Merton Share Derivations: What's in your denominator? Elm Partners

May 2, 2024

Investment Theory

Merton Share Derivations: What’s in your denominator?

By Jeffrey M. Rosenbluth and James White ¹

1. Introduction

The Review of Economics and Statistics published a pair of companion papers in 1969. “Lifetime Portfolio Selection by Dynamic Stochastic Programming”, by Paul Samuelson and “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case”, by Robert Merton. Both deal with the question of how to allocate one’s portfolio between a risk-less and risky asset in a multi-period setting. The Samuelson paper considers the discrete time case and Merton’s the continuous time one. Merton solves this problem and provides a closed form solution of the highly stylized case where the risky asset rate of return follows a Brownian Motion (so that its price follows a Geometric Brownian Motion), the risk-less rate is constant, the utility function is CRRA (Constant Relative Risk Aversion) and the investor re-balances the portfolio continuously. Under these assumptions, he also shows that portfolio selection is myopic, that is, independent of the investment horizon and in fact the investor keeps a constant fraction of her wealth in the risky asset. We call this fraction the Merton Share. It is important to note that the Merton Share formula would be different under alternative sets of assumptions and that some of Merton’s assumptions are unrealistic: in particular, continuous re-balancing and Geometric Brownian Motion for asset prices. Nevertheless, we believe that using the Merton Share as a rule of thumb makes good sense and will often be close to the correct solution.

Merton used the theory of optimal control, and the Bellman principle of optimality in particular, to derive a partial differential equation (the Hamilton-Jacobi-Bellman equation) to solve the problem. In general, finding a closed-form solution to the HJB equation is rare and solutions are typically found numerically. In this note, we motivate and derive the Merton Share several different ways that are hopefully easier mathematically and provide more intuition as to why the formula makes sense. In doing so, we often deal with a single-period model and sometimes need to use approximations to derive the formula. The framework we will be using throughout to derive this formula is Expected Utility maximization; we will also often be assuming CRRA utility.

\[\begin{align*}
U(W) =
\begin{cases}
\frac{W^{1-\gamma}}{1 – \gamma } & \text{if } \gamma \neq 1 \\
\log(W) & \text{if } \gamma = 1
\end{cases}
\end{align*}\]

The defining feature of CRRA utility functions and hence its name, Constant Relative Risk Aversion, is that relative risk aversion is constant:

\[\begin{align*}
R(W) = \frac{-WU”(W)}{U'(W)} =\gamma
\end{align*}\]

We denote by k̂ the optimal fraction of wealth to invest in the risky asset. The Merton Share formula is:

\[\begin{align*}
\hat{k} = \frac{\mu}{\gamma \sigma^2}
\end{align*}\]

where μ is the expected excess return,² that is the return on the risky asset minus the risk-less rate and σ is its standard deviation. This formula certainly passes the smell test, the Merton Share is higher when excess return is higher, and lower when standard deviation and risk aversion are higher. The variance term in the denominator σ² as opposed to perhaps σ may seem less intuitive though.

2. Motivation

Let’s start to reveal why the denominator in the Merton Share is variance as opposed to standard deviation. We actually have another relation for k that relates it to the standard deviation of the portfolio. If we invest a fraction k of wealth in the risky asset with standard deviation of return σ, then the standard deviation σ_p of the portfolio is kσ.

\[\begin{align*}
\sigma_p = k\sigma
\end{align*}\]

Rearranging, we have:

\[\begin{align*}
k = \frac{\sigma_p}{\sigma}
\end{align*}\]

If we choose k = k̂ (the Merton Share), and let σ̂_p denote the standard deviation of the portfolio at k̂, the we obtain:

\[\begin{align*}
\frac{\hat{\sigma}_p}{\sigma} = \frac{\mu}{\gamma \sigma^2}
\end{align*}\]

so that:

\[\begin{align*}
\hat{\sigma}_p = \frac{\mu}{\gamma \sigma}
\end{align*}\]

This, hopefully, provides some intuition for why variance in the denominator of our Merton Share formula makes sense. It says that the risk (standard deviation) of the optimal portfolio is the ratio of excess return to standard deviation of the risky asset divided by the coefficient of relative risk aversion γ. The ratio of excess return to standard deviation is called the Sharpe Ratio, and is a commonly-used metric of the quality of a risky asset or trade.

