, which equals i. Calculate i. ASM p.273 Answer: 15%

11. Problem 2 • A person deposits 1000 on January 1, 1997. Let the following be the accumulated value of the 1000 on January 1, 2000:P: under the investment year methodQ: under the portfolio yield methodR: where the balance is withdrawn at the end of every year and is reinvested at the new money rateDetermine the ranking of P, Q, and R ASM p.284 Answer: R>P>Q

12. Problem 3 • The one-year forward rate for year 2 is 4%. The four-year spot rate is 10%. The expected spot rate at the end of year two on a zero-coupon bond maturing at the end of year 4 is 7%. Determine the one-year spot rate. ASM p.435 Answer: 22.96%

13. Problem 4 • The real rate of interest is 4%. The expected annual inflation rate over the next two years is 5%. What is the net present value of the following cash flows? ASM p.433Year 0 1 2Cash Flow -300 160 160 Answer: -19.30

14. Problem 5 • A bond with 7.5% annual coupons will mature at par on June 30, 2006. Determine the duration of the bond on December 31, 2004 if the effective rate of interest is 5.5% per annum. ASM p.452 Answer: 1.43

15. Problem 6 • A $100 par value bond with 7% annual coupons and maturing at par in 4 years sells at a price to yield 6%. Determine the modified duration of the bond. ASM p.452 Answer: 3.43

16. Problem 7 • An annuity-immediate has payments of $1,000, $3,000, and $7,000 at the end of one, two and three years, respectively. Determine the convexity of the payments evaluated at i=10%.ASM p.472 Answer: 7.63

17. Problem 8 • A company must pay liabilities of $1,000 due one year from now and another $2,000 due two years from now. There are two available investments: one year zero coupon bonds and two-year bonds with 10% annual coupons maturing at par. The one year spot rate is 8% and the one-year forwarrd rate is 9%. What is the company’s total cost of the bonds required to exactly (absolutely) match the liabilities?ASM p.472 Answer: 2,625

18. Problem 9 • A company must pay a benefit of $1,000 to a customer in two years. To provide for this benefit, the company will buy a one-year and three-year zero-coupon bonds. The one-year and three-year spot rates are 8% and 10%, respectively. The company wants to immunize itself from small changes in the interest rates on either side of 10% (Redington immunization). What amount should it invest in the one-year bonds?ASM p.472 Answer: 420