Time Value of Money

The present value question

Time value of money compares cash flows that occur at different dates. For CFA Level I, the core idea is that a dollar available now can be invested, while a future dollar cannot be used until it arrives. Present value moves a future cash flow back to today. Future value moves a current amount forward. Both operations depend on a rate per period and the number of periods.

For one future cash flow, use PV = FV ÷ (1 + r)^n. FV is the amount received later, r is the return per period, and n is the number of periods. The matching future value equation is FV = PV × (1 + r)^n. Write the timeline first. Label the date of each amount, then check whether the question asks for today’s value, a future balance, a rate, or a number of periods.

Discount one cash flow

A bond will pay $1,000 exactly one year from now. If investors require 5 percent for that one-year risk, its present value is $1,000 ÷ 1.05 = $952.38. The present value is below $1,000 because the investor gives up the use of money for a year. If the same payment arrives in two years, PV = $1,000 ÷ 1.05² = $907.03. The extra year of waiting lowers value, assuming the discount rate stays 5 percent.

This is a present value calculation, not an expected return forecast. It answers what a specified future amount is worth at a stated required rate. If the rate changes, present value changes in the opposite direction: a higher discount rate produces a lower present value, all else equal.

Keep rate and period consistent

The rate must match the time step. If cash flows are annual and n is measured in years, use an annual rate. If there are monthly periods, use a monthly rate. Suppose a nominal annual rate is 12 percent compounded monthly. The periodic rate is 1 percent and a one-year amount has 12 periods, so the one-year accumulation factor is 1.01^12, not 1.12. If a problem gives an effective annual rate, convert it before using monthly periods: monthly effective rate = (1 + annual effective rate)^(1/12) − 1.

A frequent error is entering 12 as the number of periods when the cash flows are annual, or using 12 percent in a monthly calculation where the stated convention calls for 1 percent each month. Mark the unit beside every input: dollars, annual periods, monthly rate. Many financial calculator errors begin as a time-unit error.

Ordinary annuities and annuities due

An annuity is a series of equal cash flows at regular intervals. An ordinary annuity pays at the end of each period. An annuity due pays at the beginning. The timing difference matters because each annuity-due payment is received one period earlier.

The present value of an ordinary annuity is PMT × [1 − (1 + r)^−n] ÷ r. For an annuity due, multiply that ordinary-annuity value by (1 + r). The future value of an ordinary annuity is PMT × [(1 + r)^n − 1] ÷ r; multiply by (1 + r) for an annuity due. In a question, do not memorize a label and stop. Identify the first payment date and count how many payments occur.

Consider four end-of-year payments of $500 with a 6 percent annual rate. At time zero, their present value is $500 × [1 − (1.06)^−4] ÷ 0.06, approximately $1,732.55. If the same four payments arrive at the beginning of each year, their value at time zero is $1,732.55 × 1.06, or approximately $1,836.50. The additional value comes from each payment arriving one year sooner.

A perpetuity is a level payment stream that continues indefinitely. For a level perpetuity with the first payment one period from now, PV = PMT ÷ r. A growing perpetuity with next period’s payment PMT1 growing at rate g has PV = PMT1 ÷ (r − g), provided r is greater than g. Check whether the problem gives the next payment or the most recent payment. If it gives PMT0, compute PMT1 = PMT0 × (1 + g) before applying the formula.

Uneven cash flows

When cash flows differ, discount each one separately and add the present values: PV = Σ CFt ÷ (1 + r)^t. Suppose an investment pays $300 after one year, $500 after two years, and $700 after three years. At 8 percent, PV = 300/1.08 + 500/1.08² + 700/1.08³, about $1,266.35. The method makes the timing visible and prevents adding amounts from different dates as though they were simultaneous.

If a series has a regular portion and a terminal value, find a useful valuation date for each component. A common equity valuation pattern is to value forecast dividends or free cash flows explicitly, then add the present value of a continuing value at the end of the forecast horizon. Discount the continuing value from that horizon back to today. Do not add a terminal value at year five directly to cash flows already expressed at time zero.

Solve for an unknown rate or period

Sometimes the amount today and amount later are known, and the rate or time is unknown. Rearrange FV = PV(1 + r)^n. For one period, r = FV/PV − 1. For multiple periods, r = (FV/PV)^(1/n) − 1. For example, $800 growing to $1,000 over five years has an annual compound rate of (1,000/800)^(1/5) − 1, or about 4.56 percent.

When the cash-flow pattern is not a single amount, a calculator or spreadsheet can solve for the internal rate of return. The internal rate of return is the discount rate that makes the net present value of the cash flows equal zero. It can have multiple solutions or fail to rank mutually exclusive projects consistently when cash flows change sign more than once. Treat it as a property of the cash-flow pattern, not automatically the best decision rule.

Nominal rates, effective rates, and compounding

A quoted nominal annual rate does not include the full effect of compounding frequency. If a nominal annual rate i is compounded m times per year, the periodic rate is i/m, and the effective annual rate is (1 + i/m)^m − 1. At 8 percent compounded quarterly, EAR = (1 + 0.08/4)^4 − 1, about 8.24 percent. Effective annual rates let you compare alternatives with different compounding schedules on a common annual basis.

