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1 Ch. 5 Mathematics of Finance 5.1 Compound Interest SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) What is the effective rate that corresponds to a nominal rate of 20% compounded quarterly? 2) How many years will it take for a principal to double at a rate of 10% compounded annually? Give your answer to the nearest year. 3) To what sum will $1000 accumulate if it is invested at 10% compounded annually for one year and then at 10% compounded semiannually for two years? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) An interest rate of 8% compounded semiannually corresponds to an effective rate of A) 8%. B) %. C) %. D) %. E) 12%. 5) A trust fund is to be established by a single payment so that at the end of 15 years, there will be $20,000 in the fund. If the fund earns interest at the rate of 8% compounded semiannually, how much should be deposited initially into the fund? A) $ B) $ C) $ D) $ E) $11, SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) Find the effective rate that corresponds to an interest rate of 5% compounded daily. 7) Over a period of 3 years, an original principal of $1000 accumulated to $1200 in an account where the interest rate was compounded monthly. Determine the rate of interest to two decimal places. 8) At what nominal rate of interest, compounded quarterly, will money double in 10 years? 9) If an initial investment of $4000 grows to $4884 in five years, find the nominal rate of interest, compounded monthly, that was earned by the money. 10) If an initial investment of $4000 grows to $5718 in six years, find the nominal rate of interest, compounded quarterly, that was earned by the money. 11) If an initial investment of $3000 grows to $18,000 in ten years, find the nominal rate of interest, compounded monthly, that was earned by the money. 12) If an initial investment of $3000 grows to $18,000 in five years, find the nominal rate of interest, compounded quarterly, that was earned by the money. 13) Suppose you leave an initial amount of $315 in a savings account for 10 years. If interest is compounded monthly, use a graphing calculator to graph the compound amount S as a function of the nominal rate of interest. Determine the nominal rate of interest so that there is $519 after 10 years. Page 200

2 14) At what nominal rate of interest, compounded monthly, will an investment double in 15 years? 15) At what nominal rate of interest, compounded quarterly, will an investment double in 15 years? 16) At what nominal rate of interest, compounded semiannually, will an investment double in 20 years? 17) At what nominal rate of interest, compounded monthly, will an investment triple in 20 years? 18) Suppose you leave an initial amount of $250 in a savings account for 20 years. If interest is compounded daily (use 365 times per year), use a graphing calculator to graph the compound amount S as a function of the nominal rate of interest. Determine the nominal rate of interest so that the amount doubles after 20 years. 19) Suppose you leave an initial amount of $320 in a savings account for 30 years. If interest is compounded monthly, use a graphing calculator to graph the compound amount S as a function of the nominal rate of interest. Determine the nominal rate of interest so that the amount triples after 30 years. 20) An initial investment of $2600 grows at an annual rate of 7.5% compounded monthly. Find how long it takes for the investment to amount to $ ) An initial investment of $300 grows at an annual rate of 4.5% compounded bimonthly. Find how long it takes for the investment to amount to $ ) An initial investment of $240 grows at an annual rate of 5% compounded quarterly. Find how long it takes for the investment to amount to $ ) An initial investment of $10,000 grows at an annual rate of 3.5% compounded monthly. Find how long it takes for the investment to amount to $14, ) Suppose you invest an initial amount of $10,000 at an annual rate of 8.2% compounded monthly. Use a graphing calculator to graph the compound amount S as a function of the interest periods. Determine how long it takes for the investment to accumulate to $22, ) Suppose an initial investment grows from $2000 to $ over three years. First find the nominal rate compounded monthly and then find the equivalent effective rate. 26) Suppose an initial investment grows from $330 to $600 over five years. First find the nominal rate compounded monthly and then find the equivalent effective rate. 27) Suppose an initial investment grows from $12,000 to $30,000 over ten years. First find the nominal rate compounded quarterly and then find the equivalent effective rate. 28) Suppose an initial investment grows from $220 to $600 over fifteen years. First find the nominal rate compounded monthly and then find the equivalent effective rate. 29) An investment is compounded daily (use 365 times per year). Use a graphing calculator to graph the effective rate, re, as a function of the nominal rate r. Then use the graph to find the nominal rate that is equivalent to an effective rate of 5.4%. Page 201

