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Key Concepts
- Higher discount rate → lower present value (future money is worth less today)
- Longer time horizon → lower present value (more discounting periods)
- Positive NPV → investment creates value above the required return
- IRR is the discount rate that makes NPV = 0 — compare to your hurdle rate
- Use nominal cash flows with nominal discount rates, or real cash flows with real rates — never mix them
Examples
Example 1: Single Future Payment
What is $10,000 received 5 years from now worth today at a 6% discount rate?
Future Value (FV): $10,000
Discount Rate (r): 6% per year
Number of Periods (n): 5 years
Formula: PV = FV / (1 + r)^n
PV = $10,000 / (1.06)^5
PV = $10,000 / 1.3382
PV = $7,472.58
Interpretation: $7,472.58 invested today at 6% annually
will grow to $10,000 in 5 years.
Example 2: Ordinary Annuity (Payments at Period End)
What is the present value of receiving $1,000 per year for 10 years at 5% discount rate?
Payment (PMT): $1,000/year
Discount Rate (r): 5% per year
Number of Periods (n): 10 years
Annuity Type: Ordinary (end of period)
Formula: PV = PMT × [1 - (1 + r)^-n] / r
PV = $1,000 × [1 - (1.05)^-10] / 0.05
PV = $1,000 × [1 - 0.6139] / 0.05
PV = $1,000 × 7.7217
PV = $7,721.73
Discounting Schedule:
Year 1: $1,000 / 1.05^1 = $952.38
Year 2: $1,000 / 1.05^2 = $907.03
Year 3: $1,000 / 1.05^3 = $863.84
Year 4: $1,000 / 1.05^4 = $822.70
Year 5: $1,000 / 1.05^5 = $783.53
...
Year 10: $1,000 / 1.05^10 = $613.91
Total PV: $7,721.73
Example 3: Bond Pricing
What is the fair value of a bond with $1,000 face value, 5% coupon, 10-year maturity, when market rates are 6%?
Face Value: $1,000
Coupon Rate: 5% → $50/year coupon payment
Maturity: 10 years
Market Rate: 6% (discount rate)
PV of coupon payments (annuity):
PV = $50 × [1 - (1.06)^-10] / 0.06
PV = $50 × 7.3601 = $368.00
PV of face value at maturity:
PV = $1,000 / (1.06)^10
PV = $1,000 / 1.7908 = $558.39
Bond Price = $368.00 + $558.39 = $926.39
Conclusion: The bond trades at a discount ($926.39 < $1,000)
because the coupon rate (5%) is below the market rate (6%).