Implied return and implied growth

What does a price tell you?

A price is a forecast in disguise. The earlier lessons in this module took the cash flows and the rate and produced a price. Turn that around: take the price and the cash flows, and solve for the rate. The answer is the implied return of the instrument, the return the market is accepting at that price.

A market price and the expected cash flowsfind the rate that makes the present value equal the price

• Bond: implied return Memorize the yield to maturity
  • Zero (FV ÷ PV)^(1/t) − 1
  • Coupon bond solve I/Y, or RATE or IRR in a spreadsheet
  • Earned only if held to maturity, no default, coupons reinvested at the yield
• Share: required return Memorize dividend yield plus growth
  • r = D₁ ÷ P₀ + g growth known, solve for r
• Share: implied growth Memorize the growth the price needs, given r
  • g = r − D₁ ÷ P₀ return known, solve for g
  • From D₀ (rP₀ − D₀) ÷ (P₀ + D₀)
• Forward price-to-earnings (P/E) ratio Use it ties the multiple to payout, r and g
  • P₀ ÷ E₁ = payout ÷ (r − g) payout ratio = D₁ ÷ E₁

What is a bond’s implied return?

For a bond, the implied return is the yield to maturity : the single rate that makes the present value of the promised payments equal the price.

For a zero-coupon bond there is one payment, so the rate has a formula.

$$r = \left(\frac{FV}{PV}\right)^{1/t} - 1$$

For a coupon bond there is no formula for the rate. Enter the price as a negative PV, the face value as FV (future value), and compute I/Y.

The yield to maturity is the internal rate of return (IRR) of the promised cash flows: the single rate that makes their present value equal the price. It has no closed form, so a calculator or spreadsheet finds it by trial. In Excel or Google Sheets, =RATE(3, 4, −97.3, 100) returns 4.99%, and =IRR on the list −97.30, 4, 4, 104 gives the same. IRR needs at least one negative and one positive cash flow.

The yield is a promise, not an outcome

The yield to maturity is the return you earn only under the reinvestment assumption : you hold to maturity, nothing defaults and every coupon is reinvested at the yield itself. Break any of those and the realized return differs. For an amortizing loan the same assumption covers both the interest and the principal repaid along the way.

Go deeper Reinvestment risk against price risk

When rates rise, the price of a bond you hold falls (price risk) but the coupons you collect can be reinvested at higher rates (reinvestment risk works in your favor). When rates fall, the reverse. The two effects pull in opposite directions, and over a particular holding period they can cancel. Fixed Income measures this trade-off through duration.

What return and growth does a share price imply?

Rearrange the Gordon model and a share price yields either the required return or the growth rate, whichever one you do not know. The required return is the return investors demand for holding the share.

The required return

$$r = \frac{D_1}{P_0} + g$$

The required return is the dividend yield plus growth. Use the next dividend, D₁.

The implied growth rate

Suppose instead you know your required return and want the implied growth rate , the growth that the price needs. Growth is what is left of the required return after the dividend yield.

$$g = r - \frac{D_1}{P_0}$$

From the last dividend. If you only know the dividend just paid, D₁ itself depends on g, so rearrange P₀ = D₀ (1 + g) ÷ (r − g) instead.

$$g = \frac{r\,P_0 - D_0}{P_0 + D_0}$$

How does the Gordon model link to the P/E?

The dividend payout ratio , D₁ ÷ E₁, is the share of earnings paid out as dividends. (Avoid calling it b: in Equities, b is the retention ratio, the share kept.) The price-to-earnings ratio (P/E) divides a share price by earnings per share. Divide the Gordon value by next year’s earnings and you get the forward price-to-earnings ratio , the price divided by expected earnings.

$$\frac{P_0}{E_1} = \frac{D_1 / E_1}{r - g}$$

A higher payout and a higher growth rate raise the multiple. A higher required return lowers it.

The trailing price-to-earnings ratio divides the price by the last year’s earnings, E₀, instead. In the same model it is a little higher than the forward multiple, because E₀ is smaller than E₁.

$$\frac{P_0}{E_0} = \frac{(D_0 / E_0)(1 + g)}{r - g}$$

Go deeper What a high P/E says

Because P/E = payout ÷ (r − g), a high multiple means the market expects some mix of high growth, a low required return (low risk), or a high payout. A multiple alone cannot tell you which. Solving for g or r from the multiple, as above, turns the number into a statement you can challenge.