3 30) The population of a small town is growing at an effective rate of 2.1%. If the current population is 53,000, what will the population be in 8 years? 31) An investment is growing at an effective rate of 12.4%. If the amount invested is currently $12,000, what will the amount be in 6 years? 32) A house worth $150,000 ten years ago has increased in value at an effective rate of 3% due to inflation. Find the current value of the home. 33) A $6000 investment in a stock five years ago grew at an effective rate of 19.6%. Find the current value of the investment. 34) An investment grows from $600 to $642 in one year. If the investment continues to grow at that rate, find the number of years it will take for the investment to double. 35) An investment grows from $10,240 to $10, in one year. If the investment continues to grow at that rate, find the number of years it will take the investment to double. 36) The population of a city grows from 110,000 to 116,600 in one year. If the city continues to grow at that rate, find the number of years it will take for the population to double. 37) An investment grows from $5500 to $6105 in one year. If the investment continues to grow at that rate, find the number of years it will take for the investment to triple. 38) Suppose you can invest $10,000 at 4.5% compounded quarterly or at 4.7% compounded annually. Which is the better choice and how much more per year would you earn? 39) Suppose you can invest $6000 at 6.2% compounded monthly or at 6.5% compounded semiannually. Which is the better choice and how much more per year would you earn? 40) You have a choice of two banks. One bank pays interest at 4.66% compounded 360 times a year and the other bank pays interest at 4.65% compounded 365 times a year. Which is the better choice? 41) You have a choice of two banks. One bank pays interest at 5.54% compounded monthly and the other bank pays interest at 5.53% compounded daily (365 times a year). Which is the better choice? How much more would you make in one year if you deposited $1000? 42) You have two investment opportunities. You can invest $6000 at 12% compounded monthly or you can invest $6100 at 12.1% compounded quarterly. Which has the better effective rate of interest? Use a graphing calculator to graph both amounts as a function of time in years. Which is the better investment over twenty years? 43) If $1,000 is invested at a nominal rate of 4% compounded quarterly for 5 years, find the compound amount. 44) If $2,575 is invested at an A.P.R. of 7% compounded semiannually for 15 years, find the accumulated amount. 45) If $4,200 is invested at an annual rate of 5.4% compounded monthly for 10 years, find the compound amount. Page 202

4 46) If $3,100 is invested at a nominal rate of 6% compounded quarterly for 7 years, find a) the compound amount, and b) the compound interest. 47) If $14,300 is invested at an A.P.R. of 8.25% compounded semiannually for 3 years, find a) the compound amount, and b) the compound interest. 48) If $10,000 is invested at an effective rate of 4.25% for 8 years, what is the accumulated amount? 49) If $5,600 is invested at an effective rate of 2.1% for 17 years, what is the compound amount? 5.2 Present Value SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Determine the present value of $4000 due in 5 years if the interest rate is 10% compounded semiannually. 2) A debt of $2000 due four years from now is to be repaid by a payment of $1000 now and a second payment at the end of two years. How much should the second payment be if the interest rate is 5% compounded annually? 3) A person has the option of satisfying a debt by either paying $5000 now and $5000 in two years, or by paying $3000 now, $3000 a year from now, and a final payment of x dollars two years from now. Determine an equation of value that corresponds to the value of all payments at the end of two years. It is not necessary to solve the equation. Assume that interest is at the rate of 10% compounded semiannually. 4) For an initial investment of $10,000, suppose a company guarantees the following cash flows at the end of the indicated years: Year Cash Flow 1 $ $8000 Assume an interest rate of 5% compounded annually. (a) Determine the net present value of the cash flows. (b) Is the investment profitable? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) A debt of $2000 due in one year is to be repaid by a payment due two years from now and a final payment of $1000 three years from now. If the interest is at the rate of 4% compounded annually, then the payment due in two years is A) $ B) $ C) $ D) $ E) $ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) To purchase land for an industrial site, a company agrees to pay $20,000 down and $10,000 at the end of every six-month period for 10 years. If the interest rate is 10% compounded semiannually, what is the corresponding cash value of the land? 7) Find the present value of $5000 due in 3 years if the interest rate is 6 3 % compounded monthly. 4 Page 203

5 8) A bank pays 4% annual interest compounded quarterly. How large a deposit must be made now in order that the account contains $1500 at the end of 3 years? 9) Find the present value of $3000 due after five years if the interest rate is 9.6% compounded semiannually. 10) Find the present value of $300 due after six years if the interest rate is 5.4% compounded monthly. 11) How much must be invested at an interest rate of 7.25% compounded quarterly to have $10,000 in two years? 12) How much must be invested at an interest rate of 9.6% compounded monthly to have $3000 in five years? 13) A trust fund for a childʹs education is being set up by a single payment so that at the end of 17 years there will be $31,000. If the fund earns interest at the rate of 8.25% compounded monthly, how much money should be paid into the fund initially? 14) A trust fund for a 12-year-old child is being set up by a single payment so that when the child is 21 there will be $24,000. If the fund earns interest at the rate of 7.25% compounded quarterly, how much money should be paid into the fund initially? 15) A trust fund for a newborn is being set up by a single payment so that at the end of 18 years there will be $34,000. If the fund earns interest at the rate of 6.25% compounded monthly, how much money should be paid into the fund initially? 16) A trust fund for a 8-year-old child is being set up by a single payment so that when the child is 20 there will be $12,000. If the fund earns interest at the rate of 6.5% compounded semiannually, how much money should be paid into the fund initially? 17) A debt of $12,000, which is due 10 years from now, is instead to be paid off by four payments: $3000 now, $2000 in 3 years, $2000 in 6 years, and a final payment at the end of 8 years. What would this payment be if an interest rate of 5.5% compounded semiannually is assumed? 18) Suppose Mr. Takegawa owes Ms. Perez three sums of money: $1000 due in 2 years, $1500 due in 5 years, and $2000 due in 8 years. Suppose he would rather pay her $2000 now and the rest in 3 years. If the interest rate is 6% compounded annually, how much will he owe in 3 years? 19) A debt of $600 is due 3 years from now and $800 due 5 years from now, is instead to be paid off by two payments: $500 now and a final payment at the end of 6 years. What would this payment be if an interest rate of 6% compounded quarterly is assumed? 20) A debt of $1000 due 4 years from now and $1500 due 6 years from now, is instead to be paid off by a single payment 5 years from now. How much is the payment if an interest rate of 8.4% compounded monthly is assumed? 21) Miguel has the opportunity to invest $3000 in a friendʹs business such that he will be repaid $4500 in six years. On the other hand, he can put the $3000 in a savings account that pays 5.5% compounded quarterly. Which investment is better? Page 204

