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

2. 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

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

4. 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%

5. 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

6. 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

7. 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)

8. 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

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

10. 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

11. 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%

12. 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, 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

13. 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: %

14. 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 Cash Flow
Answer:

15. 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

16. 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

17. 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