Time Value of Money

This is the foundation for all investment valuation. A dollar today is worth more than a dollar in the future because it can be invested to earn a return.

This section covers the three core time value concepts: Future Value (FV), Net Present Value (NPV), and Internal Rate of Return (IRR).

Core Concepts:

• Discount rate (required rate of return) - the rate used to convert future cash flows into present value
• Compounding - earning returns on previously earned returns (interest on interest)

Future Value (FV)

The value of a current investment at a specified date in the future, assuming a certain rate of return.

FV = PV × (1 + r)^n

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

Used to project how much a lump sum or series of payments will grow over time.

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

Rule of 72

Approximates the number of years to double an investment:

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.

Net Present Value (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 - investment is worth more than it costs; accept
• Negative NPV - investment costs more than it is worth; reject
• Zero NPV - 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.

NPV Worked Example:

An investment costs $10,000 and will generate cash flows of $4,000 per year for 3 years. Your required rate of return is 6%.

• 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 (positive NPV means the investment earns more than the 6% required return)

Internal Rate of Return (IRR)

The discount rate at which the net present value (NPV) of an investment 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 (e.g., 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 > required rate of return (hurdle rate); reject if IRR < hurdle rate

NPV vs IRR Comparison

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

Why NPV is Preferred: 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's 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.

Example: If a project's IRR is 25% but your 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. When the two conflict, NPV is preferred.