Time Value of Money

Three forces put a price on waiting. The first is earning power: a dollar received now can be invested, so receiving it later means giving up the return it could have earned in the meantime, which is its opportunity cost. The second is Inflation: a dollar paid in ten years will usually buy less than a dollar today. The third is risk: a promise of future money can be broken, delayed or cut, while cash in hand cannot.

Together, these forces mean dollars from different years are not the same unit. Adding $10,000 received today to $10,000 received in 20 years gives $20,000 of nominal money, but not $20,000 of value. The time value of money is the toolkit for moving amounts to a common date so they can be compared or added honestly.

Two operations do the work. Compounding moves a sum forward by applying a rate of return for each period; it is the engine of compound interest. Discounting runs the same arithmetic backward, dividing a future sum by the growth it would have needed to get there. Loan payments, bond prices, pension lump sums and annuity quotes are all built on these two steps.

Future value asks what a sum grows to: FV = PV × (1 + r)^n. At 6% a year, $25,000 today grows to $59,913.95 in 15 years. Present value asks the reverse question, what a future sum is worth now: PV = FV ÷ (1 + r)^n. At the same 6%, $50,000 due in 15 years is worth $20,863.25 today.

Two refinements cover most real cases. When interest compounds more than once a year, divide the annual rate by the number of periods per year and multiply the years by it, so monthly compounding at 6% uses 0.5% a month for 12 × n months. When a problem involves a level series of payments, such as a pension, an Annuity or a loan, the present value of the whole stream is payment × [1 − (1 + r)^−n] ÷ r, for payments made at the end of each period. Payments at the start of each period, like rent, are worth one period’s growth more, so multiply by (1 + r).

Financial calculators and spreadsheet functions organize all of this around five inputs: the number of periods, the rate per period, present value, the payment and future value. Know any four and the fifth follows. Solving for the payment gives a mortgage payment and its amortization schedule; the Rule of 72 is a shortcut for the number of periods it takes money to double.

Net present value (NPV) handles uneven cash flows: discount each amount to today, count money you pay out as negative, and add them up. A positive NPV means a choice is worth more than it costs at your discount rate. The rate that makes NPV exactly zero is the internal rate of return (IRR).

The formulas are mechanical; the rate is the judgment call, and it can flip the answer. The right rate is the return you could earn on an alternative with similar risk and timing.

For a guaranteed payment, such as a pension, a Treasury bond or Social Security, a low-risk rate fits, because the payment itself carries little risk. Discounting a guaranteed stream at an expected stock return makes it look smaller than it is. Federal law takes this view for pension lump sums: under Internal Revenue Code section 417(e), a defined benefit plan’s lump sum generally cannot be less than the present value figured with an IRS mortality table and interest rates drawn from investment-grade corporate bond yields.

For purchasing power, discount at inflation. The result is a today’s-money figure: what a future sum would buy at current prices, so at 3% inflation $100,000 received in 10 years buys roughly what $74,400 buys today. Discount at a real rate of return when you want both effects at once, and never pair nominal cash flows with a real rate.

Treasury bills show discounting in its plainest form. A bill is sold at a discount or at face value and pays face value at maturity, so the gap is your interest. A 52-week bill with a $1,000 face value bought for $960 pays $40 at maturity, about 4.2% on the $960 you paid.

Almost every large financial decision trades money at one date for money at another. Spotting the time-value problem inside a decision tells you what to compare: not the raw sum of the dollars, but their value at a common date, discounted at a rate that fits their risk. The same lens works whether you are saving, borrowing or deciding how to draw an income in retirement. A few familiar examples:

Time-value errors rarely come from the arithmetic, which a spreadsheet handles. They come from the inputs. The rate and the periods must match: a 6% annual rate paired with 180 monthly periods treats money as earning 6% a month, and the present value comes out far too small. Payment timing counts too: money received at the start of a period has one more period to grow than money received at the end. Other frequent slips: