finance math ex

Chapter 3: Finance Mathematics

ExerciseBy Dr. Pham Huu Anh Ngoc

Annuity

Problem 1: Find the value of each ordinary annuity based on theinformation given.(a) R = 9200, 16% interest compounded semiannually for 7 years.(b) R = 800, 12% interest compounded semiannually for 12 years.

Hint: Using the following

Theorem

Amount of an annuity: The amount S of an annuity of payments of Rdollars each, made at the end of each period for n consecutive interestperiods at a rate of interest i per period, is given by

S = R

[(1 + i)n − 1

i

].

Solution (a) $ 222,777.27 (R = 9200, i = 0.08, n = 14)(b) $ 40,652.46

Exercise By Dr. Pham Huu Anh Ngoc


Annuity

Problem 1: Find the value of each ordinary annuity based on theinformation given.(a) R = 9200, 16% interest compounded semiannually for 7 years.(b) R = 800, 12% interest compounded semiannually for 12 years.Hint: Using the following

Theorem

Amount of an annuity: The amount S of an annuity of payments of Rdollars each, made at the end of each period for n consecutive interestperiods at a rate of interest i per period, is given by

S = R

[(1 + i)n − 1

i

].

Solution (a) $ 222,777.27 (R = 9200, i = 0.08, n = 14)(b) $ 40,652.46



Annuity

Problem 1a: (a) Pat Pearson deposits $12, 000 at the end of each yearfor 9 years in an account paying 8% interest compounded annually. Findthe final amount she will have on deposit.(b) Pat’s Brother-in-law works in a bank that pays 6% compoundedannually. If she deposits her money in this bank instead of the one in (a),how much she will have in her account?(c) How much would Pat lose over 9 years by using her brother-in-lawinstead of the bank in (a)?



Annuity

Problem 2: Find the periodic payments that will amount to the givensums under the given conditions(a) S = 10, 000, interest is 8% compounded annually ; payments aremade at the end of each year for 12 years.(b) S = 50, 000, interest is 12% compounded quarterly ; payments aremade at the end of each quarter for 8 years.

Hint: Solve

S = R

[(1 + i)n − 1

i

]for R.Solution: (a) $526.95 (b) $925.33



Annuity

Problem 3: In 1995, Oseola Mc Carty donated $150, 000 to theUniversity of Southern Mississippi to establish a scholarship fund. What isunusual about her is that the entire amount come from her work as awasher woman, a job she began in 1916 at the age of 8, when shedropped out of school.How much would Ms Mc Carthy have to put in her saving account at theend of every three months to accumulate $150, 000 over 79 years?Assume she received an interest rate of 5.25% compounded quarterly.

Solution: Solve the equation

S = R

[(1 + i)n − 1

i

]for R.Here S = 150, 000; n = 79× 4 = 316; i = 0.013125.

R = $32.49



Annuity

Problem 3: In 1995, Oseola Mc Carty donated $150, 000 to theUniversity of Southern Mississippi to establish a scholarship fund. What isunusual about her is that the entire amount come from her work as awasher woman, a job she began in 1916 at the age of 8, when shedropped out of school.How much would Ms Mc Carthy have to put in her saving account at theend of every three months to accumulate $150, 000 over 79 years?Assume she received an interest rate of 5.25% compounded quarterly.Solution: Solve the equation

S = R

[(1 + i)n − 1

i

]for R.Here S = 150, 000; n = 79× 4 = 316; i = 0.013125.

R = $32.49



Sinking fund

Problem 4 a: A firm must pay off $40, 000 worth of bonds in 7 years.Find the amount of each annual payment to be make into a sinking fund,if the money earns 6% compounded annually.

Problem 4 b: Tonya Mc Carley wants to buy a car that she estimateswill cost $18, 000 in 6 years. How much money must she deposit at theend of each quarter at 12% interest compounded quarterly in order tohave enough in 6 years to pay for her car.

Hint: Solve the equation

S = R

[(1 + i)n − 1

i

]for R.Solution: (4a) $4765.40 (b) $522.85



Sinking fund

Problem 4 a: A firm must pay off $40, 000 worth of bonds in 7 years.Find the amount of each annual payment to be make into a sinking fund,if the money earns 6% compounded annually.

