Stock RROI relative to bonds (theory)#

Bond returns#

Let \(B\) be a bond with coupon \(c\), face value \(f\), market price \(p\) with \(T\) payment periods to maturity. The net returns on holding this bond to maturity, \(r_B\), is

\[ \begin{align} r_B = -p + (1 + cT)f. \end{align} \]

That is, the holder pays a price \(p\), collects \(T\)-many coupons \(cf\) and the face value \(f\) at maturity.

Stock returns#

Consider a stock \(S\). It is defined by its current market price \(P\) and dividend yield \(\delta\). Let \(g\) be its expected rate of growth per period. After \(T\) many periods, assuming the stockholder sells the stock, their net return \(r_S\) is

\[\begin{split} \begin{align} r_S &= -P + \delta P\sum_{n=0}^{T-1} (1 + g)^n + P(1 + g)^T \\ &= -P - \delta P\frac{1 - (1 + g)^T}{g} + P(1 + g)^T \\ &= P \frac{(g + \delta)((1 + g)^T - 1)}{g} \end{align} \end{split}\]

Indifference equation#

The RROI/B#

Let \(I\) be an investment amount. With respect to a bond \(B\) and dividend yield \(\delta\), at what growth \(g\) will an investor be indifferent between investing in the bond or in stock? Investing in the bond till maturity, the bond holder generates returns \(r_B(I)\),

\[\begin{split} \begin{align} r_B(I) &= -\frac{I}{p}p + (1 + cT)\frac{I}{p}f \\ &= -I + (1 + cT)\frac{f}{p}I. \end{align} \end{split}\]

Net returns on the stock, holding for \(T\) periods then selling, generates returns \(r_S(I)\) where,

\[\begin{split} \begin{align} r_S(I) &= -\frac{I}{P}P + \frac{I}{P}\delta P\sum_{n= 0}^{T-1}(1 + g)^n + \frac{I}{P}P(1 + g)^T \\ &= I \frac{(g + \delta)((1 + g)^T - 1)}{g}. \end{align} \end{split}\]

Notice that \(r_B(I) = I\cdot r_B(1)\) and \(r_S(I) = I\cdot r_S(1)\). Equating \(r_B(I) = r_S(I)\) then, \(I\) factors this equation. Hence, the investment amount \(I\) is not itself relevant. Rather, we obtain the equation

\[ \begin{align} -1 + (1 + cT)\frac{f}{p} = \frac{(g + \delta)((1 + g)^T - 1)}{g}. \end{align} \]

The left-hand side above is \(r_B(1)\).

Remark. The bond yield is \(\gamma = cf/p\) so that we can write in terms of the bond yield, \(r_B(1) = -1 + (1 + \gamma)T\).

The indifference equation is the following polynomial equation for \(g\)

\[ \begin{align} g(1 + g)^T + \delta (1 + g)^T - (r_B(1) + 1)g - \delta = 0. \end{align} \]

It appears as though we have a degree \((T+1)\)-equation for \(g\). Observe however that the constant term \(\delta\) will cancel out so that \(g\) factors the above equation. Dividing through by \(g\), assuming \(g\neq 0\), will therefore give a degree \(T\) equation.

Definition. A solution \(g\) to the indifference equation is referred to as the required return on investment relative to \(B\), or RROI/B.

Interpretation#

To reiterate, the RROI/B is the required return on investment relative to a fixed bond \(B\). It is the minimum return an investor ought to expect from owning stock in order to justify their investment in stock for \(T\) time periods instead of the bond \(B\).

Note, the RROI/B does not indicate valuation of any stock in particular; it is a perhaps more useful as a measure of the stock market as a whole (e.g., market index). And so we might conclude, in order for the stock market to be investible for \(T\) periods of time, it needs to grow an an average rate of \(g\) per period at dividend yield (or, distribution) \(\delta\).

Notice that the dividend yield enters the indifference equation positively. Hence, increasing \(\delta\) entails decreasing \(g\) in order to mainain equality. This is as one might expect: dividend yield \(\delta\) is inversely related to growth \(g\).

RROI/B solutions#

Pure growth#

Pure growth stocks are those which pay no dividends. Accordingly, the dividend yield for such stocks is \(\delta = 0\). In this case we obtain from the indifference equation

\[ \begin{align} g(1 + g)^T - (r_B(1) + 1)g = 0 \end{align} \]

so that

\[ \begin{align} g = \left(r_B(1) + 1\right)^{\frac{1}{T}} - 1. \end{align} \]

This is an exact solution for the RROI/B.

Special maturities#

Note, for \(T = 0\) we get RROI/B, \(g = 0\).

For \(T = 1\)#

The indifference equation becomes

\[ \begin{align} g (1 + g) + \delta (1 + g) - (r_B(1) + 1)g - \delta = 0. \end{align} \]

Simplifying and solving for \(g\) gives,

\[ \begin{align} g = r_B(1) - \delta. \end{align} \]

For \(T = 2\)#

From the indifference equation we get,

\[ \begin{align} g^2 + (2 + \delta)g + (2\delta - r_B(1)) = 0. \end{align} \]

And so by the quadratic formula we find solutions

\[\begin{split} \begin{align} g_\pm &= \frac{-(2 + \delta) \pm \sqrt{(2 + \delta)^2 - 4(2\delta - r_B(1))}}{2} \\ &= \frac{-(2 + \delta) \pm \sqrt{(2 - \delta)^2 + 4r_B(1))}}{2}. \end{align} \end{split}\]

See that while \(g_\pm\) need not be positive, it will nevertheless be real if \(r_B(1)\geq 0\). That is, when the bond price to face value ratio is less than its returns in coupons, i.e., \(\frac{p}{f} \leq 1 + 2c\).

