1. Prices of Bonds at Issuance

Two hypothetical bonds are issued for $1,000 each at the start of Year 1. One matures in five years, and pays interest at a rate of 6% (the market rate for a bond of that maturity), while the other matures in 10 years and pays interest at 8% (also the market rate for its maturity).

The value an investor realizes from a bond is the stream of its cash flows, so that the prices of each bond at any point in time are the sums of the present values of each of the future interest and principal payments. The present value of a cash flow is a function of (1) the amount of cash to be received, (2) the timing of the payment, and (3) the interest rate (or discount rate) used to calculate the present value. For a given payment t, its present value (PV) is defined as:

The examples show each year’s cash flows for interest and principal, and the discount factors applied to each (Table B.1). Note that the present value of the each year’s interest payment declines the further ahead in time it is to be received, due to the compounding of the discount factor. Also note that the largest component in the price of each bond is the present value of its principal payment. Because it is paid at the end of the bond’s life, the principal is subject to the greatest discounting.

In these examples the coupon rate of interest is equal to the market rate at issuance, so the bonds are priced at “par,” or $1,000.

2. Bond Prices with Changing Interest Rates

Time moves ahead to the start of Year 2. The first interest payment on each bond has been paid to the bond holders. If market interest rates happen to be the same as they were at issuance—that is, equal to the coupon rate—the price of each bond would again be $1,000. Although there are fewer payments to be received, the present values of the interest and principal payments have increased, due to lower discount factors applied to each (Table B.2 shows the example of the five-year bond as of the start of Year 2). For example, the present value of the principal payment has increased from $747 to $792.

However, if interest rates have changed, the discount factors and prices of the bonds change with them. Higher interest rates bring a higher discount factor, and each interest and principal payment is divided by a greater number, lowering the present values. (The opposite holds if interest rates have fallen.)

Table B.3 shows the effect of a 1 percentage point rise in rates for the five-year bond (to a yield of 7% from 6%). Due to compounding at a higher discount rate, each payment would be divided by a greater discount factor and have a lower present value, and in this case the bond price would have fallen from $1,000 at issuance to $966 (a decline of 3.4%). Details of the 10-year bond example are not shown, but on a one percentage point rise in rates, to a yield of 9%, its value too would have dropped—to $940, for a fall of 6.0%.

If interest rates had instead fallen from the date of issuance to the start of Year 2, the prices of both bonds would have risen, because in each case the cash flows would be divided by a smaller discount factor.

3. Bond Duration

In general, the longer the maturity of a bond, the greater the sensitivity in its price to changes in interest rates. This is due in great part to the back-loading of the present value of a bond’s price in its principal payment. In the case of the five-year bond, the rise in market rates from 6 percent to 7 percent caused the present value in Year 2 of the principal to fall from $792 to $763, accounting for most of the $34 decrease in the bond’s price (Table B.2 versus Table B.3).

Bond math allows for a calculation of a bond’s “modified duration”—a weighted average of its term to maturity (weighted by the present value of the cash flow to be received in each year), which expresses the approximate sensitivity of a bond’s price to a 1 percentage point (or 100 basis point) change in market interest rates. The modified duration of the five-year bond at the start of Year 2 is 4.2 (which multiplied by the one percentage point rise in rates is approximately equal to the drop in price). Modified duration for the 10-year bond is far higher, however, at 6.6, reflecting the increased effects of discounting on its longer stream of cash flows over a greater time span.

