Business Mathematics: Annuities and Their Applications - Sinking Funds

Often a specific amount of money, needed at some future date, is accumulated by a series of equal periodic payments. Such a fund, known as a sinking fund, is frequently used to pay off debts or to replace worn-out or antiquated equipment.

The total amount of money in the fund is simply the accumulated value S of an annuity. Thus the periodic payment R can be computed from our previous formula,
S = R × accumulation factor
by algebraically solving for R



Example
The owners of a tool and die company want to accumulate $50,000 to replace worn-out equipment 8 years from now. How much should they contribute each month into a sinking fund which pays 8% compounded quarterly?

The number of payments, n = 8 × 4 = 32, i = 0.08/4 = 0.02, and S = $50,000:

Thus, the monthly payment should be $1,130.53.

A sinking fund schedule can be constructed if we wish to observe how the fund accumulates.

Example
Construct a sinking fund schedule which describes the accumulation of a sinking fund in which $10,000 is to be accumulated in 3 years if payments are made quarterly into an account which pays 5% compounded quarterly.

First we compute the quarterly payment:

For each period, the amount accumulated is increased by the quarterly payment, $766.80, and the interest on the amount at the beginning of that period.

All calculations were rounded to the nearest penny. The final amount is slightly different than the amount predicted mathematically due to rounding. This is typical of most calculations of this type.

The amount of money accumulated in a sinking fund after n payments is just the accumulated value of the annuity at that time:
S = R × accumulation factor


Example
Determine how much money is in the sinking fund of the last example at the end of 6 periods.
S = R × accumulation factor = $777.58 × 6.19065 = $4,813.73

This agrees with line 6 of the table above.


Example
Jodi and Frank estimate that they will need $100,000 to send their son to college in 15 years. Assume that interest rates will remain at 8% compounded quarterly.

(a)   How much should they save every 3 months?

The amount they have to put aside each period is computed by the formula

where S = $100,000, n = 4 × 15 = 60, and i = 0.08/4 = 0.02. The accumulation factor obtained from Table 6.1 is 114.05154:

(b)   How much will they have accumulated after 7½ years?
After 7½ years they will have made 30 payments. The accumulation factor is now 40.56808:
S = R × accumulation factor = 876.80 × 40.56808 = $35,570.09

They will have saved $35,570.09 after l\ years. Note that this is somewhat less than half the final amount. (Does this seem reasonable?)
Sinking funds are used to discharge debts. The debtor borrows money and pays off the debt by making equal periodic payments into a sinking fund that will accumulate the amount borrowed by the end of the loan’s term. At the end of the term of the loan, the borrower transfers the amount from the sinking fund to the lender.

The sum of the interest payment and the sinking fund deposit is called the periodic expense of the debt.
Periodic expense = interest on the loan + deposit into sinking fund

The book value of the debt at any time is the original principal minus the amount in the sinking fund.
Book value = original amount borrowed - amount in the sinking fund


Example
A businessman borrows $15,000 at 18% payable monthly and makes monthly deposits into a sinking fund so that his debt may be paid off at the end of one year. The sinking fund earns 9% compounded monthly.

(a)   What is the monthly expense of the debt?
The monthly interest on the loan is computed by multiplying the amount borrowed by the monthly
interest rate, 0.18/12 = 0.015:
Monthly interest = R × i = 15,000 × 0.015 = $225.00

Next we compute the amount which must be deposited into the sinking fund in order to accumulate $15,000 in one year:

Monthly expense = interest on the loan + deposit into sinking fund = $225.00 + $1,199.27 = $1,424.27

(b)   What is the book value of the debt at the end of 6 months?
The amount in the sinking fund at the end of 6 months is
S = R × accumulation factor = $1,199.27 × 6.11363 = $7,331.89

Book value = original amount borrowed - amount in the sinking fund = $15,000.00 - $7,331.89 = $7,668.11

Solved Problems:

6.16   Barbara wants to save up enough money to put a $24,000 down payment on a house in 2 years. How much money should she deposit each month into an account which pays 6% interest compounded monthly in order to save enough money for the down payment?
 

Solution
In this problem S = $24,000, n = 12 × 2 = 24, and i = 0.06/12 = 0.005. The accumulation factor (from Table 6.1) is 25.43196:

Barbara should deposit $943.69 at the end of each month.

6.17   Trevor owns a machine shop. One of his machines is rather old and will have to be replaced in
3 years. Trevor predicts that a new machine will cost $80,000. How much should he put aside into a sinking fund which pays an annual rate of 6% compounded semiannually in order to accumulate enough money to replace the machine?
 

Solution
S = $80,000, n = 3 × 2 = 6, and i = 0.06/2 = 0.03. The accumulation factor is 6.46841:

Trevor should put $12,367.80 into the bank at the end of each semiannual period.

6.18   A condominium association wants to establish a sinking fund to accumulate $250,000 in 3 years to repair the roofs. The fund earns 9% interest compounded monthly. If there are 200 units in the condominium, how much should each unit owner be assessed each month as a fair contribution into the fund? Assume that all units are of equal size and hence have equal assessments.
 

Solution
The total monthly assessment is computed from the formula R = 5/accumulation factor where 5 = $250,000, n = 12 × 3 = 36, and i = 0.09/12 = 0.0075. The accumulation factor (from Table 6.1) is 41.15272. Hence

Since the assessment is to be divided equally among the 200 unit owners, each should pay $6,074.93/200 = $30.37 each month.

6.19   How much interest does the sinking fund in Prob. 6.18 earn?
 

Solution
Since the total payments are $6,074.93 per month × 36 months = $218,697.48, the interest earned is $250,000 - $218,697.48 = $31,302.52.

6.20   A city issues $1,000,000 worth of bonds to raise capital to improve its sewage treatment system. What semiannual deposits must be made into a sinking fund earning interest at 8% compounded semiannually in order to redeem the bonds at the end of 15 years?
 

Solution
The accumulated value S must be $1,000,000 after 15 years, n = 15 × 2 = 30, and i = 0.08/2 = 0.04. The accumulation factor (Table 6.1) is 56.08494:

The city must deposit $17,830.10 every 6 months.

6.21   Construct a sinking fund schedule for the first 2 years of problem 6.20.
 

Solution


6.22   Construct a sinking fund schedule for the last 2 years of problem 6.20.
 

Solution
First we must determine the final amount after the 13th year (26th deposit).
S = R × accumulation factor = 17,830.10 × 44.31174 = $790,082.76

Now we can complete the sinking fund schedule.

The final amount is 4 cents short because of rounding.

6.23   A debt of $50,000, whose quarterly interest rate is 4%, must be repaid in 5 years. To discharge the debt, quarterly deposits are made into a sinking fund which earns interest at the rate of 10% compounded quarterly. What is the quarterly expense of the debt?
 

Solution
The quarterly interest on the debt is
R × i = $50,000 × 0.04 = $2,000.00

The quarterly deposit into the sinking fund is


6.24   What is the book value of the debt in problem 6.23 at the end of 3 years?
 

Solution
After 3 years (n = 12) the amount of money in the sinking fund is
S = R × accumulation factor = $1,957.36X 13.79555 = $27,002.86

Book value = original amount borrowed – amount in the sinking fund
= $50,000 - $27,002.86 = $22,997.14