Time Value of Money, Statistics and Ratios

Section I of the Series 66 is small, about 8 of the 100 scored questions, and it tests one thing: the analytical methods an adviser or agent uses to size up an investment. This chapter covers its four parts in NASAA's order. First comes the time value of money: compounding, discounting, net present value and internal rate of return. Next come the descriptive statistics that summarize return and risk, from the mean and median to standard deviation, correlation, beta, alpha and the Sharpe ratio. Last come the financial ratios that measure a company's liquidity and debt, and the valuation ratios that compare its stock price with its earnings, book value and dividends. Expect definition questions and short calculations that need nothing more than a basic calculator, so every example uses round numbers you can work by hand. FINRA, which delivers the exam, includes an on-screen calculator in its online exams, and its test centers provide a four-function calculator.

1.1Compound Interest, Future Value and the Rule of 72

Why money has a time value

The time value of money is the idea that a dollar today is worth more than a dollar received in the future. There are three reasons. A dollar you hold today can be invested to earn a return, so waiting means giving up that return, which is an opportunity cost. Inflation usually erodes what a future dollar will buy. And a promise of future money carries some risk that it will not be paid.

Every time value calculation uses the same four pieces. The present value is what a sum is worth today. The future value is what it will be worth at a later date. The rate is the return earned per period, and the number of periods is how long the money is invested. If you know any three, you can solve for the fourth.

Simple and compound interest

Simple interest is paid only on the original principal, never on interest already earned. Interest equals principal times rate times time. If Elena lends $8,000 to her cousin's landscaping business at 5% simple interest for 4 years, she earns $400 a year, or $1,600 in all, and gets back $9,600.

Compound interest is interest earned on both the principal and the interest already credited. Each period's interest is added to the balance, and the next period's interest is figured on that larger balance. Future value equals present value times one plus the rate, raised to the power of the number of periods. In symbols, FV = PV × (1 + r)^n. Raising to a power simply means multiplying by one plus the rate once for every period.

A higher rate and more time both make a future value larger. Time is especially powerful because the growth is exponential, which is why advisers urge young clients to start saving early.

Compounding frequency

Interest is often credited more than once a year. When it is, divide the annual rate by the number of compounding periods per year and multiply the years by that same number. Future value equals present value times one plus the annual rate divided by m, raised to the power of m times the number of years, where m is 2 for semiannual, 4 for quarterly and 12 for monthly compounding.

The more often interest compounds, the larger the future value at the same quoted rate. The $1,102.50 result shows that 10% compounded semiannually is really worth 10.25% a year. That true yearly return after compounding is the effective annual rate.

The Rule of 72

The Rule of 72 is a shortcut for compound growth: the number of years it takes money to double is about 72 divided by the annual rate, with the rate written as a whole number, such as 8 rather than 0.08. At 8%, money doubles in about 72 divided by 8, or 9 years. Turned around, the rate needed to double in a given number of years is about 72 divided by the years, so doubling in 6 years takes about 12%.

The rule is an approximation that works best for rates between about 6% and 10%. It works for anything that compounds, including inflation: at 3% inflation, prices double in about 24 years, so a fixed pension loses half its buying power over that time.

1.2Present Value, Net Present Value and Internal Rate of Return

Discounting a future sum

Present value runs the future value formula backward. Instead of asking what today's money will grow to, it asks what a future sum is worth today. Present value equals future value divided by one plus the rate, raised to the power of the number of periods. In symbols, PV = FV ÷ (1 + r)^n. Finding a present value is called discounting, and the rate used is the discount rate.

The discount rate is the return the investor requires, which is the return available on other investments of similar risk. That is why a riskier cash flow is discounted at a higher rate: an investor demands more for waiting on an uncertain payment.

Present value moves opposite to the discount rate. A higher discount rate means a lower present value, because the investor gives up more by waiting. A longer wait also lowers present value, and so does more frequent compounding at the same quoted rate, since money would grow faster elsewhere.

