Exercise 6 Find the annual interest rate if the amount after 6 years is 3 times bigger than the initial investment (3 cases).


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1 Exercise 1 At what rate of simple interest will $500 accumulate to $615 in 2.5 years? In how many years will $500 accumulate to $630 at 7.8% simple interest? (9,2%,3 1 3 years) Exercise 2 It is known that $600 invested for two years will earn $264 in interest. Find the accumulated value of $2000 invested at the same rate of compound interest for three years. ($3456) Exercise 3 Put $1 in the bank today compunded annually, semiannually, quarterly, monthly, daily, continuously. Calculate the payoff after 1 year if i = 10%. Exercise 4 An amount of $1500 is deposited in a bank paying an annual interest rate of 4.3%. Find the balance after 6 years taking into consideration 3 kinds of interest. Exercise 5 After how many years the principal amount of $1000 deposited in a bank which pays an annual interest rate of 5% equals $1200 (consider all types of interest). Exercise 6 Find the annual interest rate if the amount after 6 years is 3 times bigger than the initial investment (3 cases). Exercise 7 We borrow $100 and in 4 months we have to pay back $108. Find the annual interest rate. Exercise 8 Find an annual effective interest rate if initial capital is equal to $16000 and the payoff is $17000 after 3 months (3 cases). Exercise 9 Put $1 in the bank today. At what annual percentage rate will you get an effective return of $1.08 in 1 year. Consider annual, semiannual, quarterly, monthly, daily, continuous compounding. Exercise 10 There are 2 banks. Bank A where interest is charged every 6 months and annual interest rate equals 15% and bank B  interest charged quarterly, annual interest rate 15%. Which bank do you prefer to borrow money? Exercise 11 If the effective annual rate of interest is 6%, what is (a) the annual rate of interest convertible halfyearly? (b) the force of interest? (i=5.912%, δ=5.83%) Exercise 12 Calculate the nominal rate of interest convertible once every 4 years that is equivalent to a nominal rate of discount convertible quarterly. Exercise 13 Suppose that a fund initially containing $1000 accumulates with a force of interest δ(t) = 1/ (1 + t), for t > 0. What is the value of the fund after 5 years? Exercise 14 Find the accumulated value of $100 at the end of two years if (a) the nominal annual rate of interest is 6% convertible quarterly (b) the nominal annual rate of discount is 6% convertible once every four years. (F=112.65, F=114.71) 1
2 Exercise 15 Fund A accumulates at a simple interest rate of 10%. Fund B accumulates at a simple discount rate of 5%. Find the point at which the forces of interest on the two funds are equal. Exercise Fund A accumulates at a force of interest t at time t (t 0). Fund B accumulates at a force of interest You are given that the amount in Fund A at time zero is 1,000, the amount in Fund B at time zero is 500, and that the amount in Fund C at any time t is equal to the sum of the amount in Fund A and Fund B. Fund C accumulates at force of interest δ t. Find δ 2. (δ = 2+e0.1 ) 44+20e 0.1 Exercise 17 Gertrude deposits 10,000 in a bank. During the first year the bank credits an annual effective rate of interest i. During the second year the bank credits an annual effective rate of interest (i5%). At the end of two years she has 12, in the bank. What would Gertrude have in the bank at the end of three years if the annual effective rate of interest were (i + 9%) for each of the three years? ( ) Exercise 18 Fund X starts with 1,000 and accumulates with a force of interest δ t = 1 15 t for 0 t < 15. Fund Y starts with 1,000 and accumulates with an interest rate of 8% per annum compounded semiannually for the first three years and an effective interest rate of i per annum thereafter. Fund X equals Fund Y at the end of four years. Calculate i. (0.0777) Exercise 19 Jeff puts 100 into a fund that pays an effective annual rate of discount of 20% for the first two years and a force of interest of rate δ t = 2t, 2 t 4, for the next t 2 +8 two years. At the end of four years, the amount in Jeffs account is the same as what it would have been if he had put 100 into an account paying interest at the nominal rate of i per annum compounded quarterly for four years. Calculate i. (0.2952) Exercise 20 An investor is to pay off an installment loan with 2 payments of $6000, $9000 at the end of 3rd, 6th month, respectively. Compute the present value of the loan assuming 3month i=9%. Exercise 21 A person borrows $ for 15 years at an annual interest rate 5.5%, to be repaid in equal monthly installments. Compute the monthly payment (payments are at the end of month). Exercise 22 Calculate the terminal value of a payment stream $50, $70, $90 made at the end of month 1, 2, 3, respectively. Monthly interest rate equals 3%. (817.08) Exercise 23 Consider a loan of $ for 3 years with payments of $90000, $90000, $ per year (at the end of year). Find an annual interest rate. (215.15) Exercise 24 Consider a loan of $ for 2 years with payments of $60000, $80000 per year (at the end of year). Find an annual interest rate. (50%) Exercise 25 What rate of interest compounded quarterly is required for a deposit of 5000 today to accumulate to 10,000 after 10 years? (0.0699) 2