6 22) Miguel has the opportunity to invest $3000 in a friendʹs business such that he will be repaid $4500 in six years. If he instead wanted to put the $3000 into a savings account, what interest rate compounded monthly would be needed to equal the investment? 23) Suppose you have the opportunity to invest $6000 in a business venture such that you will be repaid $8000 in five years. On the other hand, you can put the $6000 in a savings account that pays 5.25% compounded monthly. Which investment is better? 24) Suppose you have the opportunity to invest $1000 in a business such that the value of your investment after seven years will be $1500. On the other hand, you can put the $1000 into a certificate of deposit that pays 6% compounded monthly. Which is better? 25) Suppose that you can invest $10,000 in a business that guarantees you the following cash flows: $5000 at the end of 2 years, $4000 at the end of 4 years, and $3000 at the end of 6 years. Assuming an interest rate of 7.25% compounded annually, find the net present value of the cash flows. Is the investment profitable? 26) Suppose that you can invest $11,000 in a business that guarantees you the following cash flows: $5500 at the end of 2 years, $4500 at the end of 4 years, and $4000 at the end of 5 years. Assuming an interest rate of 6.25% compounded annually, find the net present value of the cash flows. Is the investment profitable? 27) Suppose that Tori can invest $13,000 in a business that guarantees her the following cash flows: $6000 at the end of 2 years, $5000 at the end of 4 years, and $4000 at the end of 6 years. Assuming an interest rate of 6% compounded monthly, find the present value of the cash flows. Is the investment profitable? 28) Suppose that you can invest $5000 in a business that guarantees you the following cash flows: $3000 at the end of 2 years, $2000 at the end of 4 years, and $2000 at the end of 6 years. Assuming an interest rate of 6% compounded monthly, find the present value of the cash flows. Is the investment profitable? 29) Suppose that you can invest $5000 in a business that guarantees you the following cash flows: $3000 at the end of 2 years, $2000 at the end of 4 years, and $1500 at the end of 6 years. Use a graphing calculator to graph the net present value as a function of the interest rate compounded annually. Determine the interest rate for which the investment is profitable. 30) Suppose that you can invest $8000 in a business that guarantees you the following cash flows: $4000 at the end of 2 years, $3500 at the end of 4 years, and $2000 at the end of 7 years. Use a graphing calculator to graph the net present value as a function of the interest rate compounded annually. Determine the interest rates for which the investment is profitable. 31) Find the present value of a future value of $5,000 due in 7 years at a nominal rate of 3% compounded quarterly. 32) Find the present value of a future value of $325 due in 10 years at an A.P.R. of 5.3% compounded semiannually. 33) Find the present value of a future value of $4,200 due in 11 years at a nominal rate of 6% compounded monthly. 34) Find the present value of a future value of $1,750 due in 9 years at an effective rate of 3.65%. 35) Find the present value of a future value of $4,650 due in 18 months at an effective rate of 4.55%. Page 205

