Exam FM/2

1. Exam FM/2 PSU Study Session Fall 2010 Dan Sprik

2. Term Structure of Interest Rates • In the real world interest rates (on loans) often depend on the length of the loan. Typically longer time to maturity corresponds to higher interest rates (this is called a normal yield curve) • An n-year spot rate is an interest rate used to discount a single cash flow at time n back to time 0. • example

3. Term Structure of Interest Rates • Given spot rates for a variety of maturities, we can compute forward rates, which are implied future interest rates. • The way to do this relies on the law of one price, which says that if there are two different valid ways to value a financial instrument they must both give the same answer. • Example

4. Stock Price • Stock price is calculated as the present value of future dividends

5. Duration • Modified Duration (D): • Macaulay Duration (): Note that Macaulay Duration for a zero-coupon bond that matures at time n is just n.

6. Duration II • Consider a general series of payments at time 1, at time 2, etc. (up to at time n)

7. Duration III • Duration for a portfolio is the weighted average of the duration for each individual cash flow, where the weight is the percent of the total PV of the portfolio attributed to that cash flow. so So (for small)

8. Convexity • Convexity = P’’(i)/ P(i)

9. Asset/Liability Management • Corporations have assets (things with positive cash flows) and liabilities (things with negative cash flows). • It is important for companies to structure the receipt of assets so they will not lose money no matter how the interest rate changes. • Most obvious (but impractical) method is exactly matching asset cash flows with liability cash flows • E.g. if you will have to pay 5000 at time 1 and 10000 at time 4, to exact match these liabilities you would buy a zero-coupon bond that pays 5000 at time 1 another that pays 10000 at time 4.

10. Immunization • Redington Immunization- ensures cash flows from assets are sufficient to pay liabilities any interest rate relatively close to the current one. 1) PV (Assets) = PV (Liabilities) 2) ModD(Assets) = ModD(Liabilities) 3) Convexity(Assets) > Convexity (Liabilities)

11. Immunization • Full immunization – protects from any change in i: 1) PV (Assets) = PV (Liabilities) 2) ModD(Assets) = ModD(Liabilities) 3) Assets inflow before and after liability outflow

12. ASM • A $1,000 par value bond with 10% annual coupons matures at par in two years. You are given that the one year spot rate is 9% and the one year forward rate for year two (i.e. the one year effective rate during year 2) is 10.5%. Determine the price of the bond. • A) 910.73 B) 922.00 C) 992.63 D) 1,005.02 E) 1,017.59

13. ASM • A perpetuity immediate with level payments has a duration of 13.5 years at an effective rate of interest of i. Determine i. • A) 6.0% B) 6.5% C) 7.0% D) 7.5% E) 8.0%

14. ASM • A company must pay liabilities of $1000 due one year from now and $2,000 due two years from now. There are two available investments: one year zero coupon bonds and two years bonds with 10% annual coupons maturing at par. The one year spot rate is 8% and the one year forward rate is 9%. What is the company’s total cost of the bonds required to exactly match the liabilities? • A) 2,625 B) 2,670 C) 2,732 D) 2,795 E) 2,887

15. Actex (example 7.46) • You have a single liability of 120,000 payable at time 6. The valuation interest rate is . You wish to attempt to immunize this portfolio by buying two zero coupon bonds with maturities at times 2 and 12. What amount should you invest in the bond that matures at time 2?