3 Exercise 26 A loan of $ is to be repaid over 3 periods (at the end) with payments varying from period to period. They are 2 times bigger than the prior ones. An interest rate is 20% per annum. Find installments. (48554) Exercise 27 An investor purchases an investment which will pay 2000 at the end of one year and 5000 at the end of four years. The investor pays 1000 now and agrees to pay X at the end of the third year. If the investor uses an interest rate of 7% compounded annually, what is X? ( ) Exercise 28 A loan requires the borrower to repay 1000 after 1 year, 2000 after 2 years, 3000 after 3 years, and 4000 after 4 years. At what time could the borrower make a single payment of according to the method of equated time? What is the exact time a payment of should be made if the interest rate is 4% effective? (3 years, 2.98 years) Exercise 29 A note that pays 10,000 3 months from now is purchased by an investor for What is the effective annual rate of interest earned by the investor? (0.2277) Exercise 30 A three year certificate of deposit carries an interest rate 7% compounded annually. The certificate has an early withdrawal penalty which, at the investors discretion, is either a reduction in the interest rate to 5% or the loss of 3 months interest. Which option should the investor choose if the deposit is withdrawn after 9 months? After 27 months? (After 9 months  a reduction After 27 months  the loss) Exercise 31 You are given two loans, with each loan to be repaid by a single payment in the future. Each payment includes both principal and interest. The first loan is repaid by a 3,000 payment at the end of four years. The interest is accrued at 10% per annum compounded semiannually. The second loan is repaid by a 4,000 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. ( ) Exercise 32 John borrows $1,000 from Jane at an annual effective rate of interest i. He agrees to pay back $1,000 after six years and $1, after another 6 years. Three years after his first payment, John repays the outstanding balance. What is the amount of Johns second payment? ( (1+i) 3 ) Exercise 33 A loan of 10,000 carries an interest rate of 9%compounded quarterly. Equal loan payments are to be made monthly for 36 months. What is the size of each payment? (317.69) Exercise 34 The present value of a payment stream is $ Monthly interest rate is 0.5%. Payments are at the end of month. (a) Calculate a monthly payment of perpetuity (b) How long do you receive payments of $200? (c) Find the remaining amount of money after 60 payments made monthly of $300? (d) Find the payment such that 3
4 after 120 months of making it, the final value of annuities is aqual to 0. (a) $100 (b) 138 months (c) (d) 222 Exercise 35 Find the final value of an annuity with payments A, A+r, A+2r,..., A+(n1)r starting at time 0 at time 1. Exercise 36 A family wishes to accumulate $50000 in a college education fund at the end of 20 years. If they deposit $1000 in the fund at the end of each of the first 10 years and $1000+X in the fund at the end of each of the second 10 years, find X to the nearest dollar in the fund earned 7% effectively. (651) Exercise 37 The cash price of a new automobile is $ The purchaser is willing to finance the car at 18% convertible monthly and to make payments of $250 at the end of each month for four years. Find the down payment which will be necessary. ( ) Exercise 38 At what annual effective rate of interest is the present value of a series of payments of $1 every six months forever, with the first payment made immediately, equal to $10. (23.46%) Exercise 39 An annuity immediate pays an initial benefit of one per year, increasing by 10.25% every four years. The annuity is payable for 40 years. If the effective interest rate is 5% find an expression for the present value of this annuity. Exercise 40 Humphrey purchases a home with a $100,000 mortgage. Mortgage payments are to be made monthly for 30 years, with the first payment to be made one month from now. The rate of interest is 10%. After 10 years, Humphrey increases the amount of each monthly payment by $325 in order to repay the mortgage more quickly. What amount of interest is paid over the life of the loan? (148908) Exercise 41 On January 1, an insurance company has $100,000 which is due to Linden as a life insurance death benefit. He chooses to receive the benefit annually over a period of 15 years, with the first payment made immediately. The benefit he receives is based on an effective interest rate of 4% per annum. The insurance company earns interest at an effective rate of 5% per annum. Every July 1 the company pays $100 in expenses and taxes to maintain the policy. How much money does the company have remaining after 9 years? ( ) Exercise 42 A loan of 10,000 is to be repaid with equal monthly payments of p. The interest rate for the first year is 1.9%, while the interest rate for the remaining 2 years is 10.9%. What is p? What is the balance after the 6th payment? After the 15th payment? What are the principal and interest components of the 7th payment? Of the 16th payment? (p=303.49, the balance after the 6th payment , tha balance after the 15th payment , the interest portion of the 7th payment is 13.09, the principal portion of the 7th payment is 290.4, of 15th payment is and 251) Exercise 43 The present value of a series of payments of 2 at the end of every eight years, forever, is equal to 5. Calculate the effective rate of interest. (0.0429) 4
