Present Value: Discounting a Future Amount

The intuition

A dollar today is worth more than a dollar tomorrow — because today's dollar can be invested and grow. Present value (PV) flips compound interest backwards: given a future amount and a discount rate, how much do we need today to reach that amount?

The formula

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

where $FV$ = future value, $r$ = discount rate per period, $n$ = number of periods. The denominator $(1+r)^n$ is the discount factor.

Equivalent form: $PV = FV \cdot (1+r)^{-n}$.

Step-by-step solution

Setup: $FV = 10{,}000$, $r = 0.07$, $n = 5$.

Step 1 — Build the discount factor. $(1 + 0.07)^5 = 1.07^5 \approx 1.40255$

Step 2 — Divide future value by discount factor. $PV = \dfrac{10{,}000}{1.40255}$

Step 3 — Compute: $PV \approx \boxed{7129.86}$

So you'd need about $7,130 today, at 7%, to grow to $10,000 in 5 years.

Verification — compound forward

$7129.86 \cdot 1.07^5 = 7129.86 \cdot 1.40255 \approx 10{,}000 \checkmark$

What drives PV

• Higher discount rate ⟶ smaller PV (future money worth less today).
• Longer time horizon ⟶ smaller PV (more compounding to undo).
• Smaller FV ⟶ smaller PV (proportional).

Quick sensitivity: doubling $r$ from 7% to 14% over 5 years cuts PV from ~$7,130 to ~$5,194.

Non-annual compounding

If compounding is $n$ times per year for $t$ years: $PV = \dfrac{FV}{(1 + r/n)^{nt}}$

Continuous compounding: $PV = FV \cdot e^{-rt}$

Where PV lives in finance

• Valuing bonds (sum of PV of coupons + PV of face value)
• NPV and DCF analysis of investments
• Loan pricing (PV of a stream of payments = loan amount)
• Pension liabilities (PV of expected benefits)
• Evaluating "X today vs. Y in Z years" offers

Common mistakes

• Dividing instead of raising to a power. $(1+r)^n$, not $1 + rn$. Using linear growth for the discount gives a much smaller discount than the true compound one.
• Mismatching $r$ and $n$. Monthly cash flow ⟶ monthly rate and month count.
• Forgetting the sign of time. PV is always before the cash flow; if you're pulling money forward in time, you're future-valuing, not discounting.

Try it in the visualization

A shrinking bar illustrates how $10,000 five years out collapses down to the present value as you slide the discount rate up, and how it re-expands when the rate drops.