Annuities and discounted cash flow

In time value language, an annuity is a series of equal payments at regular intervals, such as $1,000 a year for 10 years. In an ordinary annuity each payment comes at the end of the period, and in an annuity due each comes at the beginning. Because every payment of an annuity due arrives one period sooner, an annuity due is always worth more than an otherwise identical ordinary annuity. Two payments of $1,000 at the end of each of the next 2 years, discounted at 10%, are worth about $909.09 plus $826.45, or $1,735.54. Paid at the start of each year instead, they are worth $1,000 plus $909.09, or $1,909.09.

The same idea values every security: a security is worth the present value of the cash it is expected to pay, which is called discounted cash flow valuation. A bond, for example, is a loan to a company or government that pays regular interest and repays its face value, usually $1,000, at maturity. Its price is the present value of those interest payments plus the present value of the face value, and a stock's value is the present value of its expected dividends or cash flows. Chapter 2 applies this to bonds and Chapter 3 to stocks.

Net present value

Net present value, or NPV, compares what an investment costs today with what its future cash flows are worth today. Net present value equals the sum of the present values of all future cash flows, minus the initial investment. The discount rate is the investor's required return, which a business calls its cost of capital or hurdle rate.

The NPV decision rule follows directly. A positive NPV means the investment earns more than the required return, so accept it. A negative NPV means it earns less, so reject it. An NPV of exactly zero means it earns exactly the required return. When an investor must pick one of several competing projects, the one with the highest NPV adds the most value.

NPV depends on the discount rate. If Marcus needed 20% instead, the same inflows would be worth only $19,250 today, and the NPV would be minus $750. The oven is a good deal for an investor who needs 10% and a bad one for an investor who needs 20%.

Internal rate of return

The internal rate of return, or IRR, is the discount rate that makes an investment's NPV exactly zero. Put another way, it is the compound annual return the investment itself earns, given its price and its cash flows. The oven's NPV is positive at 10% and negative at 20%, so its IRR lies between them; it works out to about 17%. In the simplest case you can see the IRR directly: paying $40,000 today for $48,400 in 2 years has an IRR of exactly 10%, because $48,400 discounted at 10% for 2 years is $40,000.

The IRR decision rule compares the IRR with the required return. If the IRR is higher, accept the investment; if it is lower, reject it. For a single, normal investment the two rules always agree: an IRR above the required return goes with a positive NPV at that required return. In practice IRR is found by trial and error or with a financial calculator, so exam questions either give you the IRR or use cash flows simple enough to see it.

A bond's yield to maturity is its IRR: the single discount rate that makes the present value of its interest payments and face value equal to its market price. Like every IRR, it assumes each cash flow can be reinvested at that same rate, which is not guaranteed. In the same way, an account's dollar-weighted return, which Chapter 10 covers, is the IRR of the investor's deposits, withdrawals and ending value.

IRR is popular because a percentage is easy to explain to a client, but it has limits. It assumes reinvestment at the IRR itself. It ignores size, so a 50% return on $1,000 can look better than a 20% return on $1 million. And when cash flows switch between positive and negative more than once, an investment can have more than one IRR. When NPV and IRR rank two competing projects differently, NPV is the more reliable guide.

Try it: the calculator's first tab grows or discounts a single sum and shows the Rule of 72 beside the exact doubling time, and its second tab finds the NPV and IRR of a series of cash flows. Enter the oven with two years of cash flows, then raise the discount rate until the NPV crosses zero.

1.3Mean, Median, Mode and Range

Describing a set of returns

Descriptive statistics summarize a set of numbers, such as a fund's yearly returns, with a few figures. One group describes the center of the data, which statisticians call central tendency. Another describes how spread out the data are, which is dispersion. This lesson uses one fund's five yearly returns: 2%, 5%, 5%, 8% and 30%.

Mean, median, mode and range

The mean, or arithmetic mean, is the ordinary average: add the values and divide by how many there are. The median is the middle value when the data are listed from lowest to highest; with an even number of values, it is the average of the two middle values. The mode is the value that appears most often, and a data set can have no mode, if no value repeats, or more than one. The range is the highest value minus the lowest, and it is the simplest measure of dispersion.

When each measure misleads

Each measure has a blind spot, and the exam likes to ask which one fits a given set of data.

• The mean is pulled toward extreme values. One 30% year lifted the fund's mean to 10%, even though four of its five years were 8% or less.
• The median ignores how extreme the extremes are, which makes it the better center for lopsided data such as home prices, incomes and net worth.
• The mode may not exist, may have several values, or may sit far from the center.
• The range depends only on the two most extreme values, so a single unusual year can make a steady investment look wild.

