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

Cash Flows Net Present Value method = Calculate present value of cash flows; interest rate is given Internal Rate of Return = Interest rate where Net Present Value is zero

Interest Measurements Dollar Weighted Return – Return assuming simple interest Time Weighted Return – Return using growth factors

Example 1 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%

Portfolio Methods Portfolio method Investment year method Everybody receives same interest every year Read down the final column Investment year method Interest rates are based on the investment year, then move to the portfolio rate Read across the row and then down the final column

Example 2 Calendar Year of Original Investment Investment Year Rates (in %) Portfolio Rates (in%) Calendar Year of Portfolio Rate y i1 i2 i3 i4 i5 1992 8.25 8.40 8.50 8.35 1997 1993 8.70 8.75 8.90 9.00 8.60 1998 1994 9.10 9.20 8.85 1999 1995 9.30 9.40 2000 1996 9.25 9.35 9.50 9.55 9.60 2001 9.70 10.00 9.90 9.80

Rates Spot Rate - yield rate for zero coupon bond bought now Forward Rate - yield rate for bond bought in the future Inflation Rate Consider it a negative interest rate Just divide by (inflation rate)

Duration Duration is a weighted present value of cash flow times Macaulay duration, or just duration Weight times using PV of cash flow (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

Convexity Convexity Relative second derivative of price, with respect to interest rate Approximating Price changing using application of Taylor Series

Immunization Immunization – protect large price changes from changes in interest rates Cash Flow Matching (Exact matching or Dedication) Match each liability with an asset Redington immunization PV(Assets) = PV(Liabilities) MacD(Assets) = MacD(Liabilities) Convexity(Assets)>Convexity(Liabilities) Full immunization ModD(Assets)=ModD(Liabilities) Assets straddle the Liabilities

Problem 1 Account K   Activity Date Fund Value Before Activity Deposit Withdrawal January 1, 1999 100 July 1, 1999 125 X October 1, 1999 110 2X December 31, 1999 Account L December 1, 1999 105.8 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%

Problem 2 Calendar Year of Original Investment Investment Year Rates (in %) Portfolio Rates (in%) Calendar Year of Portfolio Rate y i1 i2 i3 i4 i5 1992 8.25 8.40 8.50 8.35 1997 1993 8.70 8.75 8.90 9.00 8.60 1998 1994 9.10 9.20 8.85 1999 1995 9.30 9.40 2000 1996 9.25 9.35 9.50 9.55 9.60 2001 9.70 10.00 9.90 9.80 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 method Q: under the portfolio yield method R: where the balance is withdrawn at the end of every year and is reinvested at the new money rate Determine the ranking of P, Q, and R ASM p.284 Answer: R>P>Q

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%

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.433 Year 0 1 2 Cash Flow -300 160 160 Answer: -19.30

Problem 5 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

Problem 6 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

Problem 7 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