Present value and discounting
Present value turns future amounts into today's dollars, letting us compare cash flows that arrive at different times.
Present value
Present value (PV) is what a future amount is worth today, given a discount rate: \(PV = \frac{FV}{(1 + r)^n}\). Receiving 1,000 dollars in 5 years is worth \(\frac{1{,}000}{1.06^5} \approx 747\) dollars today at a 6% discount rate.
PV = \frac{FV}{(1+r)^n}The discount rate
The discount rate reflects the opportunity cost of money: the return available on alternative investments of similar risk. Riskier cash flows are discounted at higher rates, which lowers their present value. Choosing the right discount rate is one of the most important judgements in finance.
Discounting and time
The further away a cash flow, the lower its present value. At 10%, 1,000 dollars due in 1 year is worth 909 dollars today; in 10 years, 386; in 30 years, only 57. This is why long-term projects are so sensitive to the discount rate.
Comparing offers
Present value lets you compare options. Would you rather have 10,000 dollars now or 12,000 in 3 years? At an 8% discount rate, 12,000 in 3 years has a PV of about 9,526 dollars, so taking 10,000 now is better. At a 5% rate, the PV is about 10,366, so waiting is better.
Uneven cash flows
For a series of different cash flows, find the present value of each and add them: \(PV = \sum \frac{CF_t}{(1+r)^t}\). A project paying 2,000 dollars in year 1, 3,000 in year 2, and 4,000 in year 3, discounted at 10%, has a PV of about 1,818 + 2,479 + 3,005 = 7,302 dollars.
Solving for rate or time
The same equation can be rearranged to find the rate or the time. If 5,000 dollars grows to 8,000 dollars in 6 years, the annual rate is \(r = \left(\frac{8{,}000}{5{,}000}\right)^{1/6} - 1 \approx 8.15\%\). Financial calculators and spreadsheet functions such as RATE, NPER, PV, and FV handle these calculations.
Inflation and real returns
Nominal returns include inflation; real returns measure growth in purchasing power. Approximately, real rate = nominal rate - inflation. More precisely, \((1 + r_{real}) = \frac{1 + r_{nominal}}{1 + \text{inflation}}\). A 7% nominal return with 3% inflation gives a real return of about 3.9%.
Worked example
A lottery winner can take 1 million dollars today or 70,000 dollars a year for 20 years (1.4 million total). At a 6% discount rate, the PV of the annual payments is about 70,000 x 11.47 = 803,000 dollars, so the lump sum is worth more. At a 3% rate, the PV is about 1.04 million, and the payments become slightly better. The answer depends on the discount rate.
Key takeaways
Present value discounts future amounts: PV = FV / (1 + r)^n. The discount rate reflects opportunity cost and risk. Distant cash flows have much lower present values. PV allows fair comparison of offers. Uneven cash flows are discounted individually and summed. The equation can be rearranged to find rates or time. Real returns adjust nominal returns for inflation.
Flashcard
Today's value of future money
Check your understanding
What is the present value of 2,000 dollars received in 2 years at a 10% discount rate?