Skew

A distribution is the full pattern of how often each value occurs. In a symmetrical distribution the two sides mirror each other, and the mean, median and mode are equal. Skew describes a lopsided distribution with a longer tail on one side.

• Positive skew, also called right skew, has a long tail of unusually high values, which pull the mean above the median. The fund's returns are positively skewed: its mean of 10% sits well above its median of 5%.
• Negative skew, also called left skew, has a long tail of unusually low values, which pull the mean below the median. A strategy that collects small, steady gains but occasionally suffers a large loss tends to be negatively skewed.

1.4Standard Deviation, the Normal Distribution and Correlation

Measuring how much returns vary

Two funds can both average 10% a year and still give their owners very different rides. One might return between 8% and 12% every year, while the other swings from a 20% loss to a 40% gain. The range hints at this, but a better measure uses every return, not just the two extremes.

Variance is the average of the squared differences between each value and the mean. Each difference from the mean is called a deviation, and squaring the deviations makes them all positive so that values above and below the mean do not cancel out. Standard deviation is the square root of the variance, which brings the measure back from squared units to percentage points. It is the most widely used measure of an investment's volatility, and it is the usual meaning of "risk" in investment theory.

This example divides by the number of observations, which treats the five years as the whole population. Statisticians estimating from a sample divide by one fewer, which gives a slightly larger answer. Exam questions usually hand you the standard deviation and ask what it means.

A higher standard deviation means returns are more spread out around the mean, so the investment is more volatile and less predictable. A fund of Treasury bills, which are short-term U.S. government debt, has a standard deviation close to zero, while a fund of small, young companies can have one well above 20%. Standard deviation measures total risk: all the variability in an investment's returns, including both the market-wide risk that diversification cannot remove and the company-specific risk that it can. Chapter 9 explains those two kinds of risk.

The normal distribution

Many sets of data cluster around their mean in a symmetrical, bell-shaped pattern called the normal distribution. Treating returns as roughly normal lets an adviser turn a mean and a standard deviation into a range of likely outcomes using the 68, 95, 99.7 rule. About 68% of outcomes fall within one standard deviation of the mean, above or below. About 95% fall within two standard deviations, and about 99.7% fall within three.

Because the bell is symmetrical, the outcomes outside a band split evenly between the two tails. If 68% of returns fall within one standard deviation, the other 32% fall outside it, about 16% below the band and 16% above. So the balanced fund's client should expect a return worse than minus 3% in roughly one year out of six. Outside two standard deviations, about 2.5% of outcomes fall in each tail. Real markets have more extreme losses than a normal curve predicts, a pattern called fat tails, so the rule is a starting point for a risk conversation, not a guarantee.

Correlation

Standard deviation describes one investment on its own, but advisers build portfolios, so they also need to know how investments move relative to each other. Covariance measures whether two investments' returns tend to move together, but its size depends on the units and on how volatile each investment is. The correlation coefficient rescales covariance onto a fixed scale from minus 1 to plus 1. Correlation equals the covariance of the two investments divided by the product of their standard deviations.

A correlation of plus 1, called perfect positive correlation, means the two always move in the same direction in lockstep. A correlation of minus 1, perfect negative correlation, means they always move in exactly opposite directions. A correlation of zero means there is no linear relationship, so knowing one's return tells you nothing about the other's.

Correlation is the reason diversification works. Combining two perfectly positively correlated investments does nothing to reduce risk: the portfolio's standard deviation is just the weighted average of the two. Whenever the correlation is less than plus 1, some of one investment's ups offset the other's downs, and the portfolio's standard deviation falls below that weighted average. The lower the correlation, the bigger the risk reduction. Chapter 8 builds on this idea to construct efficient portfolios.

Correlation describes direction and consistency, not size, and it does not prove that one investment causes the other's moves. Squaring it gives R-squared, the share of an investment's movements explained by a benchmark. A fund with a correlation of 0.9 to its index has an R-squared of 0.81, so about 81% of its ups and downs track the index. An index fund's R-squared is near 100%.

