利息理论习题

1.1

1. 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 IRA 2.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 at a 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 during

the 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: A(t)?Kt2?Lt?Mfor0?t?2

A(0)?100,A(1)?110,A(2)?136

Determine the force of interest at time t?

1. 28. At time 0, 100 is deposited into Fund X and also into Fund Y. Fund X accumulates at a force of interest

?t?0.5?1?t??2. 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 .

1.2

1. At an effective annual interest rate of i,i?0, each of the following two sets of payments has present value K:

1) A payment of 121 immediately and another payment of 121 at the end of one year.

2) A payment of 144 at the end of two years and another payment of 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 10 years and a payment of Y at the end of 20 years is equal to the present value of a payment of X+Y at the end of 15 years. 2) X+Y=100

3) i?5%. Calculate X.

3.A customer is offered an investment where interest is calculated

?0.02t0?t?3according to the following force of interest :?t??

t?3?0.045The 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, six

and 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 time 2) 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 2n 2) 600 at time 10

At an annual effective interest rate of i,the present value of the two

nstreams are equal. Given v?0.75941, determine i

7. 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 X

8. 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 years 2) 400.94 at the end of 5 years

At 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 P

2.1

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。 分别计算在三种偿还方式下所支付的利息额。

例2.1.5 某人购房借款50000元，计划每年年末还款10000元，直到还完为止。设利率为7%，分三种方式求该借款人还款的整数次数以及最后的还款零头。其中零头的还款时间分别为：

（1）在时刻n与n+1之间。 （2）在时刻n。 （3）在时刻n+1。

1. Susan and Jeff each make deposits of 100 at the end of each year for 40 years. Starting at the end of the 41st year, Susan makes annual withdrawals of X for 15 years and Jeff makes annual withdrawals of Y for 15 years. Both funds have a balance of 0 after the last withdrawal. Susan’s fund earns an annual effective interest rate of 8%. Jeff’s fund earns an annual effective interest rate of 10%. Calculate Y?X

2. At an annual effective interest rate of 6.3%, an annuity immediate with 4N level annual payments of 1000 has a present value of 14113. Determine the fraction of the total present value represented by the first set of N payments and the third set of N payments combined.

3. An investment requires an initial payment of 10000 and annual payment of 1000 at the end of each of the first 10 years. Starting at the end of the 11th year, the investment returns 5 equal annual payments of X. Determine X to yield an annual effective rate of 10% over the 15-year period.

4. The present value of a series of payments of 2 at the end of every 8

years, forever, is equal to 5. Calculate the effective rate of interest.

5. Chuck needs to purchase an item in 10 years. The item costs 200 today, but its price inflates at 4% per year. To finance the purchase, Chuck deposits 20 into an account at the beginning of each year for first 6 years. He deposits an additional X at the beginning of years 4,5,6 to meet his goal. The annual effective interest rate is 10%. Calculate X

6. Deposits of 1000 are placed into a fund at the beginning of each year for 30 years. At the end of the 40th year, annual payments commence and continue forever. Interest is at an effective annual rate of 5%. Calculate the annual payment.

7. The following three series of payments have the same present value of P: 1) a perpetuity-immediate of 2 per year at an annual effective interest rate of i

2) a 20-year annuity-immediate of X per year at an annual effective interest rate of 2i

3) a 20-year annuity-due of 0.96154X per year at an annual effective interest rate of 2i. Calculate P

8. At a nominal rate of interest i, convertible semiannually, the present

value of a series of payments of 1 at the end of every 2 years, forever, is 5.89. Calculate i

2.2

例1

在20年期间，每月支付1500元，年利率为7%。求 （1）这些付款在前3年的现值；

（2）这些付款在最后一次付款后的第5年末的积累值。 例2

(1)已知某种年支付额为10000元，分期于每月月末支付一次相等金额，支付期10年，年实际利率是5%，该年金的现值是多少?