Do not confuse a continuously compounded rate with a rate compounded a finite number of times. With continuous compounding, FV = PV × e^(rt), and PV = FV × e^(−rt). The exponential form is a model convention frequently used in finance and derivatives. Follow the convention specified in the question and keep the rate and time units aligned.

Inflation and real versus nominal value

Nominal cash flows are expressed in money units at the date they are paid. Real cash flows are expressed in purchasing power at a base date. Discount nominal cash flows with a nominal rate and real cash flows with a real rate. The Fisher relation is (1 + nominal rate) = (1 + real rate)(1 + expected inflation). For modest rates, people sometimes approximate nominal rate as real rate plus inflation, but an exact calculation should use the multiplicative relation.

For example, if a nominal return is 6 percent and inflation is 2 percent, the exact real return is 1.06/1.02 − 1 = 3.92 percent. Subtracting gives the approximation 4 percent. The approximation is close but not exact. Avoid mixing an inflation-adjusted cash-flow forecast with a nominal discount rate because that counts inflation inconsistently.

Cash-flow signs and calculator setup

Assign positive and negative signs based on perspective. An investor who pays $1,000 today and receives money later has a negative initial cash flow and positive future cash flows. A borrower who receives loan proceeds and makes repayments has the reverse signs. Calculator functions such as NPV and IRR depend on that convention. If every cash flow has the same sign, an IRR may not exist.

Before entering data, identify the valuation date, payment frequency, compounding convention, and whether the first payment is immediate or delayed. Use a rough estimate to check the result. A future value should exceed present value when the rate is positive and the time period is positive. A present value should be below a positive future amount under those same conditions. A result that violates the direction check often signals a sign, mode, or period mistake.

A short CFA-style question

An analyst estimates a $2,000 payment in three years. The annual discount rate is 4 percent. What is the present value? Compute 2,000/(1.04)^3 = $1,777.98. A tempting but incorrect calculation is 2,000 × (1.04)^3, which compounds the payment forward and answers a different question. Another common wrong answer uses 2,000/(1 + 0.04 × 3), which applies simple interest rather than annual compounding.

The fastest reliable approach is to write the formula before using the calculator: PV = FV/(1+r)^n. Enter FV as 2,000, r as 0.04, and n as 3. Then check that the answer is less than $2,000. The arithmetic is short, but the setup is the skill being tested.

How to practice this topic

Mix question types after learning the formulas. A set should include present and future value, uneven cash flows, annuities, perpetuities, effective rates, inflation, and solving for an unknown. Explain why each cash flow belongs at a particular date. If an answer is wrong, label the mistake: payment timing, rate conversion, exponent, sign, or calculator mode. Redo the problem later without looking at the solution.

CFA Institute identifies time value of money within the quantitative methods knowledge base and provides curriculum-aligned practice in the Learning Ecosystem. Make sure your practice questions match the exam year you will sit. The financial mathematics does not excuse you from reading the wording closely: “today,” “at the end of year two,” and “beginning of each month” determine the equation as much as the numbers do.

Bond prices and yields

A fixed-rate bond’s value is the present value of its promised coupon payments and principal repayment, discounted at the required yield for the bond’s risk and cash-flow timing. If annual coupons are $60 for three years and $1,000 principal is repaid at maturity, then at a 5 percent required return the value is 60/1.05 + 60/1.05² + 1,060/1.05³, or about $1,027.23. The bond is above par because its 6 percent coupon rate exceeds the 5 percent required yield.

As required yields rise, the present value of fixed future payments falls. The relationship between price and yield is inverse. Higher discounting reduces the value of every future coupon and the principal. The size of the price response depends on cash-flow timing and other bond features. Always distinguish the coupon rate, which determines promised coupon cash flows, from yield, which discounts those cash flows.

Deferred and growing cash flows

A deferred annuity starts after a delay. First calculate its value one period before the first payment using the ordinary annuity formula, then discount that amount back to the valuation date. For instance, if five equal payments begin at the end of year three, the ordinary annuity value is located at year two. Discount it two years to time zero. A common mistake is to discount only one year or to treat the first payment as arriving at the end of year one.

For a growing annuity with first payment one period from now, the present value over n periods is PMT1/(r − g) × [1 − ((1+g)/(1+r))^n], assuming r is not equal to g and the rate exceeds the growth rate for the standard case. The time index matters: if the problem gives the most recent cash flow, grow it once to obtain next period’s payment. Use the formula only when payments grow at a consistent rate over the stated horizon.

Valuation date and cash-flow mapping

A valuation date is the point in time at which a cash flow or asset value is expressed. To compare two alternatives, move all their cash flows to the same date. An amount at year three can be brought back to today or carried forward to year three, but do not combine it directly with a year-one amount. Draw a number line when the wording contains “immediately,” “at the end,” or “beginning.” Those phrases set the timing convention.

For uneven cash flows, a compact table reduces errors: period, cash flow, discount factor, present value. In an exam problem with three or four amounts, show one row per date. Check that a time-zero amount has a discount factor of one. For a delayed first payment, do not create a cash flow in a period where none occurs.