1. Dividend Discount Models

These present value principles of bond valuation can be applied to the estimating the fair value of equities as well. In 1938, investment manager John Burr Williams authored The Theory of Investment Value, which sought to apply a mathematical framework to investing in stocks. His goal was not to show how to build wealth, but rather to provide investors an understanding, as he put it, of the physiology of the markets:

The wide changes in stock prices in the last eight years, when prices fell as much as 80 or 90 per cent from the 1929 peaks only to recover much of their decline later, are a serious indictment of past practice in Investment Analysis. Had there been any general agreement among analysts themselves concerning the proper criteria of value, such enormous fluctuations should not have occurred, because the long-run prospects for dividends have not in fact changed as much as the prices have. Prices have been based too much on current earning power, too little on long-run dividend paying power.¹

Williams defined the investment value of a stock as the present value of all the dividends to be paid, in the same way that a bond’s value is the present value of its interest and principal payments:

Most people will object at once to the foregoing formula for stocks by saying that it should use the present worth of future earnings, not future dividends. . . . Earnings are only a means to an end. . . . Therefore we must say that a stock derives its value from its dividends, not its earnings. In short, a stock is worth only what you can get out of it.²

He developed several models, applicable to companies in various financial situations—declining, stable and growing dividends. The mechanics of some are quite complicated (which he acknowledged): the “sudden expansion” variety entailed 11 known or assumed variables, and seven additional unknowns. He also conceded that the hardest part of valuing stocks on future cash flows would be estimating the dividends for each year over a long future.

In 1956 Myron Gordon, a professor at the University of Toronto, introduced a simpler dividend discount model that relieved analysts of forecasting earnings and dividends far into the future, and instead allowed them to substitute assumptions on the future growth rates of earnings and dividends. The Gordon growth model is best suited to situations with expected stable growth, and earnings and dividends not much affected by the business cycle, such as utilities and consumer staples companies. Here is its basic form:

D represents the dividend per share, k the company’s cost of equity capital, and g the rate of growth in dividends.

The Gordon growth model is appealing for its simplicity, but with so few inputs is highly sensitive to the analyst’s assumptions, particularly the reasonableness of expected growth in dividends (as the model assumes that such growth will last forever).

Other forms of dividend discount models have been devised to account for different stages of a company’s growth—typically with one segment showing more rapid growth early on, and a second for slowing growth as a company’s business matures. Dividend discount models can also be adjusted for firms not currently paying dividends, through estimates of dividends to be paid in the future.

One shortcoming of dividend discount models is their focus on dividends, particularly in a world where companies are making larger distributions to owners in the form of share repurchases (considered in Chapter 10). However, models can be modified by combining dividends and share repurchases, and looking at the firm’s returns to owners in the aggregate rather than on a per share basis.

2. Duration

The prices of equities are sensitive to interest rates shifts just as bonds are, but while measuring the variations of bond prices from changes in interest rates is scientific and well established, the evolution of duration measures for equity has been ad hoc.³ As with dividend discount models, a proper evaluation would call for developing long-term, detailed forecasts of future cash distributions, which lack the certainty of expected cash flows from bonds’ interest and principal payments.

The same principles apply to equities, however: as with bonds, the sooner cash is paid out, the lower the duration. Thus shares of a mature company that pays high dividends have a shorter duration than those of a growth company that pays little or no dividends. In turn, the prices of stocks that pay higher current returns to shareholders should generally be less vulnerable to rising interest rates, all else being equal.⁴

Professors Patricia Dechow, Richard Sloan, and Mark Soliman developed a two-stage duration model for equities in 2001, and found it to perform well in empirical tests. Overall they concluded that “[D]uration tends to be low for firms with high ROE, low growth and low market valuations and high for firms with low ROE, high growth and high market valuations.”⁵

A rough-and-ready duration measure, not calling for detailed cash flow forecasts, was described in 1985 by investment practitioner James Farrell.⁶ His formulation is similar to the Gordon growth model, and based on dividend growth and required return:

Where k is the stock’s required return, and g is the growth rate in dividends. Rearranging the terms makes the duration measure even simpler:

Farrell comments that stocks with low yields carry longer durations, and are relatively more sensitive to discount rate changes, and that “High-growth stocks, which are generally characterized by relatively low dividend yields, would be more subject to this risk than low-growth stocks.” Overall he concludes that because stocks have a perpetual life and growing dividends, “[They] should thus be more responsive than bonds to changes in real interest rates, and carry a correspondingly higher premium (via the discount rate).”⁷