Business Mathematics: Annuities and Their Applications - Capital Budgeting

Often we have to make a decision between two different investment alternatives. The determination as to which is the best alternative is made by comparing their present values.

To determine accurately how much a fixed amount of money will be worth at a later date, we must understand how compound interest works.

If a fixed amount of money, say, $1,000, earns simple interest at an annual rate of 8%, the interest earned in one year is 0.08 × $1,000 = $80. The original money is worth $1,080 after one year.

However, if the interest is calculated at 2% every quarter (1/4 of 8% = 2%), the amount of money at the end of the year will be somewhat greater.

This is easily explained with the help of the following table:

Our $1,000 has grown to $1,082.43 after one year.

To compute the amount of money we will have after interest is compounded, we can multiply A × (l + i) where i = annual interest rate/4.

f interest is compounded more often, say, monthly or daily, then i = annual interest rate/n, where n = 12 for monthly compounding and n = 360 for daily compounding. (Most banks use 360 days in a year.)

In our example, we compounded 4 times, so the final amount can be calculated by successive multiplication:
Final amount = A × (1 + i) × (1 + i) × (1 + i) × (1 + i)
or
Final amount = A × (1 + i)4

At the end of n compoundings , the final amount would be
S = A × (1 + i)n


Example
To compute how much $1,000 would be worth after 1 year if interest is compounded quarterly at 8%, use A = $1,000, i = 0.08/4 = 0.02, and n = 4:
S = A × (1 + i )4 = $1,000 × (1.02)4 = $1,000 × 1.08243 = $1,082.43

The amount computed, $1,082.43, is called the future value of the money. The present value of the money is the amount of money we started with.

To compute how much money is needed now so that we will have an amount S in the future, we rewrite the equation, solving for A:



Example
Let us compute how much money is needed today so that we will have $2,500 after 2 years if interest is 8% compounded quarterly.
Since interest is compounded quarterly, there will be eight compoundings in 2 years. Using S = $2,500, n = 8, and i = 0.08/4, we compute A:



Example
We wish to determine which of the following two alternatives is the better choice:
•  Alternative A yields a return of $2,700 at the end of 2 years plus $11,500 at the end of 6 years.
•  Alternative B gives a return of $500 at the end of each quarter for 6 years.

Assume that the bank interest rate on money borrowed is 12% compounded quarterly.
Alternative A consists of two components. We compute the present value of each.

Component 1: S1 = $2,700, n1 = 8, i = 0.12/4 = 0.03.

Component 2: S2 = $11.500, n2 = 24, i = 0.12/4 = 0.03.

A = A1 + A2 = $2,131.40 + $5,657.24 = $7,788.64.
 

Alternative B is an annuity. R = $500, n = 6 × 4 = 24, and i = 0.12/4 = 0.03. Its present value is
A = R × discount factor = $500 × 16.93554 = $8,467.77.

Since alternative B has the higher present value, it is the preferable choice.

Solved Problems:

6.30 Frank wants to buy a car. The dealer tells him that he can pay $500 a month for 36 months, or he can pay $15,000 cash for the car. Assuming that the prevailing interest rate is 9% monthly, which is a better deal for Frank?
 

Solution
First compute the present value of the annuity. This will tell us how much $500 a month for 36 months is worth today: R = $500, n = 36, i = 0.09/12 = 0.0075. The discount factor from the table above is 31.44681:
 

A = R × discount factor = $500 × 31.44681 = $15,723.41

Since the present value of the cost annuity is greater than $15,000, Frank would be better off paying $15,000 cash right away.

6.31   A company must decide whether to buy new equipment for $400,000 and enter into a service contract at $1,200 per month for 5 years (option 1) or lease the equipment for $9,000 each month over a 5-year period and then purchase it at the end of 5 years for $50,000 (option 2). If the company can earn 12% compounded monthly on its money, should the company buy or lease the equipment?
 