(2)如果年名义利率是5%，每季结算一次利息，其余条件相同，此时年金的现值又是多少？

1. Jerry will make deposits of 450 at the end of each quarter for 10 years. At the end of 15 years, Jerry will use the fund to make annual payments of Y at the beginning of each year for 4 years, after which the fund is exhausted. The annual effective rate of interest is 7%. Determine Y

2. The present value of a 5-year annuity immediate, with payments of 1000 each 6 times per year, is 20930. Determine the force of interest.

3. Harriet wishes to accumulate 60000 in a fund at the end of 25 years.

She plans to deposit 80 into the fund at the end of each of the first 120 months. She then plans to deposit 80+x into the fund at the end of each of the last 180 months. Assume the fund earns interest at an annual effective rate of 3.66%. Determine x

4. Jim and Sue are planning to retire on January 1,1995. Their goal is to have enough money in savings to be able to withdraw 3000 per month beginning one month after retirement and continuing for 25 years after retirement. They earn an annual effective rate of interest of 10% on their account. Determine the minimum amount needed in their savings account on January 1,1995, to accomplish their goal.

5. John wins 1000000 in a lottery and will be paid 20 equal annual installments of 50000 with the first payment due today. A bank offers to exchange John’s winnings for a perpetuity of X per month with the first payment due today. Find the value of X assuming a 10% effective rate of interest.

6. The proceeds of a 10000 death benefit are left on deposit with an insurance company for seven years at an annual effective interest rate of 5%. The balance at the end of seven years is paid to the beneficiary in 120 equal monthly payments of X, with the first payment made

immediately. During the payout period, interest is credited at an annual effective interest rate of 3%. Calculate X

7. A car dealer offers to sell a car for 10000. The current market loan rate is a nominal rate of interest of 12% per annum, compounded monthly. As an inducement, the dealer offers 100% financing at an effective annual interest rate of 5%. The loan is to be repaid in equal installments at the end of each month over a 4-year period. Calculate the cost to the dealer of this inducement.

8. A payment of 100 is made at the end of each two months for a period of 6 years. The nominal annual rate of interest is 3%, convertible every 8 months. Find the present value of this series of payments.

2.3

1. Find the PV of a 14-year annuity with continuous payments at the rate of 650 a year at an effective interest rate of 5.65%

2. You are given:

da10???33.865Calculate ? 1) a10?7.522)?d?

??n?3?13.987,??sn?1?23.6914, find s1 3. Given the informationa

4. You are given

?n0atdt?100, Calculate an2.4

1. Kathy deposits 100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i. The accumulated amount in the account at the end of 40 years is X, which is 5 times the accumulated amount in the account at the end of 20 years. Calculate X

2. The present value of a perpetuity of 6500 paid at the end of each year plus the present value of a perpetuity of 8500 paid at the end of every 5 year is equal to the present value of an annuity of k paid at the end of each year for 25 years. Interest is 6% convertible quarterly. Calculate k

3. A 20-year annuity pays 100 every other year beginning at the end of the second year, with additional payments of 300 each at the end of years 3,9,15. The effective annual interest rate is 4%. Calculate the present value of the annuity.

4. You are given:

1) The present value of a 6n-year annuity-immediate of 1 at the end of every year is 9.996

2) The present value of a 6n-year annuity-immediate of 1 at the end of every second year is 4.76

3) The present value of a 6n-year annuity-immediate of 1 at the end of every third year is X Calculate X

5. Determine the present value of 1 payable at the end of years 7,11,15,19,23 and 27

6. You have an annuity which pays 1200 every two years. The first payment is two years from now and the last payment is ten years from now. You can trade that annuity for another annuity of equivalent present value, which pays 180 per quarter starting today. The interest rate for both annuities is 4% per annum convertible quarterly. If you took the second annuity, how many quarterly payments would you receive? The last payment may be less than 180 but not more than 180

7. A perpetuity of 1 each year, with the first payment due immediately, has a present value of 25 at an annual effective rate of i. The owner exchanges it for another perpetuity with the first payment due immediately and subsequent payments due at two years intervals. What

should the payment of the second perpetuity be, in order to keep the same interest rate iand the same present value?