5 Exercise 44 An annuity immediate pays an initial benefit of one per year, increasing by 10.25% every four years. The annuity is payable for 40 years. Using an annual effective interest rate of 5%, determine an expression for the present value of this annuity. ( (1 + v 2 )a 20 ) Exercise 45 You are given an annuity immediate with 11 annual payments of 100 and a final balloon payment at the end of 12 years. At an annual effective interest rate of 3.5%, the present value at time 0 of all the payments is 1,000. Using an annual effective interest rate of 1%, calculate the present value at the beginning of the ninth year of all remaining payments.(439.08) Exercise 46 Joan has won a lottery that pays 1,000 per month in the first year, 1,100 per month in the second year, 1,200 per month in the third year, etc. Payments are made at the end of each month for 10 years. Using an effective interest rate of 3% per annum, calculate the present value of this prize. ( ) Exercise 47 Find the price of a $ 1000 par value 10year bond with coupons at 8.4% convertible semiannually which will be redeemed at $ The bond is bought to yield 10% convertible semiannually. ( ) Exercise 48 A $ 1000 bond with a coupon rate of 9% payable semiannually is redeemable after an unspecified number of years at $1125. The bond is bought to yield 10% convertible semiannually. If the present value of the redemption value is $225 at this yield rate, find the purchase price. (945) Exercise 49 A 26 week treasury bill is bought for 9600 at issue and will mature for What is the yield rate as quoted using the convention for treasury bills? What is the yield rate computed normally? (d=0.0791, i=0.085) Exercise 50 Two $1000 bonds redeemable at par at the end of the same period are bought to yield 4% convertible semiannually. One bond costs $ and has a coupon rate of 5% payable semiannually. The other bond has a coupon rate of 2.5% payable semiannually. Find the price of the second bond. (794.83) Exercise 51 An nyear $1000 par value bond matures at par and has a coupon rate of 12% convertible semiannually. It is bought at a price to yield 10% convertible semiannually. If the term of the bond is doubled the price will increase by $50. Find the price of the nth year bond.(1100) Exercise 52 A 1000 bond with coupon rate c convertible semiannually will be redeemed at par in n years. The purchaser price to yield 5% convertible semiannually is P. If the coupon rate were c0.02 the price of the bond would be P Another 1000 bond is redeemable at par at the end of 2n years. It has a coupon rate of 7% convertible semiannually and the yield rate is 5% convertible semiannually. Calculate the price of this second bond. (1375) Exercise 53 A 100 par value 6% bond with semiannual coupons is purchased at 110 to yield a 5
6 nominal rate of 4% convertible semiannually. A similar 3% bond with semiannual coupons is purchased at P to provide the buyer with the same yield. Calculate P. (95) Exercise 54 A 1000 par value 3 year bond with annual coupons of 50 for the first year, 70 for the second year, and 90 for the third year is bought to yield a force of interest δ t = 2t 1 2(t 2 t+1) for t 0. Calculate the price of this bond. (502.4) Exercise 55 Find the price which should be paid for a zero bond which matures for $1000 in 10 years to yield: (a) 10% effective, (b) 9% effective, (c) thus, a 10% reduction in the yield rate causes the price to increase by what percentage? (a , b , c9.56%) Exercise 56 Two $100 par value bonds both with 8% coupon rates payable semiannually are currently selling at par. Bond A matures in 5 years at par, while bond B matures in 10 years at par. If prevailing market rates of interest suddenly go to 10% convertible semiannually, find the percentage change in the price of: Bond A and Bond B.( 7.72%, %) Exercise 57 You are analyzing a unit fiveyear annuitydue. The following yiel curve prevails at time 0: 1 year7%, 2 years8%, 3 years8.75%, 4 years9.25%, 5 years9.5%. Your boss tells you to value the annuity using the above yield curve, but to also assume that every year from now on the entire yield curve shifts down 25 basis points in a parallel fashion. Find the accumulated value of this annuity at time 5. (6.4646) Exercise 58 An investment fund is started with a deposit of 1000 at time zero, with new deposits being made continuously at a rate of t for the next 5 years. The force of interest is δ t = (1 + t) 1. What is the accumulated amount at the end of 5 years? ( ). Exercise 59 Money is invested for 3 years at an interest rate of 4% effective. If inflation is 5% per year over this period, what percentage of purchasing power is lost? (2.84%) Exercise 60 Suppose the one year spot rate is 5% and the two year spot rate is 6%. What is the price of a 1000 par two year bond with 5% annual coupons, using these spot rates? (982.11) Exercise 61 A customer is offered an investment where interest is calculated according to the force of interest δ t = 0.02t for 0 t 3 and δ t = for t > 3. The customer invests 1000 at time t = 0. What nominal rate of interest, compounded quarterly, is earned over the first four year period? ( ) Exercise 62 Seth deposits X in an account today in order to fund his retirement. He would like to receive payments of 50 per year, in real terms, at the end of each year for a total of 12 years, with the first payment occurring seven years from now. The inflation 6