Problem 4 b: Tonya Mc Carley wants to buy a car that she estimateswill cost $18, 000 in 6 years. How much money must she deposit at theend of each quarter at 12% interest compounded quarterly in order tohave enough in 6 years to pay for her car.Hint: Solve the equation

S = R

[(1 + i)n − 1

i

]for R.Solution: (4a) $4765.40 (b) $522.85



Present value of Annuity

Problem 5: Find the present value of each ordinary annuity.(a) Payments of $5000 are made annually for 11 years at 6%compounded annually.(b) Payments of $50, 000 are made quarterly for 10 years at 8%compounded quarterly.

Hint:

Theorem

The present value P of an annuity of payments of R dollars each, madeat the end of each period for n consecutive interest periods at a rate ofinterest i per period is given by

P = R

[1− (1 + i)−n

i

]Solution: (a) $39, 434.37 (b) $1, 367, 773.96




Problem 5: Find the present value of each ordinary annuity.(a) Payments of $5000 are made annually for 11 years at 6%compounded annually.(b) Payments of $50, 000 are made quarterly for 10 years at 8%compounded quarterly.Hint:

Theorem

The present value P of an annuity of payments of R dollars each, madeat the end of each period for n consecutive interest periods at a rate ofinterest i per period is given by

P = R

[1− (1 + i)−n

i

]Solution: (a) $39, 434.37 (b) $1, 367, 773.96




Problem 6: Find the lump sum deposited today that will yield the sameamount as payments of $10, 000 at the end of each year for 15 years, atthe following interest rates. Interest is compounded annually.(a) 5%(b) 8%.

Hint: We have to find the present value of the given annuity. Use theformula:

P = R

[1− (1 + i)−n

i

]Solution: (a) $103, 796.58 (b) $85, 594.79




Problem 6a In his will the late Mr. Hudspeth said that each child in hisfamily could have an annuity of $2000 at the end of each year for 9 years,or the equivalent present value. If money can be deposited at 8%compounded annually, what is the present value?

Problem 6b You have a coin you wish to sell. A potential buyer offers topurchase the coin from you in exchange for a series of three annualpayments of $50 starting one year from today. What is the current valueof the offer if the prevailing rate of interest is 7% compounded annually?

Solution: Use the formula:

P = R

[1− (1 + i)−n

i

]with R = 50, i = 0.07, n = 3. Thus

P = $131.21




Problem 6a In his will the late Mr. Hudspeth said that each child in hisfamily could have an annuity of $2000 at the end of each year for 9 years,or the equivalent present value. If money can be deposited at 8%compounded annually, what is the present value?

Problem 6b You have a coin you wish to sell. A potential buyer offers topurchase the coin from you in exchange for a series of three annualpayments of $50 starting one year from today. What is the current valueof the offer if the prevailing rate of interest is 7% compounded annually?Solution: Use the formula:

P = R

[1− (1 + i)−n

i

]with R = 50, i = 0.07, n = 3. Thus

P = $131.21



Amortization

Problem 7: Find the payments necessary to amortize each loan.(a) 2500, 16% compounded quarterly ; 6 quarterly payments.(b) 45, 000, 18% compounded monthly ; 36 monthly payments.

Hint: We find R from the equation

P = R

[1− (1 + i)−n

i

]Solution: (a) $476.90 (b) $1626.86



Amortization

Problem 9: A used car costs $6000. After a down payment of $1000, thebalance will be paid off in 36 monthly payments with interest of 12% peryear, compounded monthly. Find the amount of each payment.

Solution: A single lump sum payment of $5000 today would pay off theloan, so $5000 is the present value of an annuity of 36 monthly paymentswith interest of 12%/12 = 1% per month. We find R using the formula:

P = R

[1− (1 + i)−n

i

]=⇒ R =

P[1−(1+i)−n

i

] .Replacing P with 5000, n with 36 and i with 0.01, we get

R = 166.07



Amortization

General Problem: Assume that you borrowed P dollars from a bankwith a rate of interest i per period. You plan to pay R dollars at the endof each period for n consecutive periods in order to pay off the debt. Findthe payment R.

Solution: At the end of the first period, the interest on P dollars is givenby

P.i .1 = Pi .

The amount is applied to the reduction of the original debt is

R − Pi .

Therefore, the amount in the account at the end of the first period is

P − (R − Pi) = P(1 + i)− R.

