Present value and discounting

◈ 6 cards

PV = FV ÷ (1 + r)ⁿ — what a future amount is worth today, why =PV returns it negative, and the direction PV moves when the rate or the wait changes.

Running the clock backwards

Present value is the amount you would need today, at a given rate, to have a stated amount at a stated future date. It is future value in reverse: instead of multiplying by $(1 + r)^{n}$ you divide by it.

$P V = \frac{F V}{( 1 + r ) ^{n}}$

The factor $1/ (1 + r)^{n}$ is the discount factor, the reciprocal of the FV factor. Dividing by something bigger than 1 makes the result smaller, so a present value is always less than the future amount it stands for whenever the rate is positive. That is the eyeball rule for this lesson.

Excel's =PV(rate, nper, pmt, fv, type) does the division. Enter the future amount positive (money you will receive) and Excel returns the present value negative — the deposit you must make now. Most people wrap the call: =-PV(…) reads positive. Decide which you are doing, and do it every time.

Worked example — Maritime Ferries' refit

Maritime Ferries must pay $25,000 for a hull refit in 6 years. How much set aside today, earning 5 % compounded annually, will grow to exactly that?

$P V = \frac{25 , 000}{1.0 5 ^{6}} = \frac{25 , 000}{1.340096} = 18, 655.38$

=-PV(0.05,6,0,25000) → 18,655.38; by the formula =25000/(1+0.05)^6. Check it forwards: 18,655.38 × 1.05⁶ = 25,000.

Now move one input at a time.

Higher rate. At 7 %: =-PV(0.07,6,0,25000) = 16,658.56. A better return means less needs to be set aside — the discount factor 1/1.07⁶ = 0.666342 is smaller than 1/1.05⁶ = 0.746215.
Longer wait. At 5 % but 12 years out: =-PV(0.05,12,0,25000) = 13,920.94. Twice the time is not half the money — it is the 6-year factor squared, 0.746215² = 0.556837.

So: PV falls as r rises, and PV falls as n rises. Both dials turn the same way, opposite to FV's.

One more, then rank without computing

At 4 % for 9 years: =-PV(0.04,9,0,25000) = 17,564.67. Before you compute, you should already be able to rank these. A $25,000 amount in 6 years at 5 % is worth more today than the same amount in 6 years at 7 % (higher rate), and more than the same amount in 12 years at 5 % (longer wait). The 4 %/9-year case sits between: lower rate pushes it up, longer wait pushes it down. Ranking first is how you catch a keying error before the marker does.

Raise the rate or lengthen the wait and the discount factor shrinks, so the present value shrinks with it. The 12-year factor is the 6-year factor squared.