Time Value of Money

Interest and Cash-Flow Diagrams

Money today is worth more than the same amount later because of earning potential. Draw timelines with arrows: outflows down, inflows up. Compound amount:

Present outflow compounds forward to future worth at rate $i$.

\[ F = P(1+i)^n \]

Present worth: $P = F/(1+i)^n$. Use effective rate per compounding period — monthly, quarterly, or continuous ($F = Pe^{rn}$) when stated.

Simple vs. Compound Interest

Simple interest adds a fixed amount each period; compound interest grows on accumulated balance.

\[ I_{\text{simple}} = P \cdot r \cdot t, \quad F_{\text{compound}} = P(1+i)^n \]

Effective Interest Rate

When compounding is $m$ times per year:

\[ i_e = \left(1 + \frac{r}{m}\right)^m - 1 \]

Continuous compounding: $i_e = e^r - 1$. Always compare alternatives at the same effective rate.

Rule of 72

Approximate years to double: $n \approx 72/i$ percent. Useful sanity check on the exam.

Inflation and Real Dollars

Market interest rate combines real rate and inflation approximately: $i \approx i_{\text{real}} + f$. Deflate future dollars to constant purchasing power when the problem specifies real analysis.

Exam habit

Identify $i$ and $n$ in the same period (annual rate with annual periods). Use handbook factor tables when provided instead of recomputing $(1+i)^n$.

Example: Future Worth

Compound amount

Single present outflow compounds to future worth at $i = 6\%$.

Invest $5,000 at 6% compounded annually for 4 years. Find $F$.

Solution.

\[ F = 5000(1.06)^4 = 5000(1.262) = \$6{,}310 \]

Future worth is approximately $6,310.

Example: Present Worth Discount

Single payment

Discount a future inflow to present worth at $i = 8\%$.

Receive $10,000 in 5 years with $i = 8\%$. Find present worth.

Solution.

\[ P = \frac{10{,}000}{(1.08)^5} = \frac{10{,}000}{1.469} = \$6{,}807 \]

Present worth is approximately $6,807.

Example: Effective Annual Rate

Monthly compounding

Nominal rate 12% compounded monthly ($m = 12$). Find effective annual rate $i_e$.

Solution.

\[ i_e = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = (1.01)^{12} - 1 = 0.1268 \]

Effective annual rate is approximately 12.68%.

Common mistakes

Using nominal rate as effective rate when compounding is more frequent than annual. Mixing monthly cash flows with an annual rate without converting. Forgetting that outflows are negative on cash-flow diagrams.

Practice compound interest, present worth factors, and cash-flow diagrams with six numeric problems.

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