7 rate will be 0.0% for the next six years and 1.2% per annum thereafter. The annual effective rate of return is 6.3%. Calculate X. (306.47) Exercise 63 A 5 year bond with 6% annual coupons has a yield rate of 10% effective and a 5 year bond with 8% annual coupons has a yield rate of 9%effective. What is the 5 year spot rate? (0.1440) Exercise 64 Find the duration and convexity of the following investments assuming the effective annual interest rate is 8%: (a) a 10year zero coupon bond (b) a 10year bond with 8% annual coupons. ((a) 10, (b) 7.25, convexity anologously) Exercise 65 Find the convexity (with respect to interest rate) of a loan repaid with equal installments over n years, at the end of each year, if i=0.((n+1)(n+2)/2) Exercise 66 A loan with interest rate 5% effective will be repaid with payments of at the end of the first year, at the end of the second year and 5000 at the end of the third year. What is the amount of the loan? What is the duration of the loan? ( , 1.81) Exercise 67 Find the duration of a common stock which pays dividends at the end of each year, if it is assumed that each dividend is 4% greater than the prior dividend and the effective interest rate is 8%. Exercise 68 Let A and L denote the present values of asset and liability cash flows with S = A L > 0 being the present value of the surplus. Let D A, D L = 1 L t>0 te it L t and D S = 1 S t>0 te it (At Lt) be the Macaulay durations of the assets, liabilities, and surplus, respectively. Show that SD S = AD A LD L, D S = A S (D A D L ) + D L, D S = D A + L S (D A D L ). Exercise 69 A 9year bond has a yield of 10% and a duration of years. If the market yield changes by 50 basis points, what is the percentage change in the bond s price? (3.27% or 3.27%) Exercise 70 Find the duration of a 6% coupon bond making annual coupon payment if it has 3 years until maturity and has a yield to maturity of 6%. What is the duration if the yield to maturity is 10%? Find the duration of the bond if the coupons are paid semiannually? Exercise 71 Show that the modified duration of a perpetuityimmediate and the present value of the same perpetuity are equal. Exercise 72 An insurance company must make a payment of $19487 in 7 years. The market interest rate is 10%, so the present value of the obligation is $ The company s portfolio manager wishes to fund the obligation using 3year and 11year zerocoupon bonds. How the manager immunize the obligation? 7
8 Exercise 73 An insurance company must make payments to a customer of $10 million in 1 year and $4 million in 5 years. The yield curve is flat at 10%. If it wants to fully fund and immunize its obligation to this customer with a single issue of a zerocoupon bond, what maturity bond must it purchase? What must be the face value and market value of that zerocoupon bond? Exercise 74 Rate of return for a share depends on the state of the economy as the table given below indicates. Find the standard deviation of its rate of return. State of the economy Probability Rate of return (%) Big Boom Small Boom Normal Small Recession Big Recession Exercise 75 Find a correlation coefficient for two shares A, B knowing that State of the economy Probability Rate of return (%) for A Rate of return (%) for B Big Boom Small Boom Normal Small Recession Big Recession Exercise 76 Calculate the expected return and variance of portfolios invested in Tbills and S&P500 index withe weights as follows (0, 1), (0.2, 0.8)(0.4, 0.6), (0.8, 0.2), (1, 0), respectively. Assume that the average annual rate of return on the S&P500 portfolio is averaged about 8.5% more than the Treasury bill return and that the S&P500 standard deviation is about 20% per year. The current Tbill rate is 5% (sure rate of return). Exercise 77 Consider a reisky portfolio. The endofyear cash flow derived from the portfolio be either $70000 or $20000 with equal probabilities of 0.5. The alternative riskfree investment in Tbills pays 6% per year. (a) If you require a risk premium of 8%, how much will you be willing to pay for the portfolio? (b) Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? (c) Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay? Exercise 78 An investor intends to construct a portfolio consisting entirely of the shares of Company A and B. Shortselling is not allowed. The prospects for the shares are as follows: Share A Share B Expected return 10% 20%. Standard deviation 20% 30% (a) Assuming that the objective is the minimisation of standard deviation of return for share B, what proportion of available funds should be invested in A and B 8
9 if the correlation coefficient betweenthe returne of investment A and B is equal to 1? To 0? To 1? (b)assuming that the objective is the maximisation of the function expected return of the portfolio minus its variance, what proportion of available funds should be invested in A and B if the returns of A and B are uncorrelated? 9
SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Interest Theory
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