Time value of money, derived

Almost everything in this course eventually reduces to one sentence: a rupee today is worth more than a rupee later, because today's rupee can be put to work and later's cannot.

Everything below follows from that. None of it is worth memorising, because all of it can be rebuilt in a minute from the first line.

The one assumption

Money can be invested at some rate $r$ per period. That is the entire assumption. If it were false — if there were genuinely nowhere to put money — a rupee now and a rupee later would be worth the same and finance would be a very short subject.

Forwards: future value

One rupee invested for one period becomes $1 + r$. Invested again, the whole amount grows:

$FV_2 = (1+r)(1+r) = (1+r)^2$

The pattern is not a rule to remember, it is just repetition:

$FV_n = PV \times (1+r)^n$

What makes this compounding rather than addition is that the second period's growth applies to the first period's growth as well. That is the only difference between $(1+r)^n$ and $1 + nr$, and over long horizons it is the difference between most of the outcome and a small part of it.

Backwards: present value

Discounting is the same equation rearranged. If $FV = PV(1+r)^n$, then

$PV = \frac{FV}{(1+r)^n}$

Nothing new has happened. Discounting is not a second technique; it is compounding read right to left. People find discounting harder only because the question is less familiar, not because the mathematics is.

The quantity $\frac{1}{(1+r)^n}$ is the discount factor: what one rupee at time $n$ is worth now. At 8% for ten years it is $1/1.08^{10} \approx 0.463$ — ten years at 8% roughly halves the value of a promise.

A stream of payments

A cash flow is a list of amounts with dates. Its value now is the sum of each one discounted:

$PV = \sum_{t=1}^{n} \frac{C_t}{(1+r)^t}$

That is a discounted cash flow. Chapter 20 of the Company analysis subject applies it to a business, and nothing is added there except opinions about what the $C_t$ are.

The annuity, derived

An annuity pays a constant $C$ for $n$ periods. Rather than summing term by term, use the fact that the sum is geometric. Let

$PV = C\left[\frac{1}{1+r} + \frac{1}{(1+r)^2} + \cdots + \frac{1}{(1+r)^n}\right]$

Multiply both sides by $(1+r)$ and subtract the original from the result. Every interior term cancels, leaving

$PV(1+r) - PV = C\left[1 - \frac{1}{(1+r)^n}\right]$

$PV = \frac{C}{r}\left[1 - \frac{1}{(1+r)^n}\right]$

That is the EMI formula, and the loan calculator uses it in exactly this form. A home loan is an annuity where the bank holds the $PV$ and you supply the $C$.

The perpetuity

Let $n \to \infty$. The term $\frac{1}{(1+r)^n}$ goes to zero, and the whole expression collapses to

$PV = \frac{C}{r}$

A payment of ₹1 a year forever, at 8%, is worth ₹12.50. This is the simplest valuation formula in finance and it is the backbone of the terminal value in a DCF — the part of a company's worth attributed to everything after the forecast ends.

If the payment grows at $g$ each period, the same derivation with a growing numerator gives

$PV = \frac{C}{r - g}$

with the obvious and important condition that $g < r$. A growth rate at or above the discount rate makes the value infinite, which is the arithmetic telling you the assumption is wrong rather than the company being priceless.

Periods are not years

The formulas care about periods, not calendars. The rate and the period must match: a 12% annual rate compounded monthly is 1% a month for twelve months, not 12% once. The RBI's deposit directions require interest on domestic term deposits to be calculated at quarterly or shorter rests, which is why a "7% fixed deposit" pays slightly more than 7% over a year. Chapter 2 is entirely about that gap.

Where this goes

Chapter	What it does with this
Fixed income, price and yield	A bond is an annuity plus a final lump sum
Company analysis, DCF	A business is a cash flow with uncertain $C_t$
Corporate finance, NPV	A project is accepted if its $PV$ exceeds its cost
Retirement, the number	A retirement is a perpetuity you hope outlives you

The point

A rupee later is worth less than a rupee now, and $(1+r)^n$ is the exchange rate between them. Compounding and discounting are the same equation read in opposite directions; an annuity is a geometric sum collapsed into a closed form; a perpetuity is that sum with $n$ taken to infinity. Every valuation method later in this course is this chapter with assumptions attached.