Time Value of Money

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 = PV × (1 + r)^n

• PV = Present Value (what you invest today)
• r = Rate of return per period
• n = Number of periods

Worked Example:

• You invest $10,000 today at 6% annually for 5 years
• FV = $10,000 × (1 + 0.06)^5
• FV = $10,000 × 1.3382
• FV = $13,382

How Does the Rule of 72 Approximate Doubling Time?

Years to double = 72 / annual rate of return

• At 8% return, money doubles in approximately 72 / 8 = 9 years
• At 6% return, money doubles in approximately 72 / 6 = 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 / 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 = FV / (1 + r)^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 payment discounted at 6% annually:

• Received in 1 year: $10,000 / (1.06)^1 = $9,434
• Received in 5 years: $10,000 / (1.06)^5 = $7,473
• Received in 10 years: $10,000 / (1.06)^10 = $5,584

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

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. Don't assume "more time" always means "more value"; the direction depends on which way you're 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 = Present Value of Cash Flows - 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 and generates cash flows of $4,000 per year for 3 years. The required rate of return is 6%, so each cash flow is discounted by (1 + 0.06) = 1.06 per year, the same factor used in the future value formula above.

• PV of Year 1: $4,000 / 1.06 = $3,774
• PV of Year 2: $4,000 / (1.06)^2 = $3,560
• PV of Year 3: $4,000 / (1.06)^3 = $3,358
• Total PV of cash flows: $3,774 + $3,560 + $3,358 = $10,692
• NPV = $10,692 - $10,000 = +$692
• Decision: Accept, because the positive NPV means the investment earns more than the 6% 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 = yield to maturity (YTM) (the exam may use these interchangeably)
• Works best when future cash flows are predictable, such as bonds with fixed coupons

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.

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

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