At the end of the second period, the interest on P(1 + i)− R dollars isgiven by

(P(1 + i)− R).i .1 = (P(1 + i)− R)i .



Amortization

General Problem: Assume that you borrowed P dollars from a bankwith a rate of interest i per period. You plan to pay R dollars at the endof each period for n consecutive periods in order to pay off the debt. Findthe payment R.Solution: At the end of the first period, the interest on P dollars is givenby

P.i .1 = Pi .

The amount is applied to the reduction of the original debt is

R − Pi .

Therefore, the amount in the account at the end of the first period is

P − (R − Pi) = P(1 + i)− R.

At the end of the second period, the interest on P(1 + i)− R dollars isgiven by

(P(1 + i)− R).i .1 = (P(1 + i)− R)i .



The amount is applied to the reduction of the debt is

R − (P(1 + i)− R)i .

Therefore, the amount in the account at the end of the second period is

(P(1 + i)− R)− (R − (P(1 + i)− R)i) = P(1 + i)2 − R(1 + i)− R.

By induction, the amount in the account at the end of the nth period is

P(1 + i)n − R(1 + i)n−1 − ...− R(1 + i)− R.

We pay off the debt at the end of nth period, so we have

P(1 + i)n − R(1 + i)n−1 − ...− R(1 + i)− R = 0,

or equivalently

P(1 + i)n = R(1 + i)n−1 + ... + R(1 + i) + R.



Note that

R(1 + i)n−1 + ... + R(1 + i) + R = R

[(1 + i)n − 1

i

].

Thus, we have

P(1 + i)n = R

[(1 + i)n − 1

i

].

This gives

P = R

[1− (1 + i)−n

i

].

Hence

R =P[

1−(1+i)−n

i

] .



Amortization

Problem 10: Leslie Ulman buys a new car costing $16, 000. She agreesto make payments at the end of each month for 4 years. If she pays 12 %interest, compounded monthly, what is the amount of each payment?Find the total amount of interest Leslie will pay.

Solution: We find R using the formula:

P = R

[1− (1 + i)−n

i

]=⇒ R =

P[1−(1+i)−n

i

] .Replacing P with 16,000, n with 48 and i with 0.01, we get

R = 421.34



Amortization

Problem 10: Leslie Ulman buys a new car costing $16, 000. She agreesto make payments at the end of each month for 4 years. If she pays 12 %interest, compounded monthly, what is the amount of each payment?Find the total amount of interest Leslie will pay.Solution: We find R using the formula:

P = R

[1− (1 + i)−n

i

]=⇒ R =

P[1−(1+i)−n

i


R = 421.34



Amortization

Problem 11a: An insurance firm pays $4000 for a new printer for itscomputer. It amortizes the loan for the printer in 4 annual payments at8% compounded annually. Prepare an amortization schedule for thismachine.

Solution: First, we find the payment of each year R, using the formula:

P = R

[1− (1 + i)−n

i

]=⇒ R =

P[1−(1+i)−n

i


R = 1207.68

The amortization schedule for this machine is as follows:



Amortization

Problem 11a: An insurance firm pays $4000 for a new printer for itscomputer. It amortizes the loan for the printer in 4 annual payments at8% compounded annually. Prepare an amortization schedule for thismachine.Solution: First, we find the payment of each year R, using the formula:

P = R

[1− (1 + i)−n

i

]=⇒ R =

P[1−(1+i)−n

i


R = 1207.68

The amortization schedule for this machine is as follows:



A test

(1) Harv’s Meats will need to buy a new deboner machine in 4 years. Atthat time, Harv expects the machine to cost $12, 000. To accumulateenough money to pay the machine, Harv decides to deposit a sum ofmoney at the end of each 6 month period in an account paying 16%compounded semiannually. How much should each payment be?(2) What amount must you invest today at 6% compounded annually sothat you can withdraw $5,000 at the end of each year for the next 5years?

(3) Find the monthly house payment necessary to amortize the loan$249, 560 at 7.75% for 25 years. Then find the unpaid balance after 5years for the loan.(4) Certain large semitrailer trucks cost $72, 000 each. Ace Trucking buyssuch a truck and agrees to pay for it with a loan that will be amortizedwith 9 semiannual payments at 9.5% compounded semiannually. Preparean amortization schedule for this truck.



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