A higher return can conceal greater risk, and a cheaper stock can reflect weaker prospects. Analytical methods such as the Sharpe ratio and price-to-earnings ratio help explain these differences to a customer.
You cannot compare money available today with a future payment by looking only at their dollar amounts. Today's money can earn a return before the future payment arrives. At a positive discount rate, receiving the same amount later therefore has less value today.
The time value of money expresses this relationship between dollars and time. Compounding moves a value forward by adding returns, including returns earned on earlier returns. Discounting moves it backward by removing the growth needed to reach a future amount.
Your present value (PV) is the amount today equivalent to a specified future payment. Your future value (FV) is the amount at a later date. The discount rate is the rate used to translate future money into today's equivalent.
Here, is the rate per period, and is the number of periods. Write 6% as 0.06 when using .
Common mistakes
- Mixing time units. At 8% nominal interest compounded quarterly for two years, use 2% for eight periods. Using 8% for eight periods compounds an annual rate every quarter.
- Misreading zero NPV. Zero NPV at a 7% required return means the investment meets 7%. It does not mean the investor earns zero.
- Skipping the risk-free subtraction. A 10% return, 2% risk-free return, and 16% standard deviation produce Sharpe of 0.50. The tempting 10 ÷ 16 = 0.625 uses the wrong numerator.
Bottom line
- Compounding moves value forward; discounting moves it backward using matching rates and periods.
- Positive NPV exceeds the required return; IRR makes NPV zero, and repeated cash-flow sign changes can produce multiple IRRs.
- Mean averages observations, median locates the sorted middle, mode identifies the most frequent value, and range subtracts the extremes.
- Standard deviation measures total volatility; normal distributions place approximately 68%, 95%, and 99.7% within one, two, and three standard deviations.
Exam shortcut
Identify the requested output before calculating: today's dollars, future dollars, a discount rate, a return comparison, or a company ratio. Mark each cash-flow date and sign. For compounding questions, write the periodic rate beside the period count before entering numbers. Inspect the risk denominator. Circle standard deviation for Sharpe and beta for systematic-risk comparisons; compute the excess return before dividing. Sort observations before locating the median.
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Learning objectives
- A1
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