1.5Beta, Alpha and the Sharpe Ratio

Beta

Beta measures how sensitive an investment's returns are to the returns of the overall market, usually represented by the S&P 500 index of large U.S. companies. The market itself has a beta of 1.0. A stock with a beta of 1.5 tends to move about 1.5 times as much as the market in either direction, and a stock with a beta of 0.6 tends to move about 60% as much. A beta near zero, as with Treasury bills, means little relationship with the stock market, and a negative beta means the investment tends to move opposite to it.

Beta measures systematic risk, the market-wide risk that diversification cannot remove, while standard deviation measures total risk. That difference is the key to telling the risk measures apart.

Beta is built from the same pieces as correlation. Beta equals the investment's covariance with the market divided by the variance of the market. An equivalent version is the investment's correlation with the market times its standard deviation, divided by the market's standard deviation.

Alpha

Alpha is the return an investment earned above or below the return its beta predicts. A positive alpha means the manager added value beyond what the fund's market risk alone would have produced, and a negative alpha means the manager fell short.

The beta-predicted return starts with the risk-free rate, the return on a safe, short-term investment such as a Treasury bill. It adds beta times the market's return in excess of that risk-free rate. Alpha is the actual return minus that predicted return. Chapter 8 develops the predicted-return formula as the capital asset pricing model.

You will sometimes see "alpha" used loosely for a fund's return minus its benchmark's return. The risk-adjusted version above, often called Jensen's alpha, is more precise because it accounts for how much market risk the manager took. When a question gives you a beta, a risk-free rate and a market return, compute the risk-adjusted version.

The Sharpe ratio

The Sharpe ratio measures return per unit of total risk. It equals the portfolio's return minus the risk-free rate, divided by the portfolio's standard deviation. The return above the risk-free rate is called the excess return, or risk premium, because it is the reward for taking risk. A higher Sharpe ratio is better: the investor received more excess return for each unit of volatility.

The Treynor ratio is the Sharpe ratio's companion: it divides the same excess return by beta instead of standard deviation. Because the Sharpe ratio uses total risk, it suits a client's whole portfolio or one that is not fully diversified, where company-specific risk still matters. The Treynor ratio suits a fund that is one piece of a larger, well-diversified portfolio. Chapter 10 shows how advisers use alpha and these ratios to judge managers.

1.6Financial Statements, Liquidity and Leverage

A financial ratio divides one figure from a company's statements by another so that companies of different sizes can be compared. To compute ratios you need just two statements, and this lesson walks through them for one fictional company before turning to the ratios that measure short-term safety and debt.

The balance sheet

The balance sheet shows what a company owns and owes on one date. It rests on one equation: assets equal liabilities plus shareholders' equity. Assets are what the company owns, liabilities are what it owes, and shareholders' equity, also called net worth, is what would be left for the owners if every asset were sold at its accounting value and every liability paid.

Both sides are split by time. Current assets are cash and items expected to become cash within a year: marketable securities, accounts receivable, which is money customers owe the company, and inventory, the goods held for sale. Other assets include plant and equipment and intangible assets such as patents and goodwill. Current liabilities are due within a year, such as accounts payable, the money owed to suppliers. Long-term debt, such as bonds, comes due later.

Lakemont Industries makes industrial pumps. At December 31, 2025, its current assets total $800 million: cash of $120 million, marketable securities of $80 million, accounts receivable of $300 million and inventory of $300 million. Plant and equipment are carried at $1,400 million and intangibles at $200 million, for total assets of $2,400 million. Current liabilities are $400 million, and the only long-term debt is $750 million of 4% bonds, so total liabilities are $1,150 million. Shareholders' equity is the remaining $1,250 million: $100 million of 7% preferred stock, which pays a fixed dividend and is paid before common stock, and $1,150 million of common equity, which is the money investors paid in for Lakemont's 50 million common shares plus its retained earnings, the profits it has kept over the years.

The income statement

The income statement shows revenue, expenses and profit over a period. Lakemont's 2025 net sales, or revenue, were $3,000 million. Subtracting the $1,800 million cost of the pumps it sold and $870 million of operating expenses leaves operating income of $330 million, also called EBIT, earnings before interest and taxes. Interest on the bonds, 4% of $750 million, was $30 million, leaving $300 million before taxes. Taxes at 21% took $63 million, leaving net income of $237 million. After $7 million of preferred dividends, $230 million was available to common shareholders, and Lakemont paid them a dividend of $1.38 a share.

