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Corporate Finance Unit 2: Time Value of Money

A dollar today is worth more than a dollar tomorrow because it can earn a return. This unit teaches compounding and discounting, the tools behind every valuation in finance.

Covers: Future value and compounding, Present value and discounting, Solving for the rate or the number of periods, APR, EAR and compounding frequency, Multiple and uneven cash flows, Continuous compounding and real vs nominal rates

Study guide

The time value of money starts with one idea: money you have now can be invested to earn interest, so it is worth more than the same amount later. Future value (FV) is what a sum grows to: FV = PV × (1 + r)ᵗ, where r is the rate per period and t is the number of periods. The key is compounding. With simple interest you earn interest only on the original principal, but with compound interest each period's interest is added to the balance and earns interest too. Over long horizons, interest on interest becomes the biggest part of the growth, which is why starting early matters so much.

Present value (PV) runs the process backward. To find what a future amount is worth today, divide by the growth factor: PV = FV ÷ (1 + r)ᵗ. The rate used here is called the discount rate, and it reflects the return you could earn on an investment of similar risk. PV falls when the discount rate rises or when the cash flow is further away. Because every investment decision compares money at different dates, discounting everything back to today puts all cash flows on equal footing. For several cash flows, discount each one separately by its own number of periods and add them up.

The same equation can be solved for any unknown. To find the rate of return, compute r = (FV ÷ PV)^(1/t) − 1. To find the time needed, use logarithms: t = ln(FV ÷ PV) ÷ ln(1 + r). The rule of 72 is a handy check: money doubles in about 72 ÷ (rate in percent) years, so at 8% it doubles in about 9 years. The rule works best for rates between about 5% and 12%. A common mistake is dividing a total percentage gain by the number of years, which ignores compounding and overstates the annual rate.

Interest is often quoted as an annual percentage rate (APR) that compounds more than once a year. The periodic rate is APR ÷ m, where m is the number of compounding periods per year. The effective annual rate, EAR = (1 + APR/m)ᵐ − 1, is the true yearly return or cost. More frequent compounding raises the EAR above the APR, and in the limit of continuous compounding EAR = e^(APR) − 1 and FV = PV × e^(rt). Always compare loans or savings options on their EARs, and always match the periodic rate with the number of periods.

Rates can be nominal or real. A nominal rate is stated in dollars; a real rate measures growth in purchasing power. The Fisher relation says (1 + nominal) = (1 + real)(1 + inflation), so the real rate is about the nominal rate minus inflation. Discount nominal cash flows at nominal rates and real cash flows at real rates; mixing them is a classic error. Together, these tools let you compare any set of cash flows, which is the foundation for valuing loans, bonds, stocks and projects in the rest of the course.

Key ideas

• Money today is worth more than money later because it can earn a return.
• Compounding means earning interest on previously earned interest, so growth accelerates over time.
• Present value discounts a future amount at the rate of return on investments of similar risk.
• Higher rates and longer horizons both lower present value and raise future value.
• Compare rates with different compounding frequencies by converting them to EAR.
• Cash flows at different dates can be added only after moving them to the same date.
• Real rates measure purchasing power; nominal cash flows must be discounted at nominal rates.

Formulas and rules

Common mistakes

• Using simple interest (PV × (1 + rt)) when the problem calls for compound interest.
• Mixing an annual rate with monthly or quarterly periods instead of matching rate and periods.
• Comparing APRs with different compounding frequencies instead of comparing EARs.
• Dividing total percentage growth by years to get an annual rate, which ignores compounding.
• Discounting a cash flow received today, or using the wrong number of periods for each flow.

Before the test, make sure you can

• Calculate the future value of a single sum with annual or non-annual compounding.
• Calculate the present value of a single sum and of several uneven cash flows.
• Solve for the interest rate or number of periods using roots and logarithms.
• Convert between APR and EAR for any compounding frequency, including continuous.
• Use the rule of 72 to estimate doubling time and check answers.
• Explain why higher rates or longer horizons lower present value.
• Calculate a real rate of return from nominal returns and inflation.

Words to know

Time value of money

The principle that a dollar today is worth more than a dollar in the future because it can be invested.

Future value (FV)

The amount a sum invested today will grow to after a given time at a given rate.

Present value (PV)

The value today of a future cash flow, found by discounting it at the appropriate rate.

Compounding

Growing a value forward in time with interest earned on both principal and prior interest.

Discounting

Moving a future value back to the present by dividing by the growth factor.

Discount rate

The rate used to compute present values; the return available on investments of similar risk.

Simple interest

Interest earned only on the original principal, with no interest on interest.

Compound interest

Interest earned on principal plus all interest previously credited.

APR

Annual percentage rate: the periodic rate times the number of periods per year, ignoring compounding.

EAR

Effective annual rate: the actual annual return after compounding, (1 + APR/m)ᵐ − 1.

Continuous compounding

Compounding infinitely often; FV = PV × e^(rt).

Rule of 72

Shortcut: years to double ≈ 72 divided by the percentage interest rate.

Nominal rate

An interest rate stated in dollars, not adjusted for inflation.

Real rate

The rate of growth in purchasing power: (1 + nominal) ÷ (1 + inflation) − 1.

Fisher relation

(1 + nominal rate) = (1 + real rate)(1 + inflation rate).

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.