Solution
If we purchase the equipment, we must compute the present value of the service contract. In this problem n = 60, and i = 0.12/12 = 0.01. The discount factor is 44.95504 (Table 6.2).

Present value of service contract:
A = R × discount factor = $1,200 × 44.95504 = $53,946.05
Including the cost of purchasing the equipment, the total present value of option 1 is then
$400,000 + $53,946.05 = $453,946.05.

Present value of leasing:
A = R × discount factor = $9,000 × 44.95504 = $404,595.36

Present value of the purchase of the equipment after 5 years:

The total present value of option 2 is $404,595.36 + $27,522.48 = $432,117.84. Since this amount is lower than the $453,946.05 in option 1, the company should lease its equipment.

6.32   Which would you prefer to receive: (a) $15,000 at the end of 3 years plus $30,000 at the end of 5 years, or (b) $2,250 at the end of each quarter for the next 5 years?

Assume that money is worth 10% annually and is compounded quarterly.
Solution
The present value of $15,000 3 years from now is

The present value of $30,000 5 years from now is

The total present value under option (a) is $29,461.42.
Under option (b) the present value of $2,250 per quarter for 5 years is
A = R × discount factor = $2,250 × 15.58916 = $35,075.61
Option (b) is preferable, since the present value of the money is larger.

More Problems:

6.33 Compute the accumulated value of an ordinary annuity of $100 per month for 5 years assuming that the interest rate is 9% compounded monthly.

6.34 Compute the accumulated value of an annuity due of $100 per month for 5 years assuming that the interest rate is 9% compounded monthly.

6.35 Compute the accumulated value of an ordinary annuity of $500 per quarter for 5 years assuming that the interest rate is 10% compounded quarterly.

6.36 Compute the accumulated value of an annuity due of $500 per quarter for 5 years assuming that the interest rate is 10% compounded quarterly.

6.37 Compute the accumulated value of an ordinary annuity of $1,200 every six months for 3 years assuming that the interest rate is 6% compounded semiannually.

6.38 Compute the accumulated value of an annuity due of $1,200 every 6 months for 3 years assuming that the interest rate is 6% compounded semiannually.

6.39 Compute the present value of an ordinary annuity of $150 a month for 3 years. Interest is paid at 15% compounded monthly.

6.40 Compute the present value of an annuity due of $150 a month for 3 years. Interest is paid at 15% compounded monthly.

6.41 You have just won the million-dollar lottery! You will get $100,000 at the end of each year for the next 10 years. Assuming 4% interest compounded annually at the end of each year, how much did you really win?

6.42 Bunny gets paid at the end of each month. If she deposits $50 from each paycheck into an account that pays 12% compounded monthly, how much will she have saved at the end of a year?

6.43 How much interest is accumulated at the end of 2 years if $90 per month is deposited into a savings account paying 9% compounded monthly?

6.44 How much money must Sidney save each quarter to accumulate $500 at the end of 2 years? Assume that the interest rate is 5% compounded quarterly and that Sidney makes his deposit at the end of each quarter.

6.45 Repeat Prob. 6.44 assuming that Sidney makes his deposit at the beginning of each quarter.

6.46 Pan American bank pays 6% compounded monthly on its passbook savings account. How much would you have to deposit each month in order to save $500 at the end of one year?

6.47 How much money should you put in a savings account paying 6% interest compounded monthly in order to have enough money to receive a monthly stipend of $500 per month for 10 years?

6.48 Mrs. Spector is updating her will. She wants to leave her granddaughter $1,000 a year for 10 years after her death. How much money will her executors need to invest at 4% compounded annually to accomplish this?

6.49 Mr. Bailey was sued by Mr. Smith and was ordered by the court to pay $5,000 in damages. Since Mr. Bailey didn’t have the money, he agreed to pay $1,000 now and the rest in equal monthly installments over a period of one year. If interest is compounded monthly at 6% per annum, how much must Mr. Bailey pay Mr. Smith each month?

