Clients rarely ask for a formula. They ask what their savings will grow to, whether a deal is worth its price, how rough the ride will be and whether their manager earned the fee. Each question has a number behind it, and the exam expects you to find or read it.
The time value of money rests on one idea. A dollar today is worth more than a dollar later, because today's dollar can be invested. Future value (FV) is what a sum invested today grows to by compound interest. Each period's interest joins the balance, and from then on that interest earns interest too.
For future value, each period multiplies the balance by (1 + rate), so a present sum grows to present value (PV) times (1 + rate) raised to the number of periods.
Suppose a client invests $10,000 at 6% for 10 years. Because , the future value is about $17,908. Simple interest of $600 a year would reach only $16,000.
Common mistakes
- Compounding without dividing the rate: Monthly compounding of 6% means 0.5% for 120 periods, about $18,194 on $10,000. Using 6% per month, or simple interest ($16,000), gives a wrong answer.
- Calling present value the NPV: In Example 1 the NPV is $308.39, not the $10,308.39 sum of present values. Subtract the initial cost, and never discount it.
- Alpha as return minus market: Manager A's alpha is 1.2%, not 14 − 10 = 4%. The CAPM return built on beta is the hurdle.
Bottom line
- FV = PV × (1 + r)^n rises with more frequent compounding, a higher rate or a longer horizon, and the Rule of 72 puts years to double at about 72 ÷ rate.
- NPV is future cash flows discounted at the required return minus the initial cost; positive earns more than required, negative less, and NPV falls as the rate rises.
- IRR makes NPV zero and is a bond's YTM or a portfolio's dollar-weighted return; accept above the required return, but rank by NPV when size, timing or sign changes differ.
- Mean is the average, median the middle of sorted values (it resists outliers), mode the most frequent value and range the high minus the low.
Exam shortcut
Read the denominator before you compute. A stem stressing standard deviation, total risk or an undiversified account points to Sharpe. A stem stressing beta or a well-diversified portfolio points to Treynor or alpha. If an alpha choice equals the fund's return minus the market's return, eliminate it. For NPV and IRR, compare the required return with the IRR first.
The full lesson (about 4,842 words, 32 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- A3
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