7 5.3 Interest Compounded Continuously SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) If $1000 is deposited into a savings account that earns interest at an annual rate of 6% compounded continuously, find the value of the account at the end of seven years. Give your answer to the nearest dollar. 2) If $200 is deposited into a savings account that earns interest at an annual rate of 8% compounded continuously, find the value of the account at the end of two years. 3) Determine the effective rate equivalent to an annual rate of 8% compounded continuously. 4) Determine the effective rate equivalent to an annual rate of 10% compounded continuously. 5) At an annual rate of 4% compounded continuously, in how many years would it take for a principal to double? 6) At an annual rate of 8% compounded continuously, in how many years would it take for a principal to double? 7) In five years a company will purchase equipment costing $100,000. The company decides to place a single deposit into a savings account now so that its future value will equal the cost of the equipment. If the account earns interest at an annual rate of 10% compounded continuously, determine the deposit to the nearest dollar. 8) A trust fund is to be set up by a single payment so that at the end of 10 years there will be $1,000,000 in the fund. If interest is compounded continuously at an annual rate of 9%, to the nearest dollar, how much money should be paid into the fund initially? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 9) If an investment of $20,000 earns interest at an annual rate of 9% compounded continuously, then the value (in dollars) of the investment six years from now is A) 20,000(1.09)6 B) 20,000(1.09)-6 C) 20,000e0.54 D) 20,000e-0.54 e0.54 E) 20,000 10) If an investment of $12,000 earns interest at an annual rate of 7% compounded continuously, then the value (in dollars) of the investment ten years from now is A) 12,000(1.07)-10 B) 12,000(1.07)10 C) 12, D) 12,000e0.7 e0.7 E) 12,000 Page 206

8 11) If money earns interest at an annual rate of 8% compounded continuously, then the value (in dollars) of $10,000 due at the end of five years is A) 10,000e-0.4 B) 10,000e0.4 e0.4 C) 10,000 D) 10,000(1.08)5 E) 10,000(1.08)-5 12) At an annual rate of 10% compounded continuously, the number of years in which a principal triples is A) ln ln B). C). D) 0.10 ln 3 3 ln E) e SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 13) Determine the effective rate equivalent to an annual rate of 7 3 % compounded continuously. 4 14) Given that the effective rate is 9%, determine the interest rate r which is compounded continuously to give an effective rate of 9%. 15) If a person deposits $1000 in a savings account that pays an interest rate of r% compounded continuously, and the account has $1400 at the end of 4 years, find the interest rate. 16) A person deposits $1000 in a savings account that pays an interest rate of 4 3 % compounded continuously. 4 Find the balance in the account at the end of years. 17) If $100 is invested at a rate of 5% compounded continuously, the amount in the account is given by: S = 100e0.05t. If the same principal is invested at an account earning 5% compounded semiannually, the amount is given by: S = x. Consider the difference in these two investments by graphing both functions on your graphing calculator and looking at the years 5 through 7. (Use the window 5, 7 128,142.) What do you notice about the two graphs? 18) If $100 is invested at a rate of 6% compounded continuously, the amount in the account is given by: S = 100e0.06t. If the principal is invested at an account earning 6% compounded monthly, the amount is given by: S = x. Consider the difference in these two investments by graphing both functions on your graphing calculator and looking at the years 5 through 6. (Use the window 5, 6 135,141.) What do you notice about the two graphs? 19) The function that gives the effective rate that corresponds to an annual rate of x interest compounded continuously is y = ex - 1. Graph this on your graphing calculator in the window 0,0.5 0,1 and discuss what the behavior means. Page 207

9 20) Suppose that you want to invest some money in order to have $100 available at some later time. If you invest it at 7% interest compounded continuously, the amount you need to invest now, P, is related to the number of years from now that you need the money, t, by: P(t) = 100e-0.07t. Graph this on your graphing calculator in the window 0,20 0,100 Discuss the behavior of this graph. 21) Suppose that you want to invest some money in order to have $1000 available at some later time. If you invest it at 7% interest compounded continuously, the amount you need to invest now, P, is related to the number of years from now that you need the money, t, by: P(t) = 1000e-0.07t. Graph this on your graphing calculator in the window 0,20 0,1000. Discuss the behavior of this graph. 5.4 Annuities MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the first five terms of the geometric sequence. 1) a1 = 4, r = 2 A) 4, 8, 16, 32, 64 B) 8, 16, 32, 64, 128 C) 4, 6, 8, 10, 12 D) 2, 8, 32, 128, 512 2) a1 = 5, r = 1 4 A) 5, 5 4, 5 16, 5 64, C) 5 4, 5 16, 5 64, 5 256, B) 5, 20, 80, 320, 1280 D) 5, 21 4, 11 2, 23 4, 6 3) a1 = 7, r = -3 A) 7, -21, 63, -189, 567 B) 7, 21, 63, -189, 567 C) -3, -21, 63, -189, 567 D) 7, 4, 1, -2, -5 Find the value. a 4) A) B) C) D) ) a A) B) C) D) ) a A) B) C) D) ) s A) B) C) D) Page 208