6.50 Scott and Ronnie need to save $30,000 to put a down payment on a house in 3 years. If interest at a local bank is compounded quarterly at 5% per year, how much must they save each quarter if they make their deposit (a) at the end of the quarter or (b) at the beginning of the quarter?

6.51 A washing machine sells for $450. If Mae puts $150 down, how much should she pay each month for the next 12 months if she finances the appliance? Assume that the finance charges are 15% per year compounded monthly.

6.52 Use Table 6.3 to compute the monthly payment on a 12-year loan of $17,000 at an annual interest rate of 6.5%.

6.53 Use Table 6.3 to compute the savings in interest between a 7.5% loan of $100,000 and an 8% loan of the same amount. Both have 20-year terms and are payable monthly.

6.54 A 30-year mortgage of $50,000 has a monthly payment due of $420.43. Use Table 6.3 to compute the annual interest rate.

6.55 A store advertises a refrigerator for $100 down and $100 a month for 8 months. If the interest rate is 15% compounded monthly, what is the actual value of the refrigerator?

6.56 A company wants to accumulate $200,000 in 5 years in order to purchase new equipment. How much must they deposit each month into an account which pays an interest rate of 9% per year compounded monthly?

6.57 Suppose that the interest rate in Prob. 6.56 drops to 6% after the second year. How much must the company now save each month in order to accomplish their goal?

6.58 If the annual premium on a life insurance policy is $1,200, and the interest rate is 10%, what would be a fair quarterly premium?

6.59 How much must you save each month for the next 10 years to withdraw $50 per month for the following 20 years? Assume that the interest rate stays constant at 6% compounded monthly for the entire 30-year period.

6.60 The monthly rent on a condominium is $600, payable at the beginning of the month. If the current interest rate is 6%, what would be a fair amount to charge someone who wishes to pay the yearly rent in advance?

6.61 Dr. Payne is a dentist. He estimates that his x-ray machine will have to be replaced in 5 years at a cost of $75,000. What should be his quarterly contribution into a sinking fund which pays 12% compounded quarterly so that he will have enough money to purchase a new machine?

6.62 Construct a sinking fund schedule which describes the accumulation of a sinking fund in which $20,000 is to be accumulated in 3 years if payments are made semiannually into an account which pays 8% compounded every 6 months.

6.63 The Ace plumbing supply company anticipates that they will need a new truck in 5 years which will cost $40,000. They contribute monthly into a sinking fund which pays interest at the rate of 9% compounded monthly. After 2 years, however, an emergency arises so they withdraw the entire balance from the fund. How much did they withdraw?

6.64 A tennis club has 300 members. They want to build a new clubhouse at a cost of $750,000 so they establish a sinking fund which will accumulate this amount of money in 2 years. The fund earns 12% interest compounded every two months. If all members are to pay equally, how much will their bimonthly assessment be?

6.65 A sinking fund of equal monthly payments is established to accumulate $100,000 in 5 years. If the annual interest rate is 15%, how much interest will the fund accumulate in that period of time?

6.66 The voters of the town of Brookville pass a $20,000,000 bond issue to be redeemed in 10 years. What semiannual deposits (to the nearest dollar) must the town officials make into a sinking fund paying 8% compounded semiannually in order to have enough money to redeem the bonds on time?

6.67 To have enough capital to buy new networking equipment, the Compact Computer Corporation takes out a $500,000 loan whose 12% interest is payable monthly. To pay off the debt in 5 years, monthly payments are made into a sinking fund which pays 9% interest compounded monthly. What is the monthly expense of the debt?

6.68 What is the book value of the debt, to the nearest dollar, in Prob. 6.67 at the end of 2 years?

6.69 In order to buy a new fire engine, the city of Pinewood takes out a $1,200,000 loan, whose 16% interest is payable quarterly. To pay back the debt in 4 years, quarterly payments are made to a sinking fund which pays 12% interest compounded quarterly. What is the quarterly expense of the debt (to the nearest dollar)?

