Stat 274 Theory of Interest. Chapter 2: Equations of Value and Yield Rates. Brian Hartman Brigham Young University

Transcription

1 Stat 274 Theory of Interest Chapter 2: Equations of Value and Yield Rates Brian Hartman Brigham Young University

2 Equations of Value When using compound interest with a single deposit of c at time 0, the equation of value is A c (t) = c(1 + i) t 2

3 Examples 1 You win a prize and then invest that prize at 3% annually compounding interest. In 7 years, you have 1300 in the account. How much was the prize worth? [ ] 2 You invest 1000 at 2% annually compounding interest. You now have How long was the money in the account? [9.207] 3 You invest 100 for five years and end up with 110. What interest rate (annually compounding) did you earn? [1.924%] 4 At 8% annually compounding interest, how long will it take to double your money? [9.006] 3

4 Multiple Investments Multiple investments (can be positive or negative), each at time t k, will have a time τ value of B at time T according to this equation, k C tk a(τ) a(t k ) = B a(τ) a(t ) With the following special cases, k C tk a(t ) a(t k ) = B C tk v(t k ) = Bv(T ) k (time τ equation of value) (time T equation of value) (time 0 equation of value) 4

5 Examples 1 You borrow 1000 at an annual rate of 10% (note that for the rest of the class we will assume annual compound interest unless otherwise stated) and pay it back with one payment of 600 at the end of the first year and another payment of X at the end of the second. Calculate X. [550] 2 You deposit 500 at time 0, another 300 at time 1, and a final 200 at time 2. Immediately after your last deposit, your account has Calculate the interest rate. [ ] 5

6 Calculator Examples You deposit 500 at time 0, another 300 at time 1, and a final 200 at time 2. At time 3, your account has Calculate the interest rate. [4.2%] A loan can be paid off by two different payment streams. The first is 100 at times 5 and 10 and another 200 at time 15. The second is a single payment of 400 at time t. Interest is 4.5%. Calculate t. [ ] 6

7 Yield Rates When calculating unknown interest rates, we are finding the yield rate for the investment. These yield rates are sometimes called dollar-weighted yield rates (we will get to time-weighted yield rates soon). As long as the sign of the payments only changes once, we are guaranteed a unique interest rate i > 1. 7

8 Examples 1 You invest 1000 at time 0 and 600 at time 2 and receive 600 at time 1 and 1265 at time 3. Show that 10% is the unique yield rate. 2 You receive 180 at time 1 in exchange for 100 at time 0 and 100 at time 2. Show that the yield rate is undefined. 8

9 Reinvestment Sometimes the yield rate of the borrower and the lender will be different. Example: Abby pays Brandon 100, in return he pays 20 at times 1, 2, 3, 4, 5, and 6. Each time Abby gets a payment, she invests it at 2% interest and then withdraws all the money at the end of the 6 years. Calculate the interest rate paid by Brandon and Abby s yield rate. [A = , B = ] 9

10 Dollar-weighted Yield Rate On exam FM, the dollar-weighted yield rate is the simple interest approximation of the yield rate (this is different from the book). We can derive the yield rate starting from this equation of value V t = V 0 (1 + it) + Some rearrangement leads to: i = n C j (1 + i(t t j )) j=1 V t V 0 n j=1 C j V 0 t + n j=1 (t t j)c j 10

11 Time-weighted Yield Rate Instead of focusing on the dollars, we can divide the growth into years (1 through r + 1 with yields of j k ) and get the time-weighted yield as follows: [ r+1 j tw = ] (1 + j k ) 1 k=1 With the annual time-weighted yield rate i tw = (1 + j tw ) 1/T 1 = [ r+1 1/T (1 + j k )] 1 k=1 11

12 Examples 1 You deposit 200 at time 0, it grows to 212 at time 1, then you add 50, and that grows to 273 at time 2. Calculate the dollar-weighted and time-weighted yield rates. [DW = , TW = ] 2 You deposit 2000 at time 0, it grows to 2040 at time 1, then 1000 is withdrawn, the 1040 grows to 1050 at time 2, then 1000 is deposited and that 2050 grows to 2150 at time 3. Calculate the dollar-weighted and time-weighted yield rates. [DW = 0.03, TW = 0.026] 12

13 SOA Examples Bruce deposits 100 into a bank account. His account is credited interest at a nominal rate of interest of 4% convertible semiannually. At the same time, Peter deposits 100 into a different account. Peter s account is credited at a force of interest of δ. After 7.25 years, the value of each account is the same. Calculate δ. [0.0396] , , , ,

14 SOA Examples Cont. Eric deposits 100 into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually. Mike deposits 200 into a different savings account at time 0, which pays simple interest at an annual rate of i. Eric and Mike earn the same amount of interest during the last 6 months of the 8th year. Calculate i. [9.46%] 9.06%, 9.26%, 9.46%, 9.66%, 9.86% 14

15 SOA Examples Cont. An association had a fund balance of 75 on January 1 and 60 on December 31. At the end of every month during the year, the association deposited 10 from membership fees. There were withdrawals of 5 on February 28, 25 on June 30, 80 on October 15, and 35 on October 31. Calculate the dollar-weighted (money-weighted) rate of return for the year. [11%] 9.0%, 9.5%, 10.0%, 10.5%, 11.0% 15

16 SOA Examples Cont. Jeff deposits 10 into a fund today and 20 fifteen years later. Interest is credited at a nominal discount rate of d compounded quarterly for the first 10 years, and at a nominal interest rate of 6% compounded semiannually thereafter. The accumulated balance in the fund at the end of 30 years is 100. Calculate d. [4.53%] 4.33%, 4.43%, 4.53%, 4.63%, 4.73% 16