Implied Return and Growth

Implied Return for Fixed-Income Instruments

The growth rate is the rate at which the market expects an asset to grow. On the other hand, implied return reflects a return based on the current price and future security cash flows.

Consider a fixed-income instrument. If we have its present value and assume all future cash flows happen as expected, the discount rate, $r$, or yield-to-maturity, YTM, shows the implied return under these assumptions for the cash flow pattern.

Now, take an equity investment. If we have the present value, future value, and discount rate, we can find the implied growth rate that aligns with these values.

The implied return or growth rate provides a view of the market expectations incorporated into an asset’s market price. Understanding these expectations is critical for investors when making investment decisions.

Calculating the Implied Return for Fixed-Income Instruments

Discount Bond

In the case of a discount bond or instrument, recall that an investor receives a single principal cash flow \((FV)\) at maturity, with \((FV – PV)\) representing the implied return.

To solve for the implied return earned over the life of an instrument (N periods), we can rearrange the single cash flow present value formula.

Recall that the single cash flow present value formula is:

$$ PV =FV_t (1+r)^{-t} $$

Where:

\(FV\) = Future value.

\(PV\) = Present value.

\(r\) = Stated discount rate per period.

\(t\) = Number of compounding periods.

To solve for \(r\), we can rearrange this formula as follows:

$$ r=\sqrt[t]{\frac {FV_t}{PV}}-1= \left( \frac {FV_t}{PV}\right)^{\frac {1}{t}}-1 $$

We use this formula to calculate the periodic return earned during the instrument’s life (t periods) based on its present value (or price) and future value.

Example: Calculating the Implied Return for a Discount Bond

Consider a zero-coupon bond with price of $900, a future value of $1,000, and a maturity of 5 years. Calculate the implied annualized return \(r\).

Solution

Recall that,

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

In this case, \(t\)=5, \(FV_t\)=$1,000, \(PV\)=$900

So,

$$ r= \left(\frac {1000}{900} \right)^{\frac {1}{5}}-1=2.13\% $$

This means that an investor who purchases this zero-coupon bond at a price of $900 and holds it for five years would earn an annualized return of 2.13%.

Coupon Bonds

Recall that fixed-income instruments that pay periodic interest have cash flows throughout their life until maturity. The yield-to-maturity (YTM) is a single implied market discount rate for all cash flows, regardless of timing. It assumes an investor expects to receive all promised cash flows through maturity and reinvest any cash received at the same YTM.

The present value of a fixed-income instrument with periodic interest can be calculated using the following formula:

$$ PV=\frac {PMT_1}{(1+r)^1} + \frac {PMT_2}{(1+r)^2} + \cdots + \frac {(PMT_N+FV_N)}{(1+r)^N} $$

Where:

\(PV\) = Present value (or price) of the instrument.

\(PMT\) = Periodic payment.

\(FV\) = Bond’s principal.

\(N\) = Number of periods to maturity.

\(r\) = Discount rate (or internal rate of return) (YTM).

Example: Implied Return for Fixed-income Instruments With Periodic Interest

Consider a five-year corporate bond issued in 2023 with a 4.00 percent annual coupon and a price of USD110.00 per USD100 principal three years later. If Milka can reinvest periodic interest at the original YTM of 4.00 percent, the implied three-year return is closest to:

Solution

We can calculate the future value (FV) after three years, including the future price of 110.00 and all cash flows reinvested to that date:

$$ \begin{align*}
FV_3 & =PMT_1 (1+r)^2+PMT_2 (1+r)+PMT_3+PV_3 \\
&=4 \times (1.04)^2+4 \times (1.04)+4+110.00 \\
&= \$122.49
\end{align*} $$

We can then solve for Milka’s annualized return, r, using the formula for implied return since we have \(PV\)=100, \(FV\)=122.49, \(N\)= 3 as follows:

$$ r=\sqrt[t]{\frac {FV_t}{PV}}-1=\sqrt[3]{\frac {122.49}{110}}-1=3.65\% $$

This means that Milka, who purchased the corporate bond at a price of 100 and held it for three years, would earn an annualized return of 3.65%.

