EXAM FM/2 REVIEWCASH 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

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

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 8.25 8.40 8.50 8.50 8.35 19971993 8.50 8.70 8.75 8.90 9.00 8.60 19981994 9.00 9.00 9.10 9.10 9.20 8.85 19991995 9.00 9.10 9.20 9.30 9.40 9.10 20001996 9.25 9.35 9.50 9.55 9.60 9.35 20011997 9.50 9.50 9.60 9.70 9.701998 10.00 10.00 9.90 9.801999 10.00 9.80 9.702000 9.50 9.502001 9.00

Example 2

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 PV(Assets) = PV(Liabilities) ModD(Assets)=ModD(Liabilities) Assets straddle the Liabilities

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%

Account K Activity

Date Fund Value Before Activity Deposit WithdrawalJanuary 1, 1999 100

July 1, 1999 125 XOctober 1, 1999 110 2X

December 31, 1999 125

Account L Activity

Date Fund Value Before Activity Deposit WithdrawalJanuary 1, 1999 100

July 1, 1999 125 X

December 1, 1999 105.8

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 8.25 8.40 8.50 8.50 8.35 19971993 8.50 8.70 8.75 8.90 9.00 8.60 19981994 9.00 9.00 9.10 9.10 9.20 8.85 19991995 9.00 9.10 9.20 9.30 9.40 9.10 20001996 9.25 9.35 9.50 9.55 9.60 9.35 20011997 9.50 9.50 9.60 9.70 9.701998 10.00 10.00 9.90 9.801999 10.00 9.80 9.702000 9.50 9.502001 9.00

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

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 2Cash 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