For a child's education, a family decides to invest ₹3,000 at the end of each 6-month period in a fund paying

Worked Examples · Example 17

Q. For a child's education, a family decides to invest ₹3,000 at the end of each 6-month period in a fund paying 8% per year compounded semi-annually. Find the amount of the investment at the end of 18 years.

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Concept understanding — Annuities, Calculating Value of Regular Annuity

Annuities: The Intuition First

Imagine you're saving for retirement. You decide to put ₹10,000 into a fixed deposit every year for 20 years. At the end of each year, the bank gives you 8% interest on whatever is in your account. What will your total savings be at the end of 20 years?

That's an annuity — a series of equal payments made at regular intervals. The key idea is that each payment earns interest for a different length of time. The first ₹10,000 earns interest for 20 years, the second for 19 years, and the last one earns interest for just 1 year (or zero, depending on when you count).

Note

An annuity is not a single lump sum. It's a stream of identical cash flows spaced equally in time. The "value" of an annuity is what that entire stream is worth at a particular point in time, given a certain interest rate.

Two Kinds of Annuity Value

There are two questions you can ask:

Future Value (FV) — "If I deposit ₹P every year for n years at r% interest, how much will I have at the end?"
Present Value (PV) — "If someone promises to pay me ₹P every year for n years, and I can earn r% elsewhere, what is that promise worth right now?"

Both are built from the same core idea: each payment is a separate compound-interest problem, and you add them up.

The Precise Mathematics

Let:

$P$ = the regular payment (the annuity amount)
$r$ = interest rate per period (as a decimal; e.g., 8% = 0.08)
$n$ = number of payments

Future Value of an Ordinary Annuity

"Ordinary" means payments happen at the end of each period. The first payment earns interest for $n-1$ periods, the second for $n-2$ , and the last earns no interest at all.

FV = P(1+r)^{n-1} + P(1+r)^{n-2} + \cdots + P(1+r) + P

This is a geometric series. Factor out $P$ and sum it:

FV = P \cdot \frac{(1+r)^n - 1}{r}

FV_{\text{ordinary}} = P \cdot \frac{(1+r)^n - 1}{r}

Present Value of an Ordinary Annuity

Here we discount each future payment back to today. The first payment (one period away) is discounted by $(1+r)^1$ , the second by $(1+r)^2$ , and so on.

PV = \frac{P}{(1+r)} + \frac{P}{(1+r)^2} + \cdots + \frac{P}{(1+r)^n}

Again, a geometric series:

PV = P \cdot \frac{1 - (1+r)^{-n}}{r}

PV_{\text{ordinary}} = P \cdot \frac{1 - (1+r)^{-n}}{r}

A Concrete Example

You want to know how much ₹10,000 saved every year for 20 years at 8% will grow to.

$P = 10000$ , $r = 0.08$ , $n = 20$

FV = 10000 \cdot \frac{(1.08)^{20} - 1}{0.08}

$(1.08)^{20} \approx 4.66096$

$FV = 10000 \cdot \frac{4.66096 - 1}{0.08} = 10000 \cdot \frac{3.66096}{0.08} = 10000 \cdot 45.762 = 457,620$ …

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Convert the annual figures to semi-annual periods and apply the future-value-of-annuity formula.

FV=R\cdot\dfrac{(1+i)^{n}-1}{i}

Given: $R=₹3{,}000$ every 6 months; nominal rate $8\%$ p.a. compounded semi-annually $\Rightarrow i=\dfrac{8}{2}\%=4\%=0.04$ per half-year; time $=18$ years $\Rightarrow n=18\times2=36$ half-year periods.

Compute $(1.04)^{36}$ by squaring: $1.04^2=1.0816$ , $1.04^4=1.169859$ , $1.04^8=1.368569$ , $1.04^{16}=1.873026$ , $1.04^{32}=3.508083$ , $1.04^{36}=1.04^{32}\times1.04^4=3.508083\times1.169859\approx4.103939$ .
Substitute into the formula: …

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Q.For a child's education, a family decides to invest ₹3,000 at the end of each 6-month period in a fund paying 8% per year compounded semi-annually. Find the amount of the investment at the end of 18 years.

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