2.5

1. Find the AV of a 12-year annuity-immediate with annual payments of 1225, 2575, 3925, 5275, ……

2. An annuity provides for 12 annual payments. The first payment is 100, paid at the end of the first year, and each subsequent payment is 5% more than the one preceding it. Calculate the present value of this annuity if i?0.07

3. Joe can purchase one of two annuities:

1) A 10-year decreasing annuity-immediate, with annual payments of 10, 9, 8, …, 1

2) A perpetuity-immediate with annual payments. The perpetuity pays 1 in year 1, 2 in year 2, 3 in year 3,… , and 11 in year 11. After 11, the payments remain constant at 11

At an annual effective interest rate of i, the present value of 2) is twice the present value of 1). Calculate the present value of 1)

4. Jake inherits a perpetuity that will pay him 10000 at the end of the

first year increasing by 10000 per year until a payment of 150000 is made at the end of the 15th year. Payments remain level after the 15th year at 150000 per year. Determine the present value of this perpetuity, assuming a 7.5% annual interest rate

5. Brian buys a 10-year decreasing annuity-immediate with annual payments of 10, 9, 8, …, 1. On the same date, Jenny buys a perpetuity-immediate with annual payments. For the first 11 years, payments are 1, 2, 3,…, 11. After year 11, payments remain constant at 11.At an annual effective interest rate of i, both annuities have a present value of X. Calculate X

6. The first payment of a perpetuity-immediate is 60. Subsequent payments decrease by 1 per year until they reach a level of k. Payments remain constant at k thereafter. The present value of the perpetuity is equal to the present value of a perpetuity-immediate paying 44 each year. The annual effective interest rate is 5%. Calculate k

7. An annuity provides for 10 annual payments. The first of these payments is 100 and each subsequent payment is 10 higher than the one preceding it. Find the present value of this annuity at the time one year prior to the first payment if i?10%

8. You are given an annuity-immediate paying 10 for 10 years, then decreasing by one per year for 9 years and paying one per year thereafter, forever. The annual effective rate of interest is 4%. Calculate the present value of this annuity

3.123

1. On January 1,1997, an investment account is worth 100000. On April 1,1997, the value has increased to 103000 and 8000 is withdrawn. On January 1,1999, the account is worth 103992. Assuming a dollar weighted method for 1997 and a time weighted method for 1998, the annual effective interest rate was equal to x for both 1997 and 1998. Calculate x

3. On January 1,1997, Brian’s stock portfolio is worth 100000. On September 30,1997,5000 is withdrawn from the portfolio, and immediately after this withdrawal the portfolio has a value of 105000. 12 months later, the value of the portfolio is 108000,and Brian adds 3000 worth of stock to his portfolio. On December 31,1998, the portfolio is worth 100000. What is the time-weighted rate of return for Brian’s stock portfolio over the two-year period?

4. On January 1,1999, Lucy deposits 90 into an investment account. On

April 1,1999, when the amount in Lucy’s account is equal to X, a withdrawal of W is made. No further deposits or withdrawals are made to Lucy’s account for the remainder of the year. On December 31, 1999, the amount in Lucy’s account is 85. The dollar-weighted return over the 1-year period is 20%. The time-weighted return over the 1-year period is 16%. Calculate X

5. On January 1, an investment account is worth 100. On May 1,the value has increased to 120 and D is deposited. On November 1, the value is 100 and 40 is withdrawn. On January 1 of the following year, the investment account is worth 65. The time-weighted rate of interest is 0%. Calculate the dollar-weighted rate of interest.

6. On January 1, an investment account is worth 300000. M months later, the value has increased to 315000 and 15000 is withdrawn. 2M months prior to the end of the year, the account is again worth 315000 and 15000 is withdrawn. On December 31, the account is worth 315000. The annual effective yield rate, using the dollar-weighted method, is 16%.Calculate M

7. A fund earned investment income of 9200 during 1991. The beginning and ending balances of the fund were 100000 and 129200, respectively.

