Present and future value

In depth

Moving an amount to any date

Present value is a special case of a more general rule. An amount $X$ at period $j$ is worth $X(1 + r)^{k - j}$ at period $k$. If $k$ is later than $j$, the exponent is positive and the amount is compounded forward; if $k$ is earlier, the exponent is negative and the amount is discounted back. The date you move everything to is the comparison date (some textbooks call it the focal date). It need not be today.

Why choose another date? Because that is sometimes where the question lives. A business rescheduling a debt wants the single payment, at a new date, that is worth the same as the payments it replaces; a saver wants to know what a deposit will be worth at retirement. As long as one rate applies throughout, the choice of comparison date never changes which of two options is worth more. Moving every amount from one date to another multiplies all their values by the same factor.

Each value at year 4 is the value at year 0 multiplied by 1.06⁴ = 1.262477.

The $9,000 in two years is worth the most at either date, by a whisker: $9.97 more than $8,000 today and $12.58 more at year 4, the same gap grown by 1.262477. The $10,000 in four years, the largest sum, is worth the least. The values change with the date; the ranking does not.

Replacing payments: the equation of value

When a borrower and a lender agree to replace one set of payments with another, the new payments must be worth the same as the old ones at the agreed rate. Writing that condition at a comparison date gives an equation of value: the old payments, each moved to the date, add up to the same total as the new payments, each moved to the same date. Any date gives the same answer. The convenient one is usually the date of the unknown payment, because it then needs no factor.

Suppose a borrower owes $3,000 in one year and $4,000 in three years, and the lender agrees to take one payment $X$ at year 2 instead, at 5% a year. At year 2 the first debt is a year overdue, so it is compounded once; the second is a year early, so it is discounted once: $X = 3{,}000 \times 1.05 + 4{,}000 \div 1.05 = 3{,}150 + 3{,}809.52 = 6{,}959.52$. Written at time 0 instead, the equation is $X \div 1.05^2 = 3{,}000 \div 1.05 + 4{,}000 \div 1.05^3$, which gives the same $6,959.52. Never add the two debts as they stand: $7,000 mixes dollars from different dates.

Amounts due between compounding dates

Nothing in the formula requires $n$ to be a whole number. An amount due in 7 months, discounted at 6% compounded quarterly, is $7 \div 3 = 2\tfrac{1}{3}$ quarters away, so its discount factor is $1.015^{-7/3} = 0.965856$. The same factor comes from the effective annual rate, $1.015^4 - 1 = 6.1364\%$, applied for 7/12 of a year, because $(1.015^4)^{7/12} = 1.015^{7/3}$.

In practice lenders do not always compound over part of a period. Some charge simple interest for the fraction, and the day-count convention (actual/365, 30/360 and others) varies by market and by contract, as the lesson on simple and compound interest explains. When a contract fixes the convention, follow it to the cent. For valuing and comparing amounts, the fractional exponent is the standard approach and the one this course uses.

When the rate changes over time

The single-rate formula assumes the same rate in every period. Often the rate changes: a bank resets a variable savings rate, and markets charge different rates for different horizons. The rule generalises directly. An amount is multiplied by the growth factor of each period it passes through, whatever that period’s rate. Over periods with rates $r_1, r_2, \ldots, r_n$, the future value is $\text{PV}(1 + r_1)(1 + r_2)\cdots(1 + r_n)$, and a present value divides by the same product.

The Sandpiper Online savings account Ana plans to use shows why it matters. Its variable rate was 3.2% compounded monthly in 2026 and is now 3.6% (both fictional). A fictional saver who left $10,000 in an account like it for six months at 3.2% and then six months at 3.6% would end the year with $10{,}000 \times (1 + 0.032/12)^6 \times (1.003)^6 = 10{,}345.35$ dollars. No single quoted rate describes that year; the right summary is the constant rate that gives the same product of factors.

That constant rate is a geometric mean: $\big[(1 + r_1)(1 + r_2)\cdots(1 + r_n)\big]^{1/n} - 1$. For a three-year deposit paying 4%, 5% and 6%, the growth factor is $1.04 \times 1.05 \times 1.06 = 1.15752$, and the equivalent constant rate is $1.15752^{1/3} - 1 = 4.9968\%$, a shade below the simple average of 5%. When interest rates move gently the gap is tiny. When returns swing, it is large: a fund that gains 20% and then loses 10% averages 5% a year arithmetically, but $1,000 becomes $1{,}000 \times 1.2 \times 0.9 = 1{,}080$ dollars, the same as 3.923% a year for two years. Growth factors multiply; rates do not add.

Markets often quote a different rate for each horizon: one rate for money borrowed for a year, another for five years. Discounting each amount at the rate for its own horizon is the same idea, and later courses on bonds develop it.

Solving for the unknown

Because the equation links four quantities, there are four kinds of single-amount problem. The table collects them, with the rate and the number of periods always measured in the same period.

Module 1 explains the root and the logarithm. Two habits matter here. First, check every answer by substituting it back: $9{,}400 \times 1.042113^{1.5}$ should return $10,000. Second, check its size. A rate above 50% a year or a wait of three hundred years usually means a slip, such as a percentage typed as a whole number or months mixed with years.

The calculator mindset

Financial calculators and spreadsheets solve the same equation. A financial calculator has keys for the number of periods (N), the rate per period (I/Y, entered as a percentage), the present value (PV), a level payment (PMT) and the future value (FV): enter any four and compute the fifth. For a single amount the payment is zero. Calculators also have settings for payments and compounding periods a year; this course always works with the rate per period, so set both to 1 or follow your model’s manual.

Two conventions catch people out. Calculators and spreadsheets treat money you receive as positive and money you pay as negative, so the present value and the future value of one deal carry opposite signs; enter both as positive when solving for a rate or a time and you get an error. In a spreadsheet, =PV(0.02, 4, 0, 10000) returns −9,238.45: the amount you would pay today to receive $10,000 after four quarters. =FV(rate, nper, pmt, pv), =RATE(nper, pmt, pv, fv) and =NPER(rate, pmt, pv, fv) solve for the other unknowns, with pmt set to 0. Whatever the tool, the rate must be the rate per period that matches nper.

How sensitive is a present value to the rate?

The further away an amount is, the more its present value moves when the rate changes. Raise the rate from 5% to 6% a year and the present value of $1,000 due in one year falls from $952.38 to $943.40, by 0.94%. Due in ten years, it falls from $613.91 to $558.39, by 9.04%. Due in thirty years, from $231.38 to $174.11, by 24.75%. The fall grows with the wait, almost in proportion over short waits and a little less over long ones. This is why the values of long-term bonds, pension promises and distant business cash flows swing so much more with interest rates than the values of near ones.