Our intuition didn’t lead us far astray. It’s the risk of the optimal portfolio – not the optimal fraction – that is proportional to the Sharpe Ratio.

2.1 Myopic Portfolio Choice

Let’s approach the question of “Why variance in the denominator?” from another angle. First, in addition to CRRA with relative risk aversion γ, we also make the more restrictive assumption that asset returns are independent over time.³ Consider an investor with a two-period horizon. At the end of the first period the investor is faced with a single period optimization problem and since her risk aversion does not depend on wealth and returns are independent, the solution to this problem does not depend on how much was invested in the risky asset in Period 1. Now, the time 0 portfolio choice problem does not depend on time 1 wealth. Hence, the investor makes a single period portfolio choice at time 0 as well. When an investor makes the same portfolio decisions regardless of horizon, we say portfolio choice is myopic. By backward induction the above argument can be applied to any number of periods. This shows that with CRRA utility and time independent returns that portfolio choice is myopic. In our case we have in fact an even stronger result, constant portfolio choice over time, in which the investor holds the same fraction of wealth in the risky asset in each period. Let’s state this a a theorem and prove it more formally.

Theorum 1. If returns follow a stochastic process with independent increments, then for investors with CRRA utility of wealth, portfolio choice is constant over time.⁴

Proof. By the scale invariance property of CRRA utility, we know that k̂ does not depend on W_t that is wealth at any time t. From the independent increments assumption, we know that future risky asset prices do not depend on past wealth or past choices of k̂. Therefore, portfolio choice is myopic and k̂ is constant.

How does this help us to motivate the use variance in the denominator of the Merton Share? If portfolio choice is myopic, that means we would invest the same fraction of wealth in the risky asset for any horizon t. Suppose we have the function:

\[\begin{align*}
k(X_t) = \frac{\mu t}{\rho(X_t)}
\end{align*}\]

and we are choosing between standard deviation and variance for the operator ρ. We know that if our choice is myopic, then k(X_t) will not depend on t. For this to be true, its denominator must be a factor of t so that the t‘s will cancel. If X_t has independent increments, as do the majority of the stochastic processes employed to model excess returns, then StDev(X_t) = σ √t, so ρ can’t be standard deviation. On the other hand, variance Var(X_t) = σ² t works just fine.

3. Derivations

We provide core derivations (and two more in appendix) that are designed to motivate different aspects of the portfolio choice problem as it relates to the Merton Share.

3.1 Static Approximation

In this section, we assume the risky asset excess return is identically distributed over periods of the same length and that they are uncorrelated. In this case, both mean and variance are proportional to the horizon. The utility function U(W) is required to be twice differentiable and concave. We approximate this utility function with a Taylor series, resulting in a formula that is only valid for short horizons ⁵. We then specialize this result to the CRRA utility case.

Let W be the value of the initial portfolio. For a portfolio return Y, let U(W(1 + Y)) be the utility after one period with horizon t. Since U is twice differentiable we can approximate it with a second order Taylor series about Y = 0:

\[\begin{align*}
U(W(1+Y)) &\approx U(W) + U'(W)YW + \frac{1}{2} U”(W)(YW)^2\\
E[U(W(1 + Y)] &\approx U(W) + U'(W) E[Y] W + \frac{1}{2} U”(W) E[Y^2] W^2\\
& = U(W) + U'(W)E[Y]W + \frac{1}{2} U”(W)(\textrm{Var}[Y] + E[Y]^2) W^2
\end{align*}\]

Notice what is happening here, the combination of approximating utility by a Taylor series and taking its expected value introduces the moments of the probability distribution into the equation! If we take more terms of the Taylor series for a better approximation, then we need more moments. This should gives us additional comfort in choosing variance, not standard deviations, in the Merton Share formula.