Notice where interest sits: before taxes. Bond interest is a tax-deductible expense for a corporation, while preferred and common dividends are paid out of after-tax income.

Liquidity ratios

Liquidity is the ability to meet obligations coming due within a year.

• Working capital equals current assets minus current liabilities. Lakemont's is $800 million minus $400 million, or $400 million: the cushion left after paying the bills due within a year.
• The current ratio equals current assets divided by current liabilities. Lakemont's is $800 million divided by $400 million, or 2 to 1, which a common rule of thumb treats as comfortable for an industrial company.
• The quick ratio, also called the acid-test ratio, equals current assets minus inventory, divided by current liabilities. Lakemont's quick assets are $800 million minus $300 million, or $500 million, so its quick ratio is $500 million divided by $400 million, or 1.25 to 1.

Inventory is left out of the quick ratio because it is the least liquid current asset: it must be sold, and the customer must pay, before it becomes cash. A quick ratio of at least 1 to 1 means the company could pay all its current liabilities without selling any inventory; a ratio below 1 is a warning.

If both sides of a transaction are current, working capital does not change. Paying suppliers, collecting receivables and buying inventory with cash all leave it the same. Selling new bonds or stock for cash raises working capital, while buying equipment with cash, repaying long-term debt or declaring a cash dividend lowers it.

Leverage ratios

Leverage is the use of borrowed money to finance a business. Interest must be paid in good years and bad, so more debt means more risk of default, but also a bigger boost to shareholders when the business earns more than the interest rate.

• The debt-to-equity ratio equals long-term debt divided by total shareholders' equity. Lakemont's is $750 million divided by $1,250 million, or 0.6, which can also be stated as 60%.
• The debt-to-capital ratio, also called the bond ratio, equals long-term debt divided by long-term debt plus shareholders' equity. Lakemont's is $750 million divided by $2,000 million, or 37.5%.
• Interest coverage equals EBIT divided by annual interest. Lakemont's is $330 million divided by $30 million, or 11 times, so its interest is very safely covered.

Two profitability ratios round out the picture. Net profit margin equals net income divided by net sales; Lakemont's is $237 million divided by $3,000 million, or 7.9%. Return on equity equals earnings available to common divided by common equity; Lakemont's is $230 million divided by $1,150 million, or 20%.

Ratios mean the most when compared with the same company's past results and with competitors in the same industry, because norms differ widely: utilities carry far more debt than software companies.

1.7Valuation Ratios

The ratios so far describe the company. Valuation ratios compare the company's figures with the market price of its stock, so they describe what investors are paying. A wonderful company can be a poor investment at too high a price, so analysts need both. Lakemont's common stock trades at $69 a share. Chapter 3 uses these ratios to value stocks; here you learn to compute and read them.

Earnings per share and the P/E ratio

Earnings per share, or EPS, equals net income minus preferred dividends, divided by common shares outstanding. Preferred dividends come out first because preferred shareholders are paid before common shareholders. Lakemont's EPS is $237 million minus $7 million, or $230 million, divided by 50 million shares, which is $4.60. Companies also report diluted EPS, which assumes convertible securities and options are turned into common shares, so it is never higher than basic EPS.

The price-to-earnings ratio, or P/E, equals the market price per share divided by EPS. Lakemont's is $69 divided by $4.60, or 15: investors pay $15 for each dollar of annual earnings. Rearranged, price equals P/E times EPS. A trailing P/E uses the last 12 months of reported earnings, and a forward P/E uses analysts' estimates of the next 12 months. The inverse of the P/E, EPS divided by price, is the earnings yield: 1 divided by 15, or about 6.7%.

A high P/E usually means investors expect rapid earnings growth. A low P/E is typical of slower-growing companies, or of a company the market expects to struggle. A company with losses has no meaningful P/E.