Example: Calculating the Yield-to-Maturity of a Coupon Bond

CityGroup Corp. issued a corporate bond 7 years ago with a face value of $1,000 and a 20-year maturity. The bond pays annual interest at a coupon rate of 6%. Currently, the bond is trading at $1,120. The yield to maturity (YTM) of CityGroup Corp.’s bond is closest to:

Solution

Using the BA II Plus calculator, we solve the question as follows:

We have

$$ \begin{array}{l|c|c}
\textbf{Steps} & \textbf{Explanation} & \textbf{Display} \\ \hline
{[2nd] [\text{QUIT}] }& \text{Return to standard calc Mode} & 0 \\ \hline
{[2nd] [\text{CLR TVM}] } & \text{Clears TVM Worksheet} & 0 \\ \hline
13[N] & \text{Years/periods} & N = 13 \\ \hline
-1,120[PV] & \text{Set the present value} & PV = -1,120 \\
& \text{of the bond} & \\ \hline
60[PMT] & \text{Set the periodic} & PMT = 3.90 \\
& \text{coupon payment} & \\ \hline
1,000[PV] & \text{Set the face value} & FV = 1000.00 \\
& \text{of the bond} & \\ \hline
[CPT][I/Y] & \text{Compute the YTM} & I/Y = 4.74\%
\end{array} $$

Therefore, the YTM is 4.74%.

Implied Return and Growth for Equity Instruments

The value of a stock is determined by both the expected return and the growth of its cash flows. By assuming a constant growth rate for dividends, we can use the formula for the present value of an equity investment to calculate the stock’s implied return or growth rate.

Implied Return

Recall that the present value of a stock for constant growth of dividends is given by:

$$ PV_t=\frac {D_t(1+g)}{r-g}=\frac {D_{t+1}}{r-g} $$

Where:

\(PV_t\) = Present value at time \(t\).

\(D_t\) = Expected Dividend in the next period.

\(r\) = Required rate of return.

\(g\) = Constant growth rate.

\(r-g \gt 0\)

Therefore, we can calculate the implied return on a stock given its expected dividend yield and implied growth by rearranging the above formula as follows:

$$ r=\frac {D_t(1+g)}{PV_t }+g=\frac {D_{t+1}}{PV_t}+g $$

In simple terms, if we assume a stock’s dividends will grow at a steady rate forever, the implied return is the combination of its expected dividend yield and the constant growth rate.

Example: Implied Return and Growth

Assume Apple Inc. stock is trading at a share price of USD150.00, and its annualized expected dividend per share during the next year is USD2.00.

Moh, an analyst, projects that Apple’s dividend per share will increase at a constant rate of 5% per year indefinitely. The required return expected by investors on the stock is closest to:

Solution

Recall that the implied return formula is,

$$ r= \frac {D_t(1+g)}{PV_t}+g=\frac {D_{t+1}}{PV_t}+g $$

In this case, \(D\)=$2.00, \(PV\)=$150, \(g\)=5%

Therefore,

$$ r=\frac {2.00(1.05)}{150}+0.05=6.4\% $$

Implied Growth

We can also solve for a stock’s implied growth rate, which is given by the following formula:

$$ g=\frac {r \times PV_t-D_t}{PV_t}+D_t=\text{r}-\frac {D_{t+1}}{PV_t } $$

Example: Calculating the Implied

Consider the previous example. Suppose Moh believes that Apple stock investors should expect a return of 8%, calculate the implied dividend growth rate for Apple Inc.

Solution

Recall that the formula for calculating implied growth is as follows.

$$ g=\frac {r \times PV_t-D_t}{PV_t}+D_t=\text{r}-\frac {D_{t+1}}{PV_t } $$

So,

$$ g=0.08-\frac {2.00 \times 1.05}{150}=0.066=6.60\% $$

Price-to-Earnings Ratio

In equity instruments, it is common practice to compare the price-to-earnings ratio.

The price-to-earnings (P/E) ratio is a valuation metric that compares the current share price of a stock to its earnings per share. Investors and analysts use it to determine the relative value of a company’s shares compared to other companies or the market.

A stock with a higher price-to-earnings ratio is more expensive than a lower one, as investors are willing to pay more for each unit of earnings. This ratio is also a valuation metric for stock indexes, such as S&P 500.