10 8) s A) B) C) D) ) s A) B) C) D) ) s A) B) C) D) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 11) Suppose a person deposits $1000 in a savings account at the end of every six months. What is the value of the account at the end of five years if interest is at a rate of 10% compounded semiannually? 12) To purchase land for an industrial site, a company agrees to pay $20,000 down and $10,000 at the end of every six-month period for 10 years. If the interest rate is 10% compounded semiannually, what is the corresponding cash value of the land? 13) A person establishes the following retirement plan: an immediate deposit of $10,000 and quarterly payments of $1,500 at the end of each quarter into a savings account that earns 5% compounded quarterly, what is the amount of the investment after 21 years? 14) Suppose an annuity due consists of 6 yearly payments of $200 and the interest rate is 5% compounded annually. Determine (a) the present value and (b) the future value at the end of 6 years. 15) Suppose a corporation pays $50,000 for a machine that has a useful life of eight years and a salvage value of $5000. A sinking fund is established to replace the machine at the end of 8 years. The replacement machine will cost $70,000. If equal payments are made into the fund at the end of every 6 months and the fund earns interest at the rate of 10% compounded semiannually, what should each payment be? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 16) Suppose $500 is initially placed in a savings account that earns interest at the rate of 8% compounded semiannually. Thereafter, $500 is deposited in the account at the end of every six months for five years. The value of the account at the end of five years is A) $ B) $ C) $ D) $ E) $ ) Suppose a person invests $20,000 in a business venture that guarantees the same cash flow at the end of every quarter for four years. If the investment earns interest at the rate of 16% compounded quarterly, then each cash flow is A) $ B) $ C) $ D) $ E) $ Page 209

11 18) Consider the following annuity: $2000 due at the end of each year for two years, and $3000 due thereafter at the end of each year for three years. At an interest rate of 4% compounded annually, the present value of the annuity is A) $12, B) $11, C) $10, D) $10, E) $9, ) Suppose a company establishes a sinking fund to replace equipment that has a salvage value of $50,000. The company deposits $20,000 into the fund at the end of every six months. If interest is earned at the rate of 8% compounded semiannually, the value of the fund at the end of six years is A) $137, B) $187, C) $237, D) $250, E) $300, SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 20) Find the sum of the geometric series: ) Find the sum of the geometric series: ) Find the present value of an annuity of $200 per month for years at an interest rate of 7% compounded monthly. 23) Find the amount of an annuity of payments of $150 at the end of every month for 3 years at the rate of 4% compounded monthly. Also find the compound interest. 24) A rubber ball always bounces back 2 3 of its previous height. If the ball is thrown up to a height of 30 feet, give the first five heights of the ball. 25) A company earns a profit of $2000 in its first month. Suppose its profit increases by 10% each month for two years. Find the amount of profit the company earns in its sixth and sixteenth months. 26) A company repays a $50,000 loan by paying 10% of the outstanding loan each month. Find the amount the company pays in the fourth and twentieth months. 27) $200 is invested at the rate of 6% compounded monthly for 5 months. List the compound amounts at the end of each month as a geometric sequence. 28) $200 is invested at the rate of 4.5% compounded semiannually for 8 years. Find the compound amounts at the end of the 2nd, 4th, and 8th years. 29) $200 is invested at the rate of 3.5% compounded quarterly for six and a half years. How many terms are in the geometric sequence formed by the amounts at the end of each quarter? 30) A ball rebounds 3 4 of its previous height after each bounce. If the ball is tossed up to a height of 16 feet, how far has it traveled in the air when it hits the ground for the fifteenth time? Page 210

12 31) A ball rebounds 2 of its previous height after each bounce. If the ball is dropped from a height of 27 feet, how 3 far has it traveled in the air when it hits the ground for the twentieth time? 32) A ball rebounds 3 5 of its previous height after each bounce. If the ball is dropped from a height of 20 meters, how far has it traveled in the air when it hits the ground for the sixth time? 33) A company earns a profit of $5000 in its first month. Suppose its profit decreases by 10% each month for one year. Find the amount of profit the company earns in its first year. 34) A company repays a $50,000 loan by paying 10% of the outstanding loan each month. How much has the company paid back after two years? 35) A company repays a $40,000 loan by paying 20% of the outstanding loan every four months for five years and then pays off the rest. How much was the companyʹs final payment? 36) What is the present value of an annuity of $300 per quarter for five years at an interest rate of 4.5% compounded quarterly? 37) What is the present value of an annuity of $1000 per month for ten years at an interest rate of 6.3% compounded monthly? 38) After putting $20,000 down on a piece of property, a man began paying $500 a month for ten years. Given an interest rate of 8% compounded monthly, how much would the property cost if the man had paid for it in cash? 39) After putting $10,000 down on a piece of property, a woman began paying $2500 a quarter for nine years. Given an interest rate of 7.75% compounded quarterly, how much would the property cost if the woman had paid for it in cash? 40) Given a payment of $800 per quarter for five years, use a graphing calculator to graph the present value A as a function of the interest rate per quarter r. Determine the nominal interest rate if the present value of the annuity is $14, ) Given an interest rate of 7.25% compounded monthly, find the present value of the following annuity: $700 at the end of each month for three years and $900 at the end of each month for five more years. 42) Given an interest rate of 4.6% compounded semiannually, find the present value of the following annuity: $2100 at the end of every six months for six years and $3000 at the end of every six months for four more years. 43) Suppose a woman purchases a building with an initial down payment of $40,000, and then makes monthly payments: $1500 at the end of each month for four years and $2000 at the end of each month for six more years. Given an interest rate of 5.5% compounded monthly, find the present value of the payments and the list price of the building. (Round your answer to the nearest dollar.) 44) If $30,000 is used to purchase an annuity consisting of equal payments at the end of each quarter for the next 5 years and the interest rate is 8% compounded quarterly, find the amount of each payment. Page 211