The PEG ratio

A P/E of 15 is cheap for a company growing 25% a year and expensive for one that is not growing. The PEG ratio adjusts for growth: it equals the P/E divided by the expected annual earnings growth rate, with the growth rate written as a whole number. Analysts expect Lakemont's earnings to grow 10% a year, so its PEG is 15 divided by 10, or 1.5. A PEG near 1 is commonly read as a price in line with growth, below 1 as inexpensive for its growth, and well above 1 as expensive.

Price-to-book and price-to-sales

The price-to-book ratio, or P/B, equals the market price per share divided by book value per share, the accounting value of the common shareholders' stake per share. Under the conservative convention this course and the ratio explorer use, intangible assets are also subtracted, because they may be worth little if the company is liquidated. Book value per share therefore equals common equity minus intangibles, divided by common shares outstanding. For Lakemont, $1,150 million minus $200 million is $950 million, and dividing by 50 million shares gives $19. Its P/B is $69 divided by $19, or about 3.6.

Value investors look for low P/B ratios, and P/B is especially useful for banks and insurers, whose assets are mostly financial. A P/B below 1 means the stock sells for less than the accounting value of its net assets. The price-to-sales ratio divides the company's total market value by its annual sales. Because sales are positive even when earnings are not, it is often used for young companies with no profits yet.

Dividend yield and the payout ratio

The dividend yield equals the annual dividend per share divided by the market price. Lakemont's is $1.38 divided by $69, or 2%. As with bonds, the yield falls when the price rises. The dividend payout ratio equals annual dividends per share divided by EPS. Lakemont's is $1.38 divided by $4.60, or 30%, and the other 70% stays in the business to fund growth. Mature companies such as utilities tend to pay out a large share of their earnings and suit clients who need income, while young growth companies often pay little or nothing.

Growth and value signals

Growth stocks have earnings growing faster than the market's, and they tend to have high P/E and P/B ratios and low dividend yields, because they reinvest earnings to keep growing. Value stocks trade at low prices compared with their earnings and book value and tend to pay higher dividend yields. A low P/E is not proof of a bargain, though: a stock can be cheap because its earnings are about to fall, a situation investors call a value trap.

Try it: the explorer lets you edit a company's balance sheet and income statement and watch every ratio recalculate, each with its formula in words. It opens with its own sample company. To reproduce Lakemont, enter these figures in millions: cash, securities and receivables, 500; inventory, 300; current liabilities, 400; plant and equipment, 1,400; intangibles, 200; long-term debt, 750; preferred stock, 100; net sales, 3,000; cost of goods sold, 1,800; operating expenses including depreciation, 870; depreciation, 170; bond interest, 30; preferred dividends, 7; shares, 50; price, 69; and dividend per share, 1.38. It uses the same 21% tax rate, so its liquidity, leverage and per-share figures should match this chapter.

Chapter Summary

• The time value of money means a dollar today is worth more than a dollar later, because of lost earning power, inflation and risk. Future value equals present value times one plus the rate per period, raised to the number of periods, and more frequent compounding raises it.
• The Rule of 72 estimates the years to double as 72 divided by the rate, and the rate needed as 72 divided by the years; it works best between about 6% and 10%.
• Present value equals future value divided by one plus the rate, raised to the number of periods, and it falls when the discount rate rises or the wait lengthens. Every security is worth the present value of its expected cash flows.
• A positive NPV means accept and a negative NPV means reject. IRR is the rate that makes NPV zero, an IRR above the required return means accept, and a bond's yield to maturity is its IRR.
• The mean is pulled toward extremes, so the median is the best center for skewed data; when the mean is above the median, the data are positively skewed. The range is the highest value minus the lowest.
• Standard deviation measures total risk. In a normal distribution, about 68%, 95% and 99.7% of outcomes fall within one, two and three standard deviations of the mean.
• Correlation runs from minus 1 to plus 1, and any correlation below plus 1 reduces portfolio risk. Beta measures market risk against a market beta of 1.0, alpha is return above what beta predicts, and the Sharpe ratio is excess return per unit of standard deviation.
• Assets equal liabilities plus shareholders' equity. Working capital and the current and quick ratios measure liquidity, and the debt-to-equity ratio measures leverage.
• EPS is net income minus preferred dividends, divided by common shares. The P/E, PEG, P/B and dividend yield compare price with earnings, growth, book value and dividends, and the payout ratio compares dividends with earnings.

Key Numbers