A deposit was made at time K during the year. No other deposits or withdrawals were made. The fund earned 8% in 1991 using the dollar-weighted method. Determine K

8. An investment fund has a value of 1000 at the beginning and the end of the year. A deposit of 200 was made at the end of 4 months. A withdrawal of 300 was made at the end of 7 months. Find the rate of interest earned by the fund assuming simple interest during the year.

3.5

1. 1000 is deposited into Fund X, which earns an annual effective rate of 6%. At the end of each year, the interest earned plus an additional 100 is withdrawn from the fund. At the end of the tenth year, the fund is depleted. The annual withdrawals of interest and principal are deposited into Fund Y, which earns an annual effective rate of 9%. Determine the accumulated value of Fund Y at the end of year 10.

2. Victor invested 300 into a bank account at the beginning of each year for 20 years. The account pays out interest at the end of every year at an annual effective interest rate of i.The interest is reinvested at an annual

ieffective rate of .The yield rate on the entire investment over the 20

2year period is 8% annual effective. Determine i

3. Eric deposits 12 into a fund at time 0 and an additional 12 into the same fund at time 10. The fund credits interest at an annual effective rate of i. Interest is payable annually and reinvested at an annual effective rate of 0.75i. At time 20, the accumulated amount of the reinvested interest payments is 64. Calculate i

4. Payments of 1000 are invested at the end of each year for 5 years. The payments earn interest at an annual effective rate of 10%. The interest can be reinvested at an annual effective rate of 6% in the first 4 years and at an annual effective rate of k thereafter. The amount in the fund at the end of 5 years is 6090. Calculate k

5. Bill deposits 1000 into a fund for 15 years. The fund pays interest at the end of each 6-month period at a nominal annual rate of i convertible semiannually. The interest payments are reinvested in a separate fund earning interest at an annual effective rate of 6%. During the 15-year period, Bill earns an annual effective yield of 7.56%. Calculate i

6. An investor pays P for an annuity which provides payments of 100 at the beginning of each month for 10 years. These payments are invested at a nominal annual interest rate of 12% convertible monthly. Monthly interest payments are reinvested at a nominal annual interest rate of 6%

convertible monthly. The annual yield rate over the 10-year period is 8% effective. Calculate P

7. Esther invests 100 at the end of each year for 12 years at an annual effective interest rate of i.The interest payments are reinvested at an annual effective rate of 5%. The accumulated value at the end of 12 years is 1748.4. Calculate i

8. A deposit of 1 is made at the end of each year for 30 years into a bank account that pays interest at the end of each year at j per annum. Each interest payment is reinvested to earn an annual effective interest rate of j2. The accumulated value of these interest payments at the end of 30 years is 72.88. Determine j

4.1

1. Seth borrows X for 4 years at an annual effective interest rate of 8%,to be repaid with equal payments at the end of each year. The outstanding loan balance at the end of the second year is 1076.82 and at the end of the third year is 559.12. Calculate the principal repaid in the first payment.

2. Mike borrows X for 10 years at an annual effective rate of 6%. If he pays the principal and accumulated interest in one lump sum at the end

of 10 years, he would pay 356.54 more in interest than if he repaid the loan with 10 level payments at the end of each year. Calculate X

3. A 6000 loan is being repaid with regular payments of X at the end of each year for as long as necessary plus a smaller payment one year after the final regular payment. Immediately after the ninth payment, the outstanding principal is 3 times the size of the regular payment. If the annual interest rate i is 10%, what is the value of X?

