Coaching Centre
Annuities as recurrence relations
What an annuity is, modelling regular contributions with the recurrence relation A(n+1) = A(n)(1 + r) + P, building annuity tables and spreadsheets, and superannuation as an annuity.
A loan is a lump sum you pay back in regular instalments. An annuity runs the other way: you put in regular equal payments, and the account earns compound interest on everything already in it. A regular savings plan and superannuation are both annuities. A reducing-balance loan is also an annuity, seen from the bank's side.
Building an annuity one period at a time
Each period, two things happen to the balance: interest is added on what is already there, and then a new contribution goes in. In this course contributions are made at the end of each period.
$A_n$ is the value after $n$ periods, $r$ is the interest rate per period and $P$ is the regular contribution. Compare it with the loan relation in 3A: the only change is $+P$ instead of $-M$.
Isla saves $\$300$ at the end of every month in an account paying $6\%$ p.a. compounded monthly. Find the value of her savings after each of the first $4$ months.
Annuities in a spreadsheet
With the rate per period in F1 and the contribution in F2, column A counts periods and column B holds the value. Starting with $0$ in B2, the formula in B3 is =B2*(1+$F$1)+$F$2, filled down. After $120$ rows (ten years) Isla's account shows about $\$49\,163.80$.
After $10$ years Isla's annuity is worth $\$49\,163.80$. How much did she contribute, and how much interest did she earn?
Each bar shows the value at the end of a year, split into what was paid in (blue) and the interest earned (orange). Set $\$300$ a month at $6\%$ for $10$ years and check Example 2. Then stretch it to $40$ years. When does the interest part become bigger than the contributions?
Superannuation
Superannuation (super) is money saved for retirement. Employers must pay the superannuation guarantee, currently $12\%$ of your ordinary earnings, into your super fund. The fund invests it, so super grows like an annuity: regular contributions plus compound returns over a whole working life. You can usually access it from your preservation age of $60$ once you retire.
Tom earns $\$70\,000$ a year. His employer pays the $12\%$ superannuation guarantee at the end of each year, and his fund earns $7\%$ p.a. (an example return). Find the value of these contributions after $3$ years.
Your turn
$\$1000$ is invested at the end of each year at $5\%$ p.a. compounded yearly. Find the value after $3$ years and the interest earned.
Show the working
$A_1 = 1000$, $A_2 = 1000(1.05) + 1000 = 2050$, $A_3 = 2050(1.05) + 1000 = \$3152.50$. Contributions $= \$3000$, so interest $= \$152.50$.
Which of these is an annuity?
An annuity is a series of equal regular payments earning compound interest. Only the weekly deposits fit.
An annuity has $A_{n+1} = A_n(1.005) + 400$ and $A_0 = 0$. Find $A_2$.
$A_1 = 0 + 400 = 400$. $A_2 = 400(1.005) + 400 = 402 + 400 = \$802$.
$\$1000$ is invested at the end of each year at $5\%$ p.a. Find the value after $3$ years.
$A_1 = 1000$. $A_2 = 1000(1.05) + 1000 = 2050$. $A_3 = 2050(1.05) + 1000 = \$3152.50$.
Savings of $\$250$ are deposited at the end of each month at $4.8\%$ p.a. compounded monthly. Which recurrence relation models the balance?
Monthly rate $= 0.048 \div 12 = 0.004$, so multiply by $1.004$, then add the $\$250$ deposit.
Priya earns $\$65\,000$ a year. Find her employer's yearly superannuation guarantee contribution at $12\%$.
$0.12 \times 65\,000 = \$7800$.
How much is contributed in total when $\$300$ is deposited every month for $10$ years?
$300 \times 12 \times 10 = \$36\,000$.
Those contributions grow to $\$49\,163.80$. How much interest was earned?
$49\,163.80 - 36\,000 = \$13\,163.80$.
An annuity is worth $\$5000$ after $n$ months. The rate is $0.4\%$ per month and the monthly contribution is $\$250$. Find its value one month later.
$A_{n+1} = 5000(1.004) + 250 = 5020 + 250 = \$5270$.
Exam style: Tom's employer pays $12\%$ of his $\$70\,000$ salary into super at the end of each year. The fund earns $7\%$ p.a. Find the value of the contributions after $3$ years.
$P = 0.12 \times 70\,000 = \$8400$. $A_1 = 8400$. $A_2 = 8400(1.07) + 8400 = 17\,388$. $A_3 = 17\,388(1.07) + 8400 = \$27\,005.16$.
Exam style: Instead of $\$300$ a month for $10$ years (worth $\$49\,163.80$), Isla could invest the same total, $\$36\,000$, as a single deposit now at $6\%$ p.a. compounded monthly. How much more would the single deposit be worth after $10$ years?
Single deposit: $36\,000(1.005)^{120} \approx \$65\,498.28$. Difference: $65\,498.28 - 49\,163.80 = \$16\,334.48$. The lump sum earns more because all of it earns interest for the full ten years, while each monthly deposit earns interest only from when it goes in.