Present Value & the Discount Factor

Accumulation moves money forward in time; discounting moves it back. The discount factor \(v = 1/(1+i)\) pulls a future amount back to its worth today, and the effective rate of discount \(d = i/(1+i)\) measures that same shrinkage as a rate.

By the end you'll be able to compute \(v\) and \(d\) from \(i\), verify the identities \(v = 1-d\) and \(i-d = id\) that tie them together, and use \(v^t\) to find the present value (PV) of an amount due \(t\) years from now.

Predict: as you drag i higher, does the present value of a fixed future amount rise or fall? Drag the slider below to check.

A future amount of 1, due at time t, is worth \(v^t\) today — pulled back one discount factor per year. Drag i and t and watch the present-value bar shrink as either one grows; the dashed curve traces the amount's value at every point on the way back from \(t\) to today.

v = 1/(1+i) = 0.9434 · d = i/(1+i) = 0.0566

v = 1−d: 0.9434 = 1−0.0566 ✓    i−d = i·d: 0.003396 = 0.003396 ✓

Pulling 1, due at t, back to its present value \(v^t\)
future amount (at t) present value (at 0)

If a unit grows to \((1+i)\) after one year, then the amount you'd need to invest now to have 1 in a year is just the reciprocal — that's the discount factor. Run that reciprocal forward \(t\) times and you can pull any future amount all the way back to today.

Formal

You can read \(v = 1/(1+i)\) as the factor that pulls an amount \(F\) due \(t\) years from now back to its present value today: \(PV = F\cdot v^t = F/(1+i)^t\) — the exact reverse of the accumulation \((1+i)^t\) you'd use to grow it forward. The effective rate of discount \(d = i/(1+i) = i\cdot v\) lets you measure the same interest, but as a fraction of the ending balance instead of the starting one. Add the two factors and you get \(v + d = (1+i)/(1+i) = 1\), so \(v = 1-d\) — the present-value bar you saw shrink above and the discount rate are two sides of the same coin. A third identity, \(i - d = i\cdot d\), falls out of the same algebra if you work it through: \(i - i/(1+i) = i\cdot(i/(1+i)) = id\).

Applied

When you borrow, interest is quoted on the starting balance and paid at the end; when you buy at a discount, the rate is quoted on the ending balance and taken off at the beginning — the way a Treasury bill is sold below its face value today instead of paying interest later. If you buy a T-bill paying 1000 at maturity, you pay \(1000\cdot v\) for it now, so the discount you get up front is exactly \(1000\cdot d\).

Worked example

Let \(i = 6\%\). Then \(v = 1/1.06 = \) 0.943396, \(d = 0.06/1.06 = \) 0.056604 (check: \(d = 1-v = 1-0.943396\) ✓), and \(i-d = 0.06-0.056604 = 0.003396 = 0.06\cdot 0.056604 = i\cdot d\) ✓. The present value of 5000 due in 4 years is \(5000\cdot v^4 = 5000(0.943396)^4 \approx\) 3960.47.

Your turn

Same idea, different numbers: \(i = 4\%\). \(v = 1/1.04 = 0.961538\), and \(d = 0.04/1.04 = 0.038462\) (check: \(1 - v = 1 - 0.961538 = 0.038462\) ✓). An amount of 8000 is due in 5 years. Now finish it: \(PV = 8000\cdot v^5 = 8000(0.961538)^5 = \) ____

Reveal the answer

\((0.961538)^5 \approx 0.821927\), so \(PV = 8000\times 0.821927 \approx\) 6575.42. Set \(i = 4\%\) and \(t = 5\) on the slider above and check the present-value bar against this figure.

More info — why v and d have to add to 1

Write both over the same denominator: \(v = \dfrac{1}{1+i}\) and \(d = \dfrac{i}{1+i}\). Adding the numerators, \(v + d = \dfrac{1+i}{1+i} = 1\) exactly, for any \(i\) — not an approximation, so the identities \(v = 1-d\) and \(d = 1-v\) hold precisely, not just for small rates. See the Wikibooks formula sheet in Dive deeper below to check \(v = v^t\) notation and the reciprocal \(i = d/(1-d)\) against this same algebra.

Check your understanding

Question 1 of 4

At an effective annual rate \(i = 0.08\), what is the discount factor \(v\)?

Question 2 of 4

You owe a fixed amount 3 years from now. As the effective annual interest rate \(i\) rises, what happens to that amount's present value today?

Question 3 of 4

Which equation correctly relates the discount factor \(v\) and the effective rate of discount \(d\)?

Question 4 of 4

An amount of 1000 is due in 3 years and \(i = 5\%\). Which expression gives its present value today — as opposed to the accumulated value you'd get by investing 1000 today for 3 years under compound interest \(a(t) = (1+i)^t\)?

Recap

  • The discount factor \(v = 1/(1+i)\) brings 1 due in a year back to today.
  • The present value of \(F\) due in \(t\) years is \(PV = F\cdot v^t = F/(1+i)^t\) — the reverse of accumulating at \((1+i)^t\).
  • The effective rate of discount is \(d = i/(1+i) = i\cdot v\).
  • Three identities tie them together: \(v = 1-d\), \(d = 1-v\), and \(i-d = i\cdot d\).

Dive deeper

Sources

  • Present Value, the Discount Factor, and the Rate of Discount