Q.Imagine that you are planning to retire in 35 years and you think you can afford to save ₹500/- per month. Further, you believe that you can reasonably earn about 8% per year without taking too much risk. How much amount will you have accumulated at the time you retire?

Simple Applications of Regular Annuities (Up to 3 Periods)

The Intuition: What Is an Annuity?

Imagine you decide to save ₹100 every month. At the end of each month, you put that ₹100 into a box. After 3 months, you open the box — how much is there? ₹300, obviously. But what if the box grows your money? What if each ₹100 you put in earns interest until you take it all out?

That's the core idea of an annuity: a series of equal payments made at regular intervals, where each payment earns compound interest from the moment it's made until the end of the term.

The "regular" part means the payments are identical in amount and equally spaced in time. The "simple applications up to 3 periods" means we only look at cases with 3 or fewer payments — so we can work everything out by hand, step by step, without needing formulas.

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Two Types of Annuities

There are two flavours, and the difference is just when you make the payment.

For 3 periods, here's the timeline:

In an annuity immediate, the last payment goes in at the very end and earns no interest. In an annuity due, the first payment goes in at the very start and earns interest for all 3 periods.

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The Core Calculation: Future Value

The future value of an annuity is what all the payments plus interest add up to at the end of the term. Let's compute it for both types, assuming ₹100 per period at 10% per period.

Annuity Immediate (3 periods)

• Payment 1 (end of period 1): earns interest for 2 periods → $100 \times (1.10)^2 = 100 \times 1.21 = 121$
• Payment 2 (end of period 2): earns interest for 1 period → $100 \times (1.10)^1 = 110$
• Payment 3 (end of period 3): earns no interest → $100$

Total future value: $121 + 110 + 100 = 331$

Annuity Due (3 periods)

• Payment 1 (start of period 1): earns interest for 3 periods → $100 \times (1.10)^3 = 100 \times 1.331 = 133.10$
• Payment 2 (start of period 2): earns interest for 2 periods → $100 \times (1.10)^2 = 121$
• Payment 3 (start of period 3): earns interest for 1 period → $100 \times (1.10)^1 = 110$

Total future value: $133.10 + 121 + 110 = 364.10$

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The Other Side: Present Value

Sometimes you want to know how much a stream of future payments is worth today. That's the present value — you discount each payment back to the present.

Annuity Immediate (3 periods)

• Payment 1 (end of period 1): discount back 1 period → $\frac{100}{(1.10)^1} = 90.91$
• Payment 2 (end of period 2): discount back 2 periods → $\frac{100}{(1.10)^2} = 82.64$
• Payment 3 (end of period 3): discount back 3 periods → $\frac{100}{(1.10)^3} = 75.13$

Total present value: $90.91 + 82.64 + 75.13 = 248.68$

Annuity Due (3 periods)

• Payment 1 (start of period 1): already at present → $100$
• Payment 2 (start of period 2): discount back 1 period → $\frac{100}{(1.10)^1} = 90.91$
• Payment 3 (start of period 3): discount back 2 periods → $\frac{100}{(1.10)^2} = 82.64$

Total present value: $100 + 90.91 + 82.64 = 273.55$

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Why Only Up to 3 Periods?

With 3 periods, you can write out every term explicitly. There's no need for the compact formula:

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