Questions: Present Value and Discounting

Question 1

If the annual discount rate rises from 5% to 10%, what happens to the present value of $1,000 received exactly 20 years from now?

• A. PV falls slightly, by about 10-15%
• B. PV is halved, because the rate doubled
• C. PV falls dramatically, by roughly 60%
• D. PV is unchanged because the future cash flow is fixed at $1,000

Answer: (C)

At 5%: PV = 1000 / 1.05^20 ≈ $377. At 10%: PV = 1000 / 1.10^20 ≈ $149. That is a drop of about $228, or roughly 60%. This illustrates the extreme sensitivity of long-horizon present values to the discount rate — a consequence of exponential discounting. Doubling the rate does not halve PV because the relationship is exponential, not linear.

Question 2

The discount rate used to compute present value mainly needs to account for expected inflation; there is no reason to adjust for risk.

• T. True
• F. False

Answer: False

The discount rate has (at least) three components: time preference (people prefer money now even with no inflation), expected inflation (a dollar loses purchasing power over time), and a risk premium (uncertain cash flows are worth less than certain ones). Even after adjusting for inflation using real cash flows, you must still discount for time preference and risk. Ignoring the risk component leads to systematically overvaluing risky assets.

Question 3

Explain in economic terms why $1,000 promised 30 years from now is worth less than $1,000 today, even if inflation is zero and there is no default risk.

Model answer: $1,000 today can be invested to earn a real return over 30 years, growing substantially in real purchasing power. Waiting 30 years means forgoing that compounding — the opportunity cost is the future wealth the money would have accumulated. This time preference (preferring consumption sooner rather than later) is real even without inflation or default risk, and it is captured by the real risk-free discount rate.

Discounting is the inverse of compounding. If you can earn a 3% real return, $1,000 today becomes roughly $2,427 in 30 years. Equivalently, a promise of $1,000 in 30 years is worth only about $412 today — you would need to invest $412 now at 3% to have $1,000 in 30 years. The discount rate does not require inflation to be positive; even in a zero-inflation world, capital is productive and time has value.