In our case, the portfolio with a fraction k of wealth invested in the risky asset and the remainder in the risk free asset Y = (r + kX)t, where t is the horizon of the investment. The excess return X has mean μ t and variance σ² t as per our assumption, and r is the risk free rate of return. So E[Y] = (r + kμ)t and Var[Y] = k²σ² t. Since E[Y]² = (r + kμ)² t², it can be ignored for small t.

We want to maximize:

\[\begin{align*}
U(W) + (r + k\mu)tU'(W)W + \frac{1}{2} k^2\sigma^2tU”(W)W^2
\end{align*}\]

We differentiate with respect to k to obtain the first order condition:

\[\begin{align*}
0 &= \mu t U'(W)W + k\sigma^2 t U”(W)W^2\\
&=\mu U'(W) + k\sigma^2 U”(W)W
\end{align*}\]

Hence:

\[\begin{align*}
\hat{k} = \frac{-\mu U'(W)}{\sigma^2 W U”(W)}
\end{align*}\]

Recall from Section 1 the coefficient of relative risk aversion:

\[\begin{align*}
R(W) &= -W\frac{U”(W)}{U'(W)} \\
\frac{1}{R(W)} &= \frac{-U'(W)}{WU”(W)}
\end{align*}\]

Substituting this in to the above formula for k, we arrive at:

\[\begin{align*}
\hat{k} = \frac{\mu}{R(W) \sigma^2}
\end{align*}\]

This is a fairly general result, we have made very few assumptions about the utility function and asset return distribution.

Specializing to the CRRA utility case R(W) = γ so that:

\[\begin{align*}
\hat{k} = \frac{\mu}{\gamma \sigma^2}
\end{align*}\]

the Merton Share.

3.2 Asset Prices follow a Geometric Brownian Motion

In this section, we derive the Merton Share using assumptions similar to the ones Merton himself used. We assume CRRA utility, and have a risky asset S_t that follows a Geometric Brownian Motion (GBM) and a risk-less asset B_t with continuously compounded return r. That is:

\[\begin{align*}
\frac{dS_t}{S_t} &= (r + \mu)dt + \sigma dZ_t \\
\frac{dB_t}{B_t} &= r dt
\end{align*}\]

where Z_t is a Standard Brownian Motion (i.e μ = 0, σ = 1).

This setup is very common in finance. It is also very different from the derivation above, in that we are now have a dynamic optimization problem. Hence, k̂ is now a stochastic process that depends on the price path of the asset and time t, – call it k̂(S_t, t). Solving for k̂(S_t, t) is a problem in Stochastic Control⁶ which is beyond the scope of this note and requires quite a bit more mathematical machinery ⁷. But by employing Theorem 1, we know k̂ is constant and hence we can side step the stochastic control problem. Note that we still require the portfolio to be re-balanced to contain a fraction of wealth k̂ in the risky asset at every moment in time.

Given the above differential equations we can write down the stochastic differential equation (SDE) for wealth. We can think of this as saying that instantaneous returns on the wealth portfolio are k times the instantaneous return on the risky asset plus 1 – k times the return on the riskless asset.

\[\begin{align*}
\frac{dW_t}{W_t} &= (1 – k)\frac{dB_t}{B_t} + k\frac{dS_t}{S_t} \\
&= (1 – k)rdt + k(r + \mu)dt + k \sigma dZ_t\\
&= (r + k\mu)dt + k\sigma dZ_t
\end{align*}\]

Just as the risky asset is following Geometric Brownian Motion, we can see that the portfolio also is following GBM, i.e. the portfolio is also expressed as an SDE for GBM. The difference now is that the drift is r + kμ and the diffusion term is kσ, hence:

\[\begin{align*}
W_t = W \exp((r + k\mu – \frac{1}{2} k^2 \sigma^2)t + k \sigma Z_t)
\end{align*}\]

