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Present value tables, loans and superannuation
Present value interest factor tables, finding loan repayments and how much can be borrowed, and using present value to plan retirement income from superannuation.
A loan and an annuity are two views of the same thing. When a bank lends you $\$25\,000$, it is buying a stream of your future repayments. The $\$25\,000$ is the present value of those repayments. Retirement works the same way in reverse: your super balance on the day you retire is the present value of the income it can pay you.
Present value of an annuity
A present value interest factor is the lump sum needed today to fund payments of $\$1$ at the end of each period for $n$ periods. It is always less than $n$, because money held today can earn interest while it waits to be paid out.
Monthly rates (for car and personal loans):
| Periods | $0.5\%$ | $0.75\%$ | $1\%$ |
|---|---|---|---|
| $12$ | $11.6189$ | $11.4349$ | $11.2551$ |
| $24$ | $22.5629$ | $21.8891$ | $21.2434$ |
| $36$ | $32.8710$ | $31.4468$ | $30.1075$ |
| $48$ | $42.5803$ | $40.1848$ | $37.9740$ |
| $60$ | $51.7256$ | $48.1734$ | $44.9550$ |
Yearly rates (for long-term loans and retirement):
| Periods | $3\%$ | $4\%$ | $5\%$ | $6\%$ | $7\%$ | $8\%$ |
|---|---|---|---|---|---|---|
| $5$ | $4.5797$ | $4.4518$ | $4.3295$ | $4.2124$ | $4.1002$ | $3.9927$ |
| $10$ | $8.5302$ | $8.1109$ | $7.7217$ | $7.3601$ | $7.0236$ | $6.7101$ |
| $15$ | $11.9379$ | $11.1184$ | $10.3797$ | $9.7122$ | $9.1079$ | $8.5595$ |
| $20$ | $14.8775$ | $13.5903$ | $12.4622$ | $11.4699$ | $10.5940$ | $9.8181$ |
| $25$ | $17.4131$ | $15.6221$ | $14.0939$ | $12.7834$ | $11.6536$ | $10.6748$ |
| $30$ | $19.6004$ | $17.2920$ | $15.3725$ | $13.7648$ | $12.4090$ | $11.2578$ |
Loan repayments
Hamish borrows $\$25\,000$ for a car at $9\%$ p.a., with monthly repayments over $5$ years. Find the monthly repayment and the total interest.
Mia can afford $\$1500$ a month on a loan over $4$ years at $6\%$ p.a. compounded monthly. How much can she borrow?
Superannuation and retirement
From the preservation age of $60$, once retired, you can usually take your super as a lump sum, as a regular income stream (an account-based pension), or a mix. While it pays you an income, the balance stays invested and keeps earning. The PV table answers the two big retirement questions:
- "I have a balance. What yearly income can it pay?" Divide the balance by the factor.
- "I want a yearly income. What balance do I need?" Multiply the income by the factor.
Wei retires at $67$ with $\$600\,000$ in super. The fund earns $5\%$ p.a. (an example return). (a) What yearly income, paid at the end of each year, would last exactly $25$ years? (b) What balance would be needed for $\$40\,000$ a year for $25$ years?
Set $\$600\,000$ at $5\%$ and $25$ years and check Example 3. Then drag the yearly income above the steady amount. How many years does the money last now?
Switch to Loan mode and check Example 1.
Your turn
A loan of $\$30\,000$ is taken at $9\%$ p.a. with monthly repayments over $3$ years. Find the monthly repayment.
Show the working
$r = 0.75\%$, $n = 36$, factor $= 31.4468$. Repayment $= \dfrac{30\,000}{31.4468} \approx \$953.99$.
Use the monthly PV table to find the factor for $1\%$ per month and $36$ months.
Read the $1\%$ column, row $36$: $30.1075$.
Find the monthly repayment on a $\$30\,000$ loan at $9\%$ p.a. over $3$ years, to the nearest cent.
$r = 0.75\%$, $n = 36$, factor $= 31.4468$. Repayment $= \dfrac{30\,000}{31.4468} \approx \$953.99$.
How much can be borrowed at $6\%$ p.a. compounded monthly over $4$ years with repayments of $\$1500$ a month?
$r = 0.5\%$, $n = 48$, factor $= 42.5803$. $PV = 1500 \times 42.5803 = \$63\,870.45$.
The present value of an annuity is:
It is the lump sum now that, invested at the given rate, exactly funds every future payment.
A $\$25\,000$ car loan is repaid with $60$ monthly repayments of $\$518.96$. Find the total interest.
$518.96 \times 60 = \$31\,137.60$. Interest $= 31\,137.60 - 25\,000 = \$6137.60$.
A retiree has $\$500\,000$ in super earning $5\%$ p.a. What yearly income, paid at the end of each year, would last $20$ years? Answer to the nearest cent.
Factor $(5\%, 20) = 12.4622$. Income $= \dfrac{500\,000}{12.4622} \approx \$40\,121.33$.
What super balance is needed to pay $\$50\,000$ at the end of each year for $30$ years, if it earns $5\%$ p.a.?
Factor $(5\%, 30) = 15.3725$. Balance $= 50\,000 \times 15.3725 = \$768\,625$.
A $\$300\,000$ loan at $6\%$ p.a. is repaid with equal payments at the end of each year for $20$ years. Find the yearly payment, to the nearest cent.
Factor $(6\%, 20) = 11.4699$. Payment $= \dfrac{300\,000}{11.4699} \approx \$26\,155.42$.
Exam style: Hamish can repay a $\$25\,000$ loan at $9\%$ p.a. monthly over $3$ years or over $5$ years. How much interest does he save by choosing $3$ years?
$3$ years: $\dfrac{25\,000}{31.4468} \approx \$794.99$; interest $= 794.99 \times 36 - 25\,000 = \$3619.64$. $5$ years: $\dfrac{25\,000}{48.1734} \approx \$518.96$; interest $= 518.96 \times 60 - 25\,000 = \$6137.60$. Saving $= 6137.60 - 3619.64 = \$2517.96$, though the repayments are about $\$276$ a month higher.
Exam style: Grace retires with $\$650\,000$ in super, invested at $6\%$ p.a. What equal yearly income, paid at the end of each year, can she draw for $25$ years? Answer to the nearest cent.
Factor $(6\%, 25) = 12.7834$. Income $= \dfrac{650\,000}{12.7834} \approx \$50\,847.19$.