利息理论习题

1.11. Sally has two IRAs. IRA 1 earns interest at 8% effective annually and IRA 2 earns interest at 10% effective annually. She has not made any contributions since January 1, 1985, when the amount in IRA 1 was twice the amount in IRA2.The sum of the two accounts on January 1, 1993 was $75000. Determine how much was in IRA 2 on January 1, 1985?（Individual Retirement Account）2. Suppose we are given that the effective rate of interest is 5% in the first year and 6% in the second year .We invest $1 at time 0. How much is in the fund at the end of two years?3. An investor puts 100 into Fund X and 100 into Fund Y. Fund Y earns compound interest at the annual rate of j, and Fund X earns simple interest at the annual rate of 1.05j . At the end of 2 years, the amount in Fund Y is equal to the amount in Fund X. Calculate the amount in Fund Y at the end of 5 years?4. Eric deposits X into a savings account at time 0, which pays interest ata nominal rate of i , compounded semiannually. Mike deposits 2X 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 duringthe last 6 months of the 8th year. Calculate i.5. John invests 1000 in a fund which earns interest during the first year at a nominal rate of K convertible quarterly. During the 2nd year the fund earns interest at a nominal discount rate of K convertible quarterly. At the end of the 2nd year, the fund has accumulated to 1173.54. Calculate K.6. A deposit of X is made into a fund which pays an annual effective interest rate of 6% for 10 years. At the same time, X/2 is deposited into another fund which pays an annual effective rate of discount of d for 10 years. The amounts of interest earned over the 10 years are equal for both funds. Calculate d.7. You are given: 2()A t Kt Lt M =++ for 02t ≤≤(0)100,(1)110,(2)136A A A === Determine the force of interest at time 12t =. 8. At time 0, 100 is deposited into Fund X and also into Fund Y. Fund X accumulates at a force of interest ()20.51t t δ-=+. Fund Y accumulates at an annual effective interest rate of i . At the end of 9 years, the accumulated value of Fund X equals the accumulated value of Fund Y. Determine i .1.21. At an effective annual interest rate of ,0i i>, each of the following two sets of payments has present value K:1) A payment of 121 immediately and another payment of 121 atthe end of one year.2) A payment of 144 at the end of two years and another paymentof 144 at the end of three years. Calculate K.2. You are given:1)The sum of the present values of a payment of X at the end of 10years and a payment of Y at the end of 20 years is equal to thepresent value of a payment of X+Y at the end of 15 years.2)X+Y=1003)5%i=. Calculate X.3.A customer is offered an investment where interest is calculatedaccording to the following force of interest :0.02030.0453 tt ttδ≤≤⎧=⎨>⎩The customer invests 1000 at time 0. What nominal rate of interest , compounded quarterly, is earned over the first four-year period?4. Payments of 300,500 and 700 are made at the end of years five, sixand eight, respectively. Interest is accumulated at an annual effective rate of 4%. You are to find the point in time at which a single payment of 1500 is equivalent to the above series of payments. You are given:1) X is the point in time calculated by the method of equated time2) Y is the exact point in time. Calculate X+Y.5. Jones agrees to pay an amount of 2X at the end of 3 years and an amount of X at the end of 6 years. In return he will receive 2000 at the end of 4 years and 3000 at the end of 8 years. At an 8% effective annual interest rate , what is the size of Jones’ second payment?6. David can receive one of the following two payment streams:1) 100 at time 0, 200 at time n, and 300 at time 2n2) 600 at time 10At an annual effective interest rate of i ,the present value of the two streams are equal. Given 0.75941n v , determine i7. You are given two loans, with each loan to be paid by a single payment in the future. Each payment includes both principal and interest.The first loan is repaid by a 3000 payment at the end of four years. The interest is accrued at 10% per annum compounded semiannually. The second loan is repaid by a 4000 payment at the end of five years. The interest is accrued at 8% per annum compounded semiannually.These two loans are to be consolidated. The consolidated loan is to be repaid by two equal installments of X, with interest at 12% per annum compounded semiannually. The first payment is due immediately and the second payment is due one year from now. Calculate X8. At a certain interest rate the present value of the following two patterns are equal:1)200 at the end of 5 years plus 500 at the end of 10 years2)400.94 at the end of 5 yearsAt the same interest rate, 100 invested now plus 120 invested at the end of 5 years will accumulate to P at the end of 10 years. Calculate P2.1例2.1.2 一项贷款，总额为1000元，年利率为9%.设有一下三种偿还方式：（1）贷款总额以及应付利息在第10年年末一次性偿还；（2）每年年末偿还该年度的应付利息，本金在第10年年末偿还；（3）在10年中美年年末进行均衡偿付。