Conservatively, the RROI/B in this case is taken to be the larger of the two solutions, i.e.,

\[ \begin{align} g = g_+ = \frac{-(2 + \delta) + \sqrt{(2 - \delta)^2 + 4r_B(1))}}{2}. \end{align} \]

Analytic solutions exist for maturities \(T = 3, 4\); however for \(T\geq 5\) it becomes a challenging problem for the pure mathematician. We leave the case \(T = 3, 4\) to the enterprising reader.

Quadratic approximation#

Since \(T\) is integral (i.e., an integer) the binomial expansion gives

\[\begin{split} \begin{align} (1 + g)^T &= \sum_{n=0}^T \binom{T}{n}g^n \\ &= \binom{T}{0}1 + \binom{T}{1}g + \binom{T}{2}g^2 + \cdots + \binom{T}{T}g^T \end{align} \end{split}\]

where the coefficients are binomials. Set \(x^3\equiv 0\) so that, to quadratic order,

\[ \begin{align} f(x) = 1 + Tx + \binom{T}{2}x^2 && \mbox{with error $x^3$.} \end{align} \]

Applying this approximation to the indifference equation gives the approximate indifference equation

\[ \begin{align} \binom{T}{2}g^2 + \left(T + \binom{T}{2}\delta\right)g + (T\delta - r_B(1)) = 0. \end{align} \]

We obtain through the quadratic equation therefore, just as above,

\[\begin{split} \begin{align} g_\pm &= \frac{ - \left(T + \binom{T}{2}\delta\right) \pm \sqrt{ \left(T + \binom{T}{2}\delta\right)^2 - 4\binom{T}{2}(T\delta - r_B(1)) }}{2\binom{T}{2}} \\ &= \frac{ - \left(T + \binom{T}{2}\delta\right) \pm \sqrt{ \left(T - \binom{T}{2}\delta\right)^2 + 4\binom{T}{2} r_B(1) }}{2\binom{T}{2}} && \mbox{with error $g_\pm^3$.} \end{align} \end{split}\]

Observe again that \(g_\pm\) is guaranteed to be real for \(r_B(1)\geq0\), i.e., for bond price-to-face ratio \(\frac{p}{f} \leq 1 + cT\). The RROI/B in this case is therefore

\[ \begin{align} g = g_+ = \frac{ - \left(T + \binom{T}{2}\delta\right) + \sqrt{ \left(T - \binom{T}{2}\delta\right)^2 + 4\binom{T}{2} r_B(1) }}{2\binom{T}{2}} && \mbox{with error $g_\pm^3$.} \end{align} \]

Note, we can obtain more accurate estimates by using the cubic or quartic formula. An exercise again for the enterprising reader.

RROI/B-adjusted price to earnings#

Classical price-to-earnings#

The RROI/B above is a minimum measure of growth required for any company in order for it to be investible over the bond. As for a particular public company, let \(P\) be its market price and \(E\) its earnings per share per unit time. Then the price earnings ratio \(P/E\) has units of time. It represents the following: if the company were to generate earnings \(E\) each period, it would take \(P/E\)-many periods of time to recoup the initial investment \(P\). The equation being for an investor,

\[ \begin{align} - P + \frac{P}{E}E = 0 \iff - P + \overbrace{E + E + \cdots + E}^{\scriptsize \frac{P}{E}\mbox{-times}} = 0. \end{align} \]

Adjusting for the bond#

Consider instead the case where company earnings grow at the RROI/B \(g\) from above for a fixed bond \(B\). The above series is the case where \(g = 0\). Generally we have

\[ \begin{align} -P + E + (1 + g)E + (1 + g)^2 E + \cdots + (1 + g)^{T^*-1}E = 0. \end{align} \]

The earnings \(E\) above is the forward earnings. Accordingly, we refer to the time \(T^*\) as the forward RROI/B-adjusted price-earnings ratio. Solving for it gives,

\[ \begin{align} T^* = \frac{ \log\left(1 + \frac{P}{E}g \right) }{ \log(1 + g) }. \end{align} \]

Recall that \(g\) was obtained by choosing a bond \(B\) with maturity \(T\). The above says: if we were to invest at price \(P\) into a stock with forward earnings \(E\), then assuming it grows at the RROI/B \(g\), we will recoup our initial investment after \(T^*\)-many years.

Note. Observe that \(\log(1) = 0\), so setting \(g = 0\) in the expression for \(T^*\) above is ill-defined. The indeterminate form is \(0/0\) however so, by the I’Hopital rule, we can evauate the limit \(g\rightarrow 0\) by first differentiating the numerator and denominator with respect to \(g\). In doing so, \(T^*\) will limit to \(P/E\) as \(g \rightarrow 0\) as expected.

Excess#

Set \(T^{**} = T - T^*\). We refer to \(T^{**}\) as the excess time. Recall, assuming the company grows at the RROI/B, the investor will expect to recoup their investment \(P\) purely through earnings after time \(T^*\). The excess time \(T^{**}\) is therefore the period of time over which the investor expects to make net positive returns on their investment in the stock through proportionate access to earnings (e.g., by way of dividends) relative to the bond maturity.

We can conclude therefore:

  • if \(T^{**} < 0\), the investor is better off investing in the bond;

  • if \(T^{**} > 0\) the investor may prefer the stock over the bond.

Note. We do not account for the investor selling their stock to realise gains or losses.