Without loss of generality, we can let W = 1, letting R_t = (r + kμ -½ k² σ²)t + k σ Z_t. We have:

\[\begin{align*}
\mathbb{E} \left[\frac{W_t^{1 – \gamma}}{1 – \gamma}\right] &= \mathbb{E}\left[\frac{\exp((1 – \gamma)R_t)}{1 – \gamma} \right] \\
&= \exp((r + k\mu – \frac{1}{2} k^2\sigma^2)t + \frac{1}{2} (1 – \gamma) k^2 \sigma^2 t / 2)\\
\end{align*}\]

where we have used the fact that the mean of a log-normal random variable with drift m and diffusion term s is:

\[\begin{align*}
\exp \left(m + \frac{1}{2} s^2 \right)
\end{align*}\]

For γ > 1, maximizing this expression is the same as minimizing:

\[\begin{align*}
(r + k\mu – \frac{1}{2} k^2\sigma^2) + \frac{1}{2} (1 – \gamma) k^2 \sigma^2
\end{align*}\]

The first order condition is:

\[\begin{align*}
\mu – k\sigma^2 + (1 – \gamma)k\sigma^2 = \mu -\gamma k \sigma^2 = 0
\end{align*}\]

Solving for k gives:

\[\begin{align*}
\hat{k} = \frac{\mu}{\gamma \sigma^2}
\end{align*}\]

Appendix

Normal Returns and Constant Absolute Risk Aversion (CARA) Utility

CARA utility and normally-distributed returns provide the only case where the Merton Share is an exact formula in the single-period world. Normal returns are undesirable since they allow negative asset prices and can’t be used for both sub-period and total period returns. The CARA (exponential) utility function exhibits constant absolute risk aversion A, which is also unrealistic. Despite these shortcomings, this case provides an instructive example. The CARA (exponential) utility function is:

\[\begin{align*}
U(W) = \frac{-\exp(-AW)}{A}
\end{align*}\]

To maximize expected utility, we can minimize the negative of U. Letting W be the starting wealth, we have:

\[\begin{align*}
\min_k \mathbb{E}[\exp(-A(1 + r + kX)W)] &= \mathbb{E}[\exp(-A(1 + r)W)\exp(-kAXW)]\\
&=\exp(-A(1 + r)W)\mathbb{E}[\exp(-kAXW)]\\
\end{align*}\]

The expectation of the log-normal random variable:

\[\begin{align*}
\mathbb{E}[\exp(-kAXW)] = \exp\left(-kA\mu W + \frac{k^2A^2 \sigma^2 W^2}{2}\right)
\end{align*}\]

So our minimization problem becomes:

\[\begin{align*}
\min_k \exp(-A(1+r)W)\exp\left(-kA\mu W + \frac{k^2A^2 \sigma^2 W^2}{2}\right)
\end{align*}\]

which is the same as:

\[\begin{align*}
\max_k A\mu W – \frac{k^2A^2 \sigma^2 W^2}{2}
\end{align*}\]

The first order condition is:

\[\begin{align*}
0 & =A \mu W – kA^2\sigma^2 W^2\\
&= \mu – kA \sigma^2 W
\end{align*}\]

Solving for k gives:

\[\begin{align*}
\hat{k} &= \frac{\mu}{AW \sigma^2} \\
&= \frac{\mu}{R(W) \sigma^2}
\end{align*}\]

This is effectively the Merton Share formula, and is the same result we obtained in Section 3.1.

Let’s explore this result a bit further. Since we assume that a utility function U is strictly concave, we know from Jensen’s inequality that:

\[\begin{align*}
\mathbb{E}[U(W_1)] < U(\mathbb{E}[W_1]) \end{align*}\]

We can think of this as an equality:

\[\begin{align*}
\mathbb{E}[U(W_1)] = c U(\mathbb{E}[W_1])
\end{align*}\]

for some c > 1. For most combinations of utility function and wealth distribution, we do not know what c is explicitly, but for the combination of exponential utility and normal returns we do. It’s exp(k² A² σ² W² /2). This shows that our maximization problem is a trade-off between mean μ and variance σ².