6.70 What is the book value of the debt after 2\ years (to the nearest dollar)?

6.71 What are the monthly payments on a $250,000 mortgage at 9% annual interest amortized over 20 years?

6.72 A new Cadillac costs $35,000. The interest rate on a dealer-financed auto loan, payable monthly, is 15%. If Mr. Taylor wants to pay $1,000 a month for 3 years, how much of a down payment will he have to give the dealer?

6.73 What is the total interest on the auto loan in Prob. 6.72?

6.74 What is the cost (i.e., interest) of a 30-year, $100,000 mortgage, payable monthly, whose interest rate is 7½%?

6.75 A home computer was purchased for $100 down and $50 a month for 24 months. If the interest rate is 15%, what is the actual price of the computer?

6.76 What is the total finance charge for the computer purchase of Prob. 6.75?

6.77 Redwood Savings Bank charges an annual interest rate of 9% for a 30-year mortgage with monthly payments. Greenwood Federal charges only 8.5% for the same loan. How much would you save each month on a $100,000 mortgage if you took your business to Greenwood?

6.78 Suppose that you took your monthly savings from Prob. 6.77 and deposited it in a savings account paying 6% interest compounded monthly. How much money would be in the account after 10 years?

6.79 A new car can be purchased for $20,000 or can be leased for 36 months at $500 per month and then purchased at the end of the lease for $5,000. If the interest rate for the lease is 12% annually, and the interest rate on savings is 9% compounded monthly, which is the better alternative? Assume that you will definitely purchase the car one way or the other.

6.80 A growing company needs to increase its warehouse space. They anticipate that they will need a total of 150,000 ft2 in 1 year, but they need only 75,000 ft2 right away. If the cost of construction is $10/ft2 now but will be $ ll/ft2 in a year, which of the following alternatives is best? Assume that the interest rate is 12% simple interest and will not fluctuate during the year.
   (a)   Build a 150,000-ft2 warehouse immediately.
   (b)   Build a 75,000-ft2 warehouse now and add another 75,000 ft2 next year.

6.81 Assuming interest rates remain constant at 12% compounded quarterly, which of the following alternatives is preferable?
   (a)   An investment today of $20,000 with expected returns of $2,500 a quarter for the next 10 years.
   (b)   An investment today of $25,000 with expected returns of $2,625 a year for the next 10 years.

Answers to Above Problems:

6.33 $7,542.41
6.34 $7,598.98
6.35 $12,772.33
6.36 $13,091.64
6.37 $7,762.09
6.38 $7,994.95
6.39 $4,327.09
6.40 $4,381.18
6.41 $811,090
6.42 $634.13
6.43 $2,356.96
6.44 $59.82
6.45 $59.08
6.46 $40.53
6.47 $45,036.73
6.48 $8,110.90
6.49 $344.27
6.50 (a) $2,332.75, (b) $2,303.95
6.51 $27.08
6.52 $170.33
6.53 $30.85/mo
6.54 9.5%
6.55 $856.81
6.56 $2,651.67
6.57 $3,172.50
6.58 $318.98
6.59 $42.59
6.60 $7,006.21
6.61 $2,791.18
6.62 


6.63 $13,888.53
6.64 $186.40
6.65 $32,260.60
6.66 $671,635
6.67 $11,629.18
6.68 $326,392
6.69 $107,533
6.70 $517,521
6.71 $2,249.32
6.72 $6,152.73
6.73 $7,152.73
6.74 $151,717.40
6.75 $1,131.21
6.76 $168.79
6.77 $35.71
6.78 $5,852.13
6.79 Leasing the car and purchasing it at the end of the lease
6.80 Alternative (b)
6.81 Alternative (a)
 

Note: Some use the notation Sn|i to denote the accumulation factor [(1 + i)n – 1]/i. Thus S = R × Sn|i However, we shall not use this notation in this guide.