25. Annuities and Loans | Mathematics Advanced

Starting point: Years 9 and 10

Building a balance table

An annuity is a series of equal payments made at regular times, with compound interest. Savings plans, superannuation and loans are all annuities.

Suppose you deposit 1000 dollars at the end of each year into an account paying 5% per year.

Year	Start	Interest (5%)	Deposit	End
1	0	0	1000	1000.00
2	1000.00	50.00	1000	2050.00
3	2050.00	102.50	1000	3152.50

After 3 years the account holds 3152.50 dollars. You paid in 3000 dollars, so 152.50 dollars is interest.

A loan table

A loan works the other way. Interest is added each period, then the repayment is taken away.

Borrow 2000 dollars at 1% per month and repay 700 dollars a month:

Month	Start	Interest (1%)	Repayment	End
1	2000.00	20.00	700	1320.00
2	1320.00	13.20	700	633.20
3	633.20	6.33	639.53	0.00

The last payment is smaller: just enough to clear the debt.

Annuities

Regular payments

Deposits: savings
Repayments: loans

Tables

Interest then payment

Future value

FV = M((1+r)ⁿ - 1)/r

Present value

PV = FV / (1+r)ⁿ

Loans

Reducing balance
Repayment formula

Interest factor tables

Value of 1 dollar

Year 12Future and present value formulas

Future value. Deposits of $M$ at the end of each period for $n$ periods. The first deposit earns interest for $n - 1$ periods, the last for none. The total is a geometric series:

$F V = M + M (1 + r) + \dots + M (1 + r)^{n - 1} = M [\frac{( 1 + r ) ^{n} - 1}{r}]$

Check the table: $1000 \times \frac{1.0 5 ^{3} - 1}{0.05} = 3152.50$ .

Present value. Bring the future value back to today:

$P V = \frac{F V}{( 1 + r ) ^{n}} = M [\frac{1 - ( 1 + r ) ^{- n}}{r}]$

Interest factor tables. Exams often give a table of future or present values of an annuity of 1 dollar. Multiply the table value by $M$ . If the future value table shows 6.3081 for $r = 2%$ and $n = 6$ , then six deposits of 500 dollars grow to $500 \times 6.3081 = 3154.05$ dollars.

Sienna's loan. The loan amount is the present value of her repayments, with $r = \frac{0.09}{12} = 0.0075$ and $n = 60$ .

$15000 = M [\frac{1 - 1.007 5 ^{- 60}}{0.0075}] \Rightarrow M \approx 311.38 dollars per month$

She pays $60 \times 311.38 = 18 682.80$ dollars in total, so the interest is about 3683 dollars.

The same result written as one formula is

$M = \frac{P r ( 1 + r ) ^{n}}{( 1 + r ) ^{n} - 1}$

Quick summary

An annuity is a set of equal regular payments with compound interest.
Savings tables: add interest, then add the deposit. Loan tables: add interest, then subtract the repayment.
$F V = M [\frac{( 1 + r ) ^{n} - 1}{r}]$ and $P V = \frac{F V}{( 1 + r ) ^{n}}$ .
A loan amount is the present value of its repayments.
With a table of values for 1 dollar, multiply by the payment.

Quiz

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