What is discounting (quick definition)
– Discounting is the process of converting expected future payments into an equivalent value today. That today-value is called the present value. Discounting reflects two things: the time value of money (money available now is worth more than the same amount later) and the compensation investors require for risk.
Key concepts and definitions
– Present value (PV): the value today of money or cash flows that will arrive in the future.
– Future value (FV): the amount a payment will be worth at a specified future date.
– Discount rate: the interest rate used to convert future cash flows into present value. Higher discount rates reduce present value.
– Discount factor: a multiplier (dependent on time and the discount rate) applied to each future cash flow to get its present value.
– Cost of capital: the effective rate a firm pays to obtain funds; commonly used as the discount rate for valuing company projects or earnings.
– Coupon payments: periodic interest payments from a bond. When valuing a bond, each coupon and the final par (face) value are discounted to today.
– Callable bond: a bond the issuer can redeem before maturity. This feature affects valuation because it changes expected future cash flows.
– Junk (high‑yield) bond: a bond rated poorly by rating agencies with higher risk of default. These typically trade at deeper discounts and offer higher yields to compensate investors.
– Breakpoint discount (mutual funds): a reduced front‑end sales load (fee) on Class A mutual funds that applies once investment reaches specified volume thresholds.
How discount
ing works
Discounting converts future cash flows into their equivalent value today (present value). The process relies on three inputs: the future cash amount, the time until payment, and the discount rate (the return you require to forgo consumption today). The central idea is the time value of money: one dollar today is worth more than one dollar tomorrow because you can invest it and earn a return.
Key formulas and definitions
– Present value (PV) of a single future amount:
PV = FV / (1 + r)^n
where FV = future value, r = periodic discount rate, n = number of periods.
– Discount factor: DF(n) = 1 / (1 + r)^n. Multiplying a future cash flow by DF(n) yields its PV.
– PV of a level annuity (series of equal payments C for n periods):
PV = C * [1 – (1 + r)^(-n)] / r
– PV of a perpetuity (infinite level payments C starting one period from now):
PV = C / r
(Valid when r > 0 and payments are level.)
– Continuous discounting:
PV = FV * e^(-r * t)
where e is the base of natural logarithms and t is time in years.
Worked numeric examples (step‑by‑step)
1) Single future payment
– Problem: You will receive $1,000 in 3 years. Required return r = 5% annually. Find PV.
– Compute DF(3) = 1 / (1.05)^3 = 1 / 1.157625 = 0.86384.
– PV = 1,000 * 0.86384 = $863.84.
2) Bond-like cash flows (annuity + final principal)
– Problem: A bond pays $50 annually for 5 years and returns $1,000 at maturity. Required yield r = 6%.
– PV of coupons = 50 * [1 – (1.06)^(-5)] / 0.06.
Calculate (1.06)^5 = 1.3382256; inverse = 0.74726; numerator = 1 – 0.74726 = 0.25274.
PV coupons = 50 * 0.25274 / 0.06 = 50 * 4.21236 = $210.62.
– PV of principal = 1,000 / (1.06)^5 = 1,000 * 0.74726 = $747.26.
– Total
– Total PV = $210.62 + $747.26 = $957.88.
Interpretation: the bond’s price ($957.88) is below par ($1,000) because the required yield (6%) is higher than the coupon rate (50/1000 = 5%). When required yield > coupon rate, existing fixed coupons are less attractive, so price falls below face value.
Other common discounting cases
1) Uneven cash flows
– For cash flows CF1, CF2, …, CFn at times 1..n and a constant discount rate r, PV = sum_{t=1..n} CFt / (1+r)^t.
– Worked example: cash flows of $300, $400, $500 in years 1–3; r = 8%.
– PV1 = 300 / 1.08 = 277.78
– PV2 = 400 / (1.08)^2 = 400 / 1.1664 = 342.97
– PV3 = 500 / (1.08)^3 = 500 / 1.259712 = 397.08
– Total PV = 277.78 + 342.97 + 397.08 = $1,017.83
2) Perpetuity (constant indefinite cash flows)
– Definition: a perpetuity pays a fixed cash flow C each period forever.
– Formula: PV = C / r (requires r > 0 and payments start one period from now).
– Worked example: a perpetual payment of $50 annually, r = 6% → PV = 50 / 0.06 = $833.33.
– Use case: valuing preferred stock with fixed dividends or a perpetuity-style annuity.