13 45) If $5000 is used to purchase an annuity consisting of equal payments at the end of each month for the next years and the interest rate is 6% compounded monthly, find the amount of each payment. 46) If $25,000 is used to purchase an annuity consisting of equal payments at the end of each year for the next 8 years and the interest rate is 5% compounded annually, find the amount of each payment. 47) If $12,000 is used to purchase an annuity consisting of equal payments at the end of every six months for the next 7 years and the interest rate is 6.2% compounded semiannually, find the amount of each payment. 48) Given an annuity with equal payments at the end of each month for six years and an interest rate of 5.3% compounded monthly, use a graphing calculator to graph the present value A as a function of the monthly payment R. Determine the monthly payment if the present value of the annuity is $30, ) The premiums on an insurance policy are $20 a month, payable at the beginning of each month. If the policy holder wishes to pay 1 yearʹs premiums in advance, how much should be paid provided that the interest rate is 5.1% compounded monthly? 50) The premiums on an insurance policy are $80 every six months, payable at the beginning of each six -month period. If the policy holder wishes to pay 1 yearʹs premiums in advance, how much should be paid provided that the interest rate is 4.3% compounded semiannually? 51) A woman makes house payments of $4200 at the beginning of every quarter. If the woman wishes to pay yearʹs worth of payments in advance, how much should she pay provided that the interest rate is 5.4% compounded quarterly? 52) Find the amount of an annuity consisting of payments of $100 paid at the end of each quarter for five years at the rate of 5% compounded quarterly. Also find the compound interest. 53) Suppose you deposit $200 at the end of every month into a bank account that pays 6% compounded monthly. After six years, how much will you have? 54) Suppose Lena deposits $500 at the end of every month into a bank account that pays 5.4% compounded monthly. After five years, how much will she have? 55) Find the future value of an annuity due consisting of payments of $100 paid at the beginning of each quarter for five years at the rate of 5% compounded quarterly. 56) Suppose you deposit $200 at the beginning of every month into a bank account that pays 6% compounded monthly. After six years, how much will you have? 57) Suppose Lena deposits $500 at the beginning of every month into a bank account that pays 5.4% compounded monthly. After five years, how much will she have? Page 212

14 58) Suppose a truck costing $36,000 is to be replaced at the end of 10 years, at which time it will have a resale value of $12,000. In order to provide money at the time for a new truck costing $40,000, a sinking fund is set up into which equal payments are placed at the end of every six months. If the fund earns 6% compounded semiannually, what should each payment be? 59) Suppose a machine costing $12,000 is to be replaced at the end of 7 years, at which time it will have a salvage value of $6000. In order to provide money at that time for a new machine costing $15,000, a sinking fund is set up into which equal payments are placed at the end of every quarter. If the fund earns 5.6% compounded quarterly, what should each payment be? 60) In order to establish a sinking fund of $125,000, how much will have to be invested at the end of each year at the rate of 11.2% compounded annually for 8 years? 61) Suppose the Laus wish to save $36,000 for a down payment in three years. If they make payments at the end of every month into an account paying 7.5% compounded monthly, what size payments should they make? 62) Suppose a machine will yield a net of $200 per month for 5 years, after which the machine would be worthless. How much should the firm pay for the machine if it wants to earn 5% annually on its investment and also set up a sinking fund to replace the purchase price? For the fund, assume monthly payments and a rate of 4% annually. 63) Suppose you wish to purchase a mine that will yield an annual return of $32,000 for 12 years, after which the mine will have no value. You want to earn 8% annually on this investment and also set up a sinking fund to replace the purchase price. If money is placed in the fund at the end of each year and earns 6.2% compounded annually, how much should you pay for the mine? 64) Suppose a diagnostic machine will yield a net of $1000 per quarter for 5 years, after which the machine can be sold for $1000. How much should a firm pay for the machine if it wants to earn 7.5% annually on its investment and also set up a sinking fund to replace the purchase price? For the fund, assume quarterly payments and a rate of 5.5% compounded quarterly. 65) Suppose you wish to purchase a factory that will yield an annual return of $12,000 for 12 years, after which the factory will have no value. You want to earn 8.25% annually on your investment and also set up a sinking fund to replace the purchase price. If money is placed in the fund at the end of each year and earns 4.2% compounded annually, how much should you pay for the factory? 66) Find the Present Value of an ordinary annuity with semiannual payments of $450 for 30 years at 6% compounded semiannually. 67) Find the Present Value of an ordinary annuity with quarterly payments of $250 for 40 years at 7.25% compounded quarterly. 68) Find the Present Value of an ordinary annuity with monthly payments of $125 for 10 years at 4.5% compounded monthly. 69) Find the Present Value of an annuity due with semiannual payments of $350 for 35 years at 6.25% compounded semiannually. Page 213