4. A loan, at a nominal annual interest rate of 24% convertible monthly, is to be repaid with equal payments at the end of each month for 2n months. The nth payment consists of equal payments of interest and principal. Calculate n

5. A loan is being repaid in 5 annual payments. The first two payments are 200. The third and fourth payments are 400. The final payment is 500. The annual effective interest rate is 6%.Determine the interest portion of the third payment

6. Carla borrowed 100000 on January 1,1996. She will make 10 annual payments of 10000 to the lender beginning on January 1,1997. In addition, she will make monthly payments of amount X to the lender beginning February 1,1996, and continuing for 15 years. You may assume

that the annual effective interest rate is 7.5%. Determine the principal outstanding on January 1,2001, immediately after the two payments due on that date have been made

7. Warren has a loan with an effective interest rate of 5% per annum. He makes payments at the end of each year for 10 years. The first payment is 200, and each subsequent payment increases by 10 per year. Calculate the interest portion in the fifth payment

8. A 30000 bank loan is to be repaid by 30 annual installments of 1000 payable on the first day of each year beginning one year hence. In addition, interest at 4% is payable on the last day of each year on the principal outstanding at the beginning of the year. At time of issue, the bank establishes a value L for the loan, such that the yield rate is 5% annually. Find L

4.2

1. A 10-year loan of 10000 is to be repaid with payments at the end of each year consisting of interest on the loan and a sinking fund deposit. The loan is charged at a 12% annual effective rate. The sinking fund’s annual effective interest rate is 8%. However, beginning in the sixth year, the annual effective interest rate on the sinking fund drops to 6%. As a result, the annual payment to the sinking fund is then increased by X.

Calculate X

2. John borrows 10000 for 10 years at an annual effective interest rate of

i. He accumulates the amount necessary to repay the loan be using a

sinking fund. He makes 10 payments of X at the end of each year, which includes interest on the loan and the payment into the sinking fund, which earns an annual effective rate of 8%. If the annual effective rate of the loan had been 2i, his total annual payment would have been 1.5X. Calculate i

3. Glenda repays a loan of 360000 by making payments of 60000 at the end of each year for 10 years as follow:

1) She replaces the capital be means of a sinking fund

2) She pays interest on the loan at an annual effective rate of 8% The effective annual interest rate earned be the sinking fund is

13i.Calculate i

4. Joe repays a loan of 10000 by establishing a sinking fund and making 20 equal payments at the end of each year. The sinking fund earns 7% effective annually. Immediately after the fifth payment, the yield on the sinking fund increases to 8% effective annually. At that time Joe adjusts his sinking fund payment to X so that the sinking fund will accumulate to

10000 20 years after the original loan date. Determine X

5. Smith borrows 3000 and agrees to establish a sinking fund to repay the loan at the end of 10 years. Interest at 8% on the debt is paid annually as it falls due. Level annual deposits to the sinking fund are made at the end of each year, with interest accumulating at an effective annual rate of 5% for the first 4 years and 3%, thereafter. What is the size of the sinking fund payment?

6. John borrows X and repays the principal by making 10 annual payments at the end of each year into a sinking fund which earns an annual effective rate of 8%. The interest earned on the sinking fund in the 3rd year is 85.57. Calculate X

7. John has borrowed 50000 on which he is paying interest at 17.5% effective per year. He is contributing a constant amount P to a sinking fund at the end of each year. The sinking fund earns an annual effective rate of 9%. His combined payment to both the fund and the loan is 9116.82 annually. Determine the year in which the balance of the sinking fund will be sufficient to repay the loan

8. A corporation borrows 10000 for 25 years, at an effective annual

interest rate of 5%. A sinking fund is used to accumulate the principal by means of 25 annual deposits earning an effective annual interest rate of 4%. Calculate the sum of the net amount of interest paid in the 13th installment and the increment in the sinking fund for the ninth year.

5.1

1. A 1000 par value 20-year bond with annual coupons and redeemable at maturity at 1050 is purchased for P to yield an annual effective rate of 8.25%. The first coupon is 75. Each subsequent coupon is 3% greater than the preceding coupon. Determine P

2. John purchases a 1000 par value 10-year bond with coupons at 8% convertible semiannually which will be redeemed for R. The purchase price is 800 and the present value of the redemption value is 301.51. Calculate R

3. A 1000 par value bond with 9% coupons payable semiannually is purchased for 1300. The yield to the purchaser is 6%, convertible semiannually. If the same bond were redeemable at 120% of par, what price would have been paid to obtain the same yield?