3) Growing perpetuity (cash flows grow at rate g 0)
– Growing perpetuity (constant growth g < r): PV = C / (r – g)
– Multiple cash flows / irregular schedule: PV = sum_{t} CF_t / (1 + r_t)^{t} (use appropriate rate r_t for each cash flow)
– Continuous compounding: PV = FV * e^{-r t}
Worked numeric examples (step-by-step)
1) Single future sum
– Problem: What is the present value (PV) of $1,000 to be received in 3 years if the discount rate is 5% per year, compounded annually?
– Formula: PV = FV / (1 + r)^t
– Calculation: PV = 1,000 / (1.05)^3 = 1,000 / 1.157625 = 863.84
– Interpretation: At 5% annual discount rate, $1,000 in three years is worth about $863.84 today.
2) Ordinary annuity (level, end-of-period payments)
– Problem: You will receive $200 at the end of each year for 5 years. Discount rate = 6% annually. What is PV?
– Formula: PV = C * [1 – (1 + r)^{-n}] / r
– Calculation: PV = 200 * [1 – (1.06)^{-5}] / 0.06
– (1.06)^{-5} = 0.747258
– Numerator = 1 – 0.747258 = 0.252742
– PV = 200 * 0.252742 / 0.06 = 200 * 4.21237 = 842.47
– Interpretation: The 5-year stream is worth $842.47 today at 6%.
3) Annuity due (payments at beginning of period)
– Problem: Same $200 but paid at the beginning of each year for 5 years, r = 6%.
– Method: Multiply the ordinary annuity PV by (1 + r).
– Calculation: PV_due = 842.47 * 1.06 = 892.02
– Interpretation: Payments at the beginning are worth more (you receive each payment one period earlier).
4) Perpetuity and growing perpetuity
– Perpetuity: $50 annually forever, r = 8%. PV = 50 / 0.08 = 625.
– Growing perpetuity (Gordon formula): First payment C = $2 next period, growth g = 3%, r = 8%. PV = 2 / (0.08 – 0.03) = 2 / 0.05 = 40.
5) Continuous discounting
– Problem: PV of $1,000 in 3 years with continuous compounding at r = 5%.
– Formula: PV = 1,000 * e^{-0.05 * 3}
– Calculation: e^{-0.15} ≈ 0.860707 ⇒ PV ≈ 860.71
Checklist before you discount cash flows
– Choose the correct discount rate (reflects time value + risk premium).
– Match nominal vs. real: use nominal rates with nominal cash flows; use real rates with inflation-adjusted cash flows.
– Match compounding frequency: annual, monthly, continuous — adjust r and t consistently.
– Match payment timing: beginning (annuity due) vs. end (ordinary annuity).
– For multiple maturities, use the term structure: discount each cash flow with the zero-coupon (spot) rate for that maturity.
– Account for taxes, fees, and credit risk if relevant (adjust cash flows or discount rate).
– Maintain sufficient decimal precision until the final result to reduce rounding error.
Practical steps to compute PV for a stream of cash flows
1. List each cash flow CF_t and its timing t (in years or consistent periods).
2. Determine the appropriate discount rate(s) r_t for each t.
– If using a single rate, ensure it reflects required return and compounding.
– If using a term structure, obtain zero rates for each maturity.
3. Apply PV = sum CF_t / (1 + r_t)^{t} (or PV = sum CF_t * e^{-r_t t} for continuous).
4. Sum the discounted values to get total PV.
5. Check sensitivity: recompute PV with plausible alternative rates to see how results change.
Common mistakes to avoid
– Mixing nominal cash flows with real discount rates (or vice versa).
– Forgetting to adjust r when switching compounding frequency (e.g., annual vs. monthly).
– Treating an annuity due as an ordinary annuity.
– Using the same discount rate for all maturities when rates vary significantly across the yield curve.
Assumptions and limitations
– Most quick formulas assume deterministic cash flows and a constant discount rate; real-world uncertainty and changing rates require more sophisticated models.
– The discount rate encodes time preference, inflation expectations, and risk premia; selecting it involves judgment.
– Growing-perpetuity formulas require growth g < r; otherwise the formula produces invalid or infinite values.
Further reading and references
– Investopedia — Discounting definition and examples: https://www.investopedia.com/terms/d/discounting.asp
– U.S. Department of the Treasury — Daily treasury yield curve rates (useful for term structure/spot rates): https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
– Federal Reserve Education — Present value and discounting basics: https://www.federalreserveeducation.org
Educational disclaimer
This explanation is educational only and not individualized investment advice. It shows formulas and examples for learning. For decisions about specific investments or tax treatment, consult a qualified professional.