15 70) Find the Present Value of an annuity due with quarterly payments of $2,750 for 27 years at 7.8% compounded quarterly. 71) Find the Present Value of an annuity due with monthly payments of $200 for 17 years at 5.25% compounded monthly. 72) Find the Future Value of an ordinary annuity with semiannual payments of $475 for 30 years at 6.15% compounded semiannually. 73) Find the Future Value of an ordinary annuity with quarterly payments of $300 for 30 years at 7.5% compounded quarterly. 74) Find the Future Value of an ordinary annuity with monthly payments of $250 for 40 years at 6.8% compounded monthly. 75) Find the Future Value of an annuity due with semiannual payments of $6,250 for 21 years at 4.25% compounded semiannually. 76) Find the Future Value of an annuity due with quarterly payments of $550 for 32 years at 3.75% compounded quarterly. 77) Find the Future Value of an annuity due with monthly payments of $250 for 26 years at 6% compounded monthly. 78) After graduating from college and gaining employment, you decide to open an IRA to help save for retirement. If the IRA earns 6.15% compounded monthly and you deposit $250 into this IRA at the end of each month, what will be the amount you have when you retire 43 years later? 79) You wish to be a millionaire when you retire. To accomplish this you decide to open up an IRA and make equal monthly payments at the end of each month. If the IRA earns 7.5% compounded monthly and you know you will retire in 40 years, what must be the monthly payment to obtain your $1,000,000? 5.5 Amortization of Loans SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) A $5000 loan is to be repaid over three years by equal payments due at the end of every quarter. If interest is at the rate of 20% compounded quarterly, determine (a) the quarterly payment and (b) the total interest paid. 2) A 20-year loan for $100,000 is to be amortized by equal semiannual payments. If interest is at the nominal rate of 10% compounded semiannually, find (a) the semiannual payment; (b) the interest in the first payment; (c) the principal repaid in the first payment. Page 214

16 3) A debt of $600 is to be repaid by two equal yearly payments with interest at the rate of 5% compounded annually. Complete the following amortization schedule for this debt. Principal outstanding at Interest Payment at end Principal repaid Period beginning of period for period of period at end of period 1 2 Totals MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) An $800 loan is amortized by equal quarterly payments over two years. If interest is at the rate of 16% compounded quarterly, then the quarterly payment is A) $ B) $ C) $ D) $ E) $ ) A $10,000 loan is amortized by equal semiannual payments over 5 years. If the interest rate is 8% compounded semiannually, then the principal repaid in the first payment is A) $ B) $ C) $ D) $ E) $ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A person purchases a home for $130,000, makes a down payment of $30,000. Find the monthly payment if the person takes a loan for 25 years with an interest rate of 8% compounded monthly. 7) A person purchases a home for $130,000, makes a down payment of $30,000. Find the monthly payment if the person takes a loan for 15 years with an interest rate of 8% compounded monthly. 8) A person purchased a television set for $850 and agreed to pay it off by monthly payments of $50. If the store charges an interest rate of 9% compounded monthly, how many months will it take to pay off the debt? 9) A person amortizes a loan of $180,000 for a new home by obtaining a 30 -year mortgage at the rate of 8.7% compounded monthly. Find (a) the monthly payment, (b) the total interest charges, and (c) the principal remaining after 10 years. 10) The Krishnans amortize a loan of $150,000 for a new home by obtaining a 40-year mortgage at the rate of 10.2% compounded monthly. Find (a) the monthly payment, (b) the total interest charges, and (c) the principal remaining after 15 years. 11) Mary amortizes a loan of $80,000 for a new home by obtaining a 15 -year mortgage at the rate of 9.9% compounded monthly. Find (a) the monthly payment, (b) the total interest charges, and (c) the principal remaining after 8 years. 12) James amortizes a loan of $210,000 for a new home by obtaining a 40 -year mortgage at the rate of 7.5% compounded monthly. Find (a) the monthly payment, (b) the total interest charges, and (c) the principal remaining after 15 years. 13) A man bought a stereo system for $3500 and agreed to pay off the loan by making monthly payments of $79. If the store charges an interest rate of 11% compounded monthly, how many months will it take to pay off the debt? Page 215