4. A 1000 par value 18-year bond with annual coupons is bought to yield an annual effective rate of 5%. The amount for amortization of premium

in the 10th year is 20. The book value of the bond at the end of 10 is X. Calculate X

5. A 30-year 10000 bond that pays 3% annual coupons matures at par. It is purchased to yield 5% for the first 15 years and 4% thereafter. Calculate the amount for accumulation of discount for year 8.

6. A 1000 20-year 8% bond with semiannual coupons is purchased for 1014. The redemption value is 1000. The coupons are reinvested at a nominal annual rate of 6%, compounded semiannually. Determine the purchaser’s annual effective yield rate over the 20 year period

7. An investor bought a 15-year bond with par value of 100000 and 8% semiannual coupons. The bond is callable at par on any coupon date beginning with the 24th coupon. Find the highest price paid that will yield

??a rate not less than i?10%

2

8. A 1000 par value, 8% bond with quarterly coupons is callable 5 years after issue. The bond matures for 1000 at the end of 10 years and is sold to yield a nominal rate of 6% compounded quarterly under the assumption that the bond will not be called. Calculate the redemption value, at the end of 5 years, that will yield the purchaser the same nominal rate of 6% compounded quarterly.

5.2

1、Consider a $1000 par value two-year 8% bond with semiannual couponsbought to yield 6% convertible semiannually. The price of the bond is computedto be $1037.17. Compute the flat price, accrued interest, and marketprice five months after purchase of the bond.

2、1000 par value 5 year bond with semi-annual coupons of 60 is purchasedto yield 8% convertible semi-annually. Two years and two months after purchase,the bond is sold at the flat price which maintains the yield over the twoyears and two months. Calculate the flat price using the theoretical method.

3、A $100 par value 10 year bond provides 5% semiannual coupons. The yield rate is 4% convertible semiannually. What is the flat price (i.e., the money that actually changes hands if the bond is sold, ignoring expenses) 8.4 years after issue at the same yield rate?

5.3

1、Find the book value immediately after the payment of the 14th coupon of a10-year 1,000 par value bond with semiannual coupons, if r = 0.05 and theyield rate is 12% convertible semiannually.

2、A $1000 bond, redeemable at par on December 1, 1998, with 9%

couponspaid semiannually. The bond is bought on June 1, 1996. Find the purchaseprice and construct a bond amortization schedule if the desired yield is 8%compounded semiannually.

3、A $1000 bond, redeemable at par on December 1, 1998, with 9% couponspaid semiannually. The bond is bought on June 1, 1996. Find the purchaseprice and construct a bond amortization schedule if the desired yield is 10%compounded semiannually.

4、Consider a $1000 par value two-year 8% bond with semiannual couponsbought to yield 6% convertible semiannually. Compute the interest portionearned and the principal adjustment portion of the 3rd coupon payment.

5、Laura buys two bonds at time 0. Bond X is a 1000 par value 14-year bond with 10% annual coupons. It is bought at a price to yield an annual effective rate of 8%. Bond Y is a 14-year par value bond with 6.75% annual coupons and a face amount of F. Laura pays P for the bond to yield an annual effective rate of 8%. During year 6, the write-down in premium (principal adjustment) on bond X is equal to the write-up in discount (principal adjustment) on bond Y. Calculate P.

6、Bryan buys a 2n-year 1000 par value bond with 7.2% annual coupons at a price of P. The price assumes an annual effective yield of 12%. At the end of n years, the book value of the bond, X, is 45.24 greater than the

npurchase price, P. Assume v12%?0.5. Calculate X.

