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Work out what one cash flow is worth today

CFA® Level I · Quantitative Methods · The Time Value of Money in Finance · QM.1.b

The idea

Present value runs the future-value relationship backwards: PV = FV / (1 + r)^n. The discount rate is the return available on an equivalent alternative, so discounting asks what sum today would grow into the promised amount. Two things drive the answer: the rate and the number of periods, and both push in the same direction — a higher rate or a longer wait means a smaller present value. This is the single operation underneath bond pricing, net present value and every valuation model, which all reduce to discounting a set of future amounts and adding them up.

Formulas

Present value of a single cash flow

PV = FV ÷ (1 + r)ⁿ

FV
the amount received at the end
r
the discount rate per period
n
periods until it is received

The same relationship as future value, read backwards. Raising the rate or the horizon lowers the present value.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is the present value of $1,000 received in three years, discounted at 5%?

  1. $863.84 — correct
  2. $1,157.63Why not: 1,000 × 1.05³ — compounded forwards instead of discounted back. A present value is always smaller than the amount received later.
  3. $850.00Why not: Simple discounting: 1,000 × (1 − 0.05 × 3). Discounting divides by (1.05)³ rather than subtracting three lots of 5%.

1,000 ÷ 1.05³ = $863.84. Multiplying instead of dividing computes a future value and moves the cash flow the wrong way through time.

A harder one

Two cash flows of $500 arrive in one and two years. At 8%, what are they worth today?

  1. $1,000.00Why not: The two amounts added up with no discounting at all. Money arriving later is worth less today.
  2. $891.63 — correct
  3. $396.92Why not: 500 ÷ 1.08³ — one cash flow, discounted three years. There are two flows, at one and two years.

500/1.08 + 500/1.08² = $891.63. Each cash flow is discounted by its own number of periods, then the results are added.

The hardest tier

Holding the amount and the horizon fixed, what happens to a present value as the discount rate rises?

  1. It risesWhy not: Backwards. The rate sits in the denominator, so a larger rate makes the present value smaller.
  2. It is unchangedWhy not: The rate is the whole of the discounting. Change it and the present value changes with it.
  3. It falls — correct

A higher rate means a smaller sum today would suffice to reach the same future amount, so present value falls. The effect compounds with the horizon, which is why long-dated cash flows are the most rate-sensitive.

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