Relating P/E Ratio to Expected Future Cash Flows

The P/E ratio can relate to our earlier discussion on a stock’s price (PV) to the expected future cash flow relationship. Recall the following equation:

$$ PV_t=\frac {D_t \times (1+g)}{r-g} $$

By dividing both sides of the equation by \(E_t\), which represents earnings per share for period \(t\), we get the following equation:

$$ \frac {PV_t}{E_t} =\frac { \frac {D_t}{E_t} \times (1-g)}{r-g} $$

Where:

\(\frac {PV_t}{E_t}\) = Price-to-earnings (P/E) ratio.

\(\frac {D_t}{E_t}\) = Dividend payout ratio.

\(g\) = Growth rate.

\(r\) = Required rate of return.

The dividend payout ratio represents the percentage of a company’s earnings paid out to shareholders in the form of dividends.

Typically, the forward P/E ratio, which is based on a projection of a company’s earnings per share for the next period \((t+1)\), is used. This ratio is positively correlated with higher expected dividend payouts and growth rates but negatively correlated with the required return.

Therefore, the equation:

$$ \frac {PV_t}{E_t} =\frac { \frac {D_t}{E_t} \times (1-g)}{r-g} $$

This can be simplified as below to find the forward P/E ratio:

$$ \frac {PV_t}{E_{t+1}} =\frac { \frac {D_{t+1}}{E_{t+1} }}{ r-g }=\frac {D_{t+1}}{E_{t+1} } \times \frac {1}{r-g} $$

Example: Solving for Implied Dividend Growth Rate

Suppose a company has a forward P/E ratio of 15, a dividend payout ratio of 40%, and a required return of 10%. The implied dividend growth rate for this company is closest to:

Solution:

First, we can use the formula for the forward P/E ratio to solve for the implied dividend growth rate:

$$ \frac {PV_t}{E_{t+1}} =\frac {D_{t+1}}{E_{t+1}} \times \frac {1}{r-g} $$

Where:

\(PV_t\) = Present value at time \(t\).

\(E_{t+1}\)= Earnings per share for the next period.

\(D_{t+1}\)= Dividend payout for the next period.

\(r\) = Required return.

\(g\) = Implied dividend growth rate.

Substituting the given values into the formula, we get:

$$ 15=\frac {0.4}{0.1-g} $$

Solving for \(g\), we get:

$$ g=0.1-\frac {0.4}{15}=0.0733 $$

Therefore, the implied dividend growth rate for this company is 7.33%.

Example: Solving for Required Return

Let’s assume you are not given the required rate of return in the question above so that the company has a forward P/E ratio of 15, a dividend payout ratio of 40%, and an implied dividend growth rate of 7.33%. What is the required return for this company?

Solution

Recall the formula for the forward P/E ratio to solve for the required return:

$$ \frac {PV_t}{E_{t+1}} =\frac {D_{t+1}}{E_{t+1}} \times \frac {1}{r-g} $$

Substituting the given values into the formula, we get:

$$ 15=\frac {0.4}{r-0.0733} $$

Solving for \(r\), we get:

$$ r=\frac {0.4}{15}+0.0733=0.1000 $$

Therefore, the required return for this company is 10%.

Question

Edmund company’s stock trades at USD50.00. The company pays an annual dividend to its shareholders, and its most recent payment of USD 2.00 occurred yesterday. Analysts following the company expect its dividend to grow at a constant rate of 4 percent per year. What is the company’s required return?

1. 8.16%.
2. 8.48%.
3. 9.16%.

The correct answer is A.

Solution

Recall that:

$$ PV=\frac { D_{t+1} \times (1+g)}{r-g} $$

Where:

\(PV\) = Current stock price.

\(D_{t+1}\) = Recent dividend payout.

\(g\) = Expected dividend growth rate.

\(r\) = Required return.

Substituting the given value into the formula, we get:

$$ 50=\frac {2\times (1+0.04)}{r-0.04} $$

Solving for \(r\), we get:

$$ \begin{align*}
r & =\frac {2 \times (1+0.04)}{50}+0.04=0.0816 \\
r & =0.0816 \end{align*} $$

Edmund’s required return is 8.16%.