17 14) Cyndi bought a multimedia home computer system for $4500 and agreed to pay off the loan by making monthly payments of $109. If the store charges an interest rate of 9.7% compounded monthly, how many months will it take to pay off the debt? 15) A couple bought a motorboat for $11,000 and agreed to pay off the loan by making monthly payments of $559. If the dealer charges an interest rate of 8.9% compounded monthly, how many months will it take to pay off the debt? 16) A man purchased a new laser printer for his computer for $1100 and agreed to pay off the loan by making monthly payments of $59. If the store charges an interest rate of 12.2% compounded monthly, how many months will it take to pay off the debt? 17) At a car dealership, you are given two options for financing a new car worth $16,000. Option 1: You can take out a 5 year loan at 0% A.P.R. compounded monthly, or Option 2: You can get $3,000 cash back and finance the rest at 4.9% A.P.R. compounded monthly for 5 years. Which is the better option for you? Page 216

18 Ch. 5 Mathematics of Finance Answer Key 5.1 Compound Interest 1) % 2) 7 3) $ ) B 5) A 6) % 7) 6.09% 8) 6.99% 9) 4% 10) 6% 11) % 12) 37.49% 13) 5% 14) 4.63% 15) 4.65% 16) 3.5% 17) 5.51% 18) 3.47% 19) 3.67% 20) 4 years 21) 9 years 22) years 23) years 24) 10 years 25) %, 12.1% 26) %, 12.7% 27) %, % 28) %, % 29) % 30) 62,587 31) $24,198 32) $201,587 33) $14,683 34) years 35) years 36) years 37) years 38) 4.7% annually; $ ) 6.5% semiannually; $ ) 4.66% compounded 360 times a year 41) 5.53% daily. $ ) The first rate is better; the second investment earns more. 43) The compound amount is $1, ) The accumulated amount is $7, ) The compound amount is $7, ) a) The compound amount is $4, b) The compound interest is $1, Page 217

19 47) a) The compound amount is $18, b) The compound interest is $3, ) The accumulated amount is $13, ) The compound amount is $7, Present Value 1) $ ) $ ) 5000(1.05) = 3000(1.05) (1.05)2 + x 4) (a) $ (b) yes 5) B 6) $144, ) $ ) $ ) $ ) $ ) $ ) $ ) $ ) $12, ) $11, ) $ ) $ ) $ ) $ ) $ ) the friendʹs business 22) 6.78% 23) the business venture 24) the certificate of deposit 25) -$658.68; no 26) $357.00; yes 27) -$948.19; no 28) $632.36; yes 29) less than 7.918% 30) less than 4.742% 31) The present value is $4, ) The present value is $ ) The present value is $2, ) The present value is $1, ) The present value is $4, Interest Compounded Continuously 1) $1522 2) $ ) 8.33% 4) 10.52% 5) ) 8.7 7) $60,653 8) $406,570 9) C 10) D 11) A 12) A 13) % Page 218

20 14) % 15) % 16) $ ) The amount in the continuously compounded account is only slightly more that the account compounded semiannually, and although the difference is getting bigger, it is doing so very slowly. 18) The amount in the continuously compounded account is only slightly more that the account compounded monthly, and although the difference is getting bigger, it is doing so very slowly. 19) The graph looks roughly linear, which means that the effective rate increases about the same amount per incremental change in the annual rate regardless of what the initial annual rate is. 20) The graph shows that the earlier you invest the money, the less you need to invest. 21) The graph shows that the earlier you invest the money, the less you need to invest. 5.4 Annuities 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) $12, ) $144, ) The amount of the investment is $294, after 21 yrs. 14) (a) $ (b) $ ) $ ) D 17) C 18) B 19) E 20) ) ) $13, ) $5,727.23; $ ) 30 ft, 20 ft, ft, ft, ft 25) $ , $ ) $3645, $ ) $201, $202, $203.02, $204.03, $ ) $218.62, $238.97, $ ) 26 30) ft 31) ft 32) m 33) $35, ) $46,012 35) $ ) $ ) $88, ) $61, ) $74, ) 5.23% 41) $58, Page 219

21 42) $38, ) $162,788; $202,788 44) $ ) $ ) $ ) $ ) $ ) $ ) $ ) $24, ) $ ; $ ) $17, ) $34, ) $ ) $17, ) $34, ) $ ) $ ) $10, ) $ ) $10,390 63) $230,900 64) $21,689 65) $80,921 66) $12, ) $13, ) $12, ) $ ) $125, ) $27, ) $79, ) $132, ) $620, ) $426, ) $136, ) $187, ) $633, ) $ Amortization of Loans 1) (a) $ (b) $ ) (a) $ (b) $ (c) $ ) Principal outstanding at Interest Payment at end Principal repaid Period beginning of period for period of period at end of period Totals ) C 5) D 6) $ ) $ ) months, approximately Page 220

22 9) (a) $ (b) $327, (c) $160, ) (a) $ (b) $472, (c) $140, ) (a) $ (b) $73,864 (c) $51, ) (a) $ (b) $453,336 (c) $149, ) 58 months 14) 51 months 15) 22 months 16) 21 months. 17) Take the $3,000 cash back and the 4.9% A.P.R. loan Page 221

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