7、Becky buys an n-year 1,000 par value bond with 6.5% annual coupons at a price of 825.44. The price assumes an annual effective yield rate of i. The total write-up in book value of the bond during the first 2 years after purchase is 23.76. Calculate i. (i> 0)

5.4

1、John buys a bond that is due to mature at par in 1 year. It has a 100 par value and coupons at 4% convertible semiannually. John pays 98.51 to obtain a yield rate i convertible semiannually, i>0.Calculate i.

2、A 1000 20-year 8% bond with semiannually coupons is purchased for 1,014.The redemption value is 1,000. The coupons are reinvested at a nominal annual rate of 6%, compounded semiannually. Determine the purchaser’s annual effective yield rate over the 20 year period.

3、You are given a 10-year bond with semiannual coupons, where: i. ii.

The purchase price is 650. The par value is 1,000.

iii. iv.

The redemption value is 1,050. The coupon rate is 12%.

Using the bond salesman’s method, calculate the nominal yield rate, convertible semiannually.

5.5

1、A 15 year 1000 par bond has 7% semiannual coupons and is callable at parafter 10 years. What is the price of the bond to yield 5% for the investor?

2、A 100 par value 4% bond with semi-annual coupons is callable at the followingtimes:

109.00, 5 to 9 years after issue 104.50, 10 to 14 years after issue

100.00, 15 years after issue (maturity date).

What price should an investor pay for the callable bond if they wish to realizea yield rate of

(1) 5% payable semi-annually and (2) 3% payable semi-annually?

3、A $10,000 serial bond is to be redeemed in $1000 installments of principalper half-year over the next five years. Interest at the annual rate of 12%is paid semi-annually on the balance outstanding. How much

should aninvestor pay for this bond in order to produce a yield rate of 8% convertiblesemi-annually?

4、A 5000 serial bond with 10% annual coupons will be redeemed in five equalinstallments of 1100 beginning at the end of the 11th year and continuingthrough the 15th year. The bond was bought at a price P to yield 9% annualeffective. Determine P.

6.1

1. A 10-year $100,000 mortgage will be paid off with level semi-annual amortization payments. Assume that the interest rate on the mortgage is 8%, convertible semiannually, and that payments are made at the end of each half-year. Find the Macaulay duration of this mortgage.

2. (a) Find the Macaulay duration of a ten-year, $1,000 face value, 8% annual coupon bond. Assume an effective annual interest rate of 7%. (b) Find the modified duration for the bond

3. Megan buys the following bonds:

(I) Bond A with a modified duration of 7 for 1000; (II) Bond B with a modified duration of 5 for 2000; and (III) Bond C with a modified duration of 10 for 500. Calculate the modified duration of the portfolio.

4. A perpetuity-immediate with level payments has a duration of 13.5 years at an effective rate of interest i. Determine i.

6.23

1. A client deposits 100,000 in a bank, with the bank agreeing to pay 8% effective for two years. The client indicates that half of the account balance will be withdrawn at the end of the first year. The bank can invest in either one year or two year zero coupon bonds. The one year bonds yield 9% and the two year bonds yield 10%. Develop an investment program based on immunization.

2. For the assets in Example 1, find the modified duration and convexity

3. For a 30-year home mortgage with level payments and an interest rate of 10.2% convertible monthly, find the modified duration and the convexity of the payments.

4. A company must make payments of $10 annually in the form of a 10-year annuity-immediate. It plans to buy two zero coupon bonds to fund these payments. The first bond matures in 2 years and the second bond matures in 9 years, and both are purchased to yield 10% effective. What face amount of each bond should the company buy in order to be immunized

from

small

changes

in

the

interest

rate

(Redingtonimmunization)?

5. A bank agrees to pay 5% compounded annually on a deposit of 100,000 made with the bank. The depositor agrees to leave the funds on deposit on these terms for 8 years. The bank can either buy 4 year zero coupon bonds or preferred stock, both yielding 5% effective. How should the bank apportion its investment in order to immunize itself against interest rate risk?

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