How much money is needed to endure a series of lectures costing ₹2500 at the beginning of each year indefinitely, if money is worth 3% compounded annually?

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Accepted Answer

> [!TLDR]
> The key idea is to treat the series of payments as a **perpetuity due** — payments at the start of each year forever. The present value is the amount needed today, which is ₹85,833.33.

To understand why this works, think about what "money worth 3% compounded annually" means. It means that if you invest a sum today at 3% per year, the interest earned each year is exactly enough to cover the ₹2500 payment at the start of that year, without ever touching the principal. The trick is that the first payment happens *immediately* (at the beginning of year 1), so you need to have that ₹2500 ready right now, plus enough principal left over to generate the subsequent payments forever.

The standard formula for a perpetuity (payments at the *end* of each period) is $PV = \frac{C}{r}$, where $C$ is the annual payment and $r$ is the interest rate. But here, payments are at the *beginning* — that's a perpetuity due. The adjustment is simple: each payment is moved one year earlier, so the present value is larger by a factor of $(1+r)$.

Let's work through it step by step.

1. **Identify the variables.**  
   The annual payment at the start of each year is $C = ₹2500$. The annual interest rate is $r = 3\% = 0.03$.

2. **Recall the perpetuity due formula.**  
   For a perpetuity due (payments at the beginning of each period), the present value is:
   $$PV_{	ext{due}} = C + \frac{C}{r}$$
   Why? Because the first payment of $C$ is made immediately (so its present value is just $C$), and the remaining infinite stream of payments (at years 1, 2, 3, …) forms an ordinary perpetuity. Deriving the present value of the full combined stream carefully:

> [!PROOF]
   > Derivation of perpetuity due formula:
   > 
   > The present value of a perpetuity due is:
   > $$PV = C + \frac{C}{1+r} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots$$
   > This is a geometric series with first term $C$ and common ratio $\frac{1}{1+r}$. Sum to infinity:
   > $$PV = \frac{C}{1 - \frac{1}{1+r

Q.How much money is needed to endure a series of lectures costing ₹2500 at the beginning of each year indefinitely, if money is worth 3% compounded annually?

Concept understanding — Present Value Perpetuity

Present Value of a Perpetuity: The Intuition

The Precise Statement

Where Does This Formula Come From?

The Critical Assumption

Real-World Examples …

Q.How much money is needed to endure a series of lectures costing ₹2500 at the beginning of each year indefinitely, if money is worth 3% compounded annually?

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