Present Value of a Single Sum

Builds onFuture Value of a Single Sum — if this page feels steep, start there.

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

$PV$: present value — what the future payment is worth in today's money
$FV$: future value — the amount you'll actually receive later
$\frac{FV}{\cdots}$: a fraction is division: we divide the future amount by the growth it would have needed
$(1+r)^n$: the total growth factor over n periods at rate r per period

Why it must be true

A dollar tomorrow is worth less than a dollar today, because today's dollar could be invested and grow. Present value asks: how much would I need today so that, compounded at $r$ , it becomes exactly $FV$ in $n$ periods?

Since growing forward means multiplying by $(1+r)^n$ , coming back means dividing by it. Discounting is just compounding run in reverse — the two formulas are one identity read in opposite directions.

The derivation

Let $PV$ be the unknown amount invested today. After $n$ periods of compounding it becomes

PV(1+r)^n = FV

Divide both sides by the growth factor $(1+r)^n$ :

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

The factor $\frac{1}{(1+r)^n}$ is the discount factor — the price today of $1 delivered in $n$ periods.

When to reach for it

One known future amount that must be stated in today's money — pricing a zero-coupon payoff, booking a single liability, or checking what a promise is worth now.

Listen for

“worth today / value now”“fair price of a zero-coupon”“discounted at”“a payment of … due in n years”“how much must be set aside today”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Rule of 72 backwards: value halves every 72 ÷ rate% periods. \ pitfalls: [0,000 due in 18 periods at 8% ≈ two halvings ≈ $2,500 — instantly eliminates half the choices.
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PV must be below FV for any positive rate — and the compound-discount answer is always smaller than the simple-discount value $FV/(1+rn)$ , so bracket between the two.
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One period is easy: knock off roughly r%. \ pitfalls: [,000 due next period at 7% ≈ $935 (true: 934.58).

Traps in applying it

✗Discounting every cash flow by one period regardless of when it arrives — each flow gets its own exponent.
✗Using a nominal rate against inflation-adjusted (real) cash flows, or vice versa.
✗Rate/period mismatch — a semiannual discount rate with n counted in years.

Limits & criticisms

The answer is only as good as the discount rate, and the formula gives no help choosing it — risk-adjusting the rate is where the real judgement (and disagreement) lives. It also uses a single flat rate, while actual term structures slope: a 2-year and a 20-year cash flow rarely deserve the same r.

Where it came from

Discounting was born in the insurance office, not the classroom. Johan de Witt (1671) and Edmond Halley (1693) — the comet astronomer — priced life annuities by discounting future payments against mortality tables. Irving Fisher (The Rate of Interest, 1907) made present value the foundation of capital theory, and John Burr Williams (1938) built modern equity valuation on it.

Today it is the single most-used calculation in finance: every DCF valuation, bond price, lease and pension liability is a present value. If a number in finance refers to the future, someone has discounted it.

One identity, 3 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Price a future cash flow

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

The core act of valuation: divide by the growth factor to state a future amount in today's dollars.

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Implied yield from a price

r = \left(\frac{FV}{PV}\right)^{1/n} - 1

Given what you pay and what you receive, this face backs out the return the price is offering — exactly how a zero-coupon bond's yield is quoted.

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Implied horizon

n = \frac{\ln(FV/PV)}{\ln(1+r)}

Price, payoff and rate pin down the only maturity consistent with all three — logarithms unlock the exponent.

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On the BA II Plus

Worked example: A note bought today for $24,998.35 pays $32,000.00 at maturity in 10 periods. What per-period discount rate does that price imply?

1.[2ND] [CLR TVM]always clear the worksheet first
2.10 [N]
3.24,998.35 [+|-] [PV]cash paid out is negative
4.0 [PMT]no recurring payment — zero it explicitly
5.32,000 [FV]
6.[CPT] [I/Y]compute the unknown

→ 2.5%

Where it leads

Master this and the following come almost for free:

Present Value of an Ordinary Annuity Net Present Value (NPV)Bond Price & Yield to Maturity Zero-Coupon Bond & Spot Rates Put–Call Parity

Drill this formula Back to the library