Time Value of Money

Quick Answer

A dollar today is worth more than a dollar in the future because it can be invested to earn a return. Future value compounds a present sum forward; present value discounts a future sum back. Accept an investment with a positive net present value (NPV) or an internal rate of return (IRR) above the required rate.

These four tools (FV, PV, NPV, IRR) are the foundation for every valuation topic later in the exam, from bond pricing to project selection.

Core Concepts

Discount rate (required rate of return): the rate used to convert future cash flows into present value; generally set to reflect the current market rate of return on investments of comparable risk (higher risk means a higher discount rate).

Compounding: earning returns on previously earned returns (interest on interest).

How Do You Project Future Value?

Future value (FV) is the value of a current investment at a specified future date, assuming a certain rate of return. FV is used to project how much a lump sum or a series of payments will grow over time.

FV equals PV times (1 plus r) to the power n.

PV equals Present Value (what you invest today).

r equals Rate of return per period.

n equals Number of periods.

Worked Example:

You invest 10,000 dollars today at 6 percent annually for 5 years.

FV equals 10,000 dollars times (1 plus 0.06) to the power 5.

FV equals 10,000 dollars times 1.3382.

FV equals 13,382 dollars.

How Does the Rule of 72 Approximate Doubling Time?

Years to double equals 72 divided by annual rate of return.

At 8 percent return, money doubles in approximately 72 divided by 8 equals 9 years.

At 6 percent return, money doubles in approximately 72 divided by 6 equals 12 years.

Exam Tip: Gotchas

The Rule of 72 is a quick approximation, not an exact calculation. If a question asks "approximately how long to double," use 72 divided by rate.

How Does Present Value Change With Time?

Present value (PV) is the value today of a sum to be received in the future, found by discounting that future sum back at the discount rate.

PV equals FV divided by (1 plus r) to the power n.

Discounting is the mirror image of compounding. As the number of periods (n) increases, present value decreases, because the same future sum is divided by a larger factor. The further out a cash flow is received, the less it is worth today.

Worked Example: A 10,000 dollar payment discounted at 6 percent annually:

Received in 1 year: 10,000 dollars divided by (1.06) to the power 1 equals 9,434 dollars.

Received in 5 years: 10,000 dollars divided by (1.06) to the power 5 equals 7,473 dollars.

Received in 10 years: 10,000 dollars divided by (1.06) to the power 10 equals 5,584 dollars.

The same 10,000 dollars is worth less today the further out it is received.

The discount rate has the same effect. A higher rate makes the divisor larger, so present value falls. The same 10,000 dollars received in 5 years:

Discounted at 6 percent: 10,000 dollars divided by (1.06) to the power 5 equals 7,473 dollars.

Discounted at 8 percent: 10,000 dollars divided by (1.08) to the power 5 equals 6,806 dollars.

Discounted at 10 percent: 10,000 dollars divided by (1.10) to the power 5 equals 6,209 dollars.

A higher discount rate means the investor demands more return for waiting, so the payment is worth less today. It never raises present value.

Exam Tip: Gotchas

Time moves FV and PV in opposite directions. A longer time period increases future value (more compounding) but decreases present value (more discounting), holding the discount rate constant. Do not assume "more time" always means "more value"; the direction depends on which way you are moving along the timeline.

When Should You Accept a Project Based on NPV?

NPV is the difference between the present value of future cash flows and the initial cost of the investment.

NPV equals Present Value of Cash Flows minus Cost of Investment.

Decision rule:

Positive NPV: the investment is worth more than it costs; accept.

Negative NPV: the investment costs more than it is worth; reject.

Zero NPV: the investment earns exactly the required return; indifferent.

Exam Tip: Gotchas

A positive NPV means the investment earns MORE than the required rate of return. A zero NPV does not mean zero profit; it means the investment earns exactly the discount rate used.

Worked Example: An investment costs 10,000 dollars and generates cash flows of 4,000 dollars per year for 3 years. The required rate of return is 6 percent, so each cash flow is discounted by (1 plus 0.06) equals 1.06 per year, the same factor used in the future value formula above.

PV of Year 1: 4,000 dollars divided by 1.06 equals 3,774 dollars.

PV of Year 2: 4,000 dollars divided by (1.06) squared equals 3,560 dollars.

PV of Year 3: 4,000 dollars divided by (1.06) cubed equals 3,358 dollars.

Total PV of cash flows: 3,774 dollars plus 3,560 dollars plus 3,358 dollars equals 10,692 dollars.

NPV equals 10,692 dollars minus 10,000 dollars equals plus 692 dollars.

Decision: Accept, because the positive NPV means the investment earns more than the 6 percent required return.

What Is IRR and When Do You Accept It?

The internal rate of return (IRR) is the discount rate at which an investment's net present value equals zero. It represents the investment's expected annualized rate of return.

For bonds: IRR equals yield to maturity (YTM) (the exam may use these interchangeably).

For project-like investments, including direct participation programs (DPPs): IRR is commonly used to evaluate projected cash flows over the investment's expected life.

Works best when future cash flows are predictable, such as fixed bond coupons or a DPP's forecast distributions and ending value.

Exam Tip: Gotchas

If a question asks for a bond's IRR, it is asking for YTM. If a question says an investment's IRR exceeds the required return, the investment should be accepted.

Decision rule: accept if IRR is greater than the required rate of return (hurdle rate); reject if IRR is less than the hurdle rate.

IRR greater than Required return: NPV positive. Decision: Accept.

IRR equals Required return: NPV zero. Decision: Indifferent.

IRR less than Required return: NPV negative. Decision: Reject.

Why Is NPV Preferred Over IRR When They Conflict?

When NPV and IRR conflict on mutually exclusive projects, NPV is the preferred method.

What Is the Reinvestment Assumption?

The two methods make different assumptions about what happens to cash flows received during the project:

NPV assumes interim cash flows are reinvested at the discount rate (your required return): realistic, since that is the return you can actually earn elsewhere.

IRR assumes interim cash flows are reinvested at the IRR itself: often unrealistic, especially when the IRR is high.

If a project's IRR is 25 percent but the firm's typical reinvestment rate is 8 percent, IRR overstates the actual return you can capture. NPV, using 8 percent, gives the more honest picture.

NPV versus IRR:

What it measures: NPV measures dollar amount of value added. IRR measures percentage rate of return.

Decision rule: NPV accept if NPV greater than 0. IRR accept if IRR greater than hurdle rate.

Assumes reinvestment at: NPV discount rate (more realistic). IRR IRR itself (less realistic).

Preferred when conflicts arise: NPV yes. IRR no.

Best for: NPV comparing projects of different sizes. IRR quick comparison to required return.

Exam Tip: Gotchas

NPV is expressed in dollars; IRR is expressed as a percentage. NPV assumes reinvestment at the discount rate (more realistic); IRR assumes reinvestment at the IRR itself (less realistic). When the two conflict, NPV is preferred.

What Should You Check on Exam Day?

FV compounds forward, PV discounts backward; more time increases FV but decreases PV.

Accept a positive NPV or an IRR above the hurdle rate; a zero NPV still earns the required return, not zero return.

For a bond, IRR is the same figure as yield to maturity; IRR is also commonly used for project-like investments such as DPPs.

When NPV and IRR disagree on mutually exclusive projects, NPV wins because its reinvestment assumption is more realistic.

Additional Details

Published: 2026-11-02T21:59:23.883Z