Quadratic Utility

The quadratic utility function

\[\begin{align*}
U(W) = -\frac{1}{2}(a – W)^2
\end{align*}\]

is not very realistic in that it has increasing absolute risk aversion and a “satisfaction” point beyond which more wealth lowers utility.

Its Arrow-Pratt Measure of Absolute Risk Aversion is:

\[A(W) = \frac{1}{a – W}\]

It’s often used to demonstrate a utility function whose portfolio selection fraction depends only on mean and variance regardless of the distribution of returns.

As usual, we start with the expected utility maximization problem:

\[\begin{align*}
\max_k \mathbb{E}[-\frac{1}{2}(a – (1+r+kX)W)^2]
\end{align*}\]

Differentiating with respect to k and setting to 0:

\[\begin{align*}
0 &= \mathbb{E}[WX (a -(1 + r – kX)W)] \\
&= a\mu W -(1 + r)\mu W^2 – k(\sigma^2 + \mu^2)W^2 \\
&= \mu (a -(1 + r)W) – k(\sigma^2 + \mu^2)W \\
k(\sigma^2 + \mu^2) W &= \mu(a -(1 + r)W)\\
k &= \frac{\mu(a -(1 + r)W)}{(\sigma^2 + \mu^2)W} \\
&= \frac{\mu}{\sigma^2 + \mu^2}\left(\frac{1 – rWA(W)}{WA(W)}\right)
\end{align*}\]

Static Approximation Revisited

When we derived the Merton Share back in Section 3.1, we made the assumptions that excess returns are identically distributed over periods of the same length and that they are uncorrelated. We needed to do this to ensure that both portfolio return and variance scale with horizon t. This is what allowed us to approximate the solution for small t. It turns out we can drop this restriction if instead we assume that the mean excess return is small. We can always write the excess return X as the sum of its expected return and a random variable with zero mean and the same standard deviation as X, say Z:

\[\begin{align*}
X = \mu + Z
\end{align*}\]

This lets us take k̂ to be a function of μ, k̂(μ) then we can use a first order Taylor expansion about 0 to estimate it.

\[\begin{align*}
\hat{k}(\mu) \approx \hat{k}(0) + \mu \hat{k}'(0)
\end{align*}\]

And since the optimal investment in a risky asset with zero return is 0.

\[\begin{align*}
\hat{k}(\mu) \approx \mu \hat{k}'(0)
\end{align*}\]

Let W₁ = (1 + r)W and w̃ = W₁ + k̂(μ)(μ + Z)W. At the optimum, k̂, the first order condition must be 0.

\[\begin{align*}
\mathbb{E}[(\mu + Z)WU'(W(1 + r + \hat{k}(\mu)(\mu + Z)))] = \mathbb{E}[(\mu + Z)WU'(\tilde{W})] = 0
\end{align*}\]

We use this to calculate k̂'(0) by implicit differentiation. Differentiating the first order condition with respect to μ, then setting μ = 0:

\[\begin{align*}
0 &= \mathbb{E}[(\mu + Z)W(\hat{k}(\mu)W + \hat{k}'(\mu)(\mu + Z)W)U”(\tilde{W}) + WU'(\tilde{W})] \\
&= \mathbb{E}[Z^2W^2\hat{k}'(0)U”(W_1) + WU'(W_1)] \\
&= \mathbb{E}[Z^2W\hat{k}'(0)U”(W_1) + U'(W_1)] \\
&= \sigma^2 W \hat{k}'(0)U”(W_1) + U'(W_1)\\
\hat{k}'(0) &= \frac{-U'(W_1)}{\sigma^2 W U”(W_1)} \\
\mu \hat{k}'(0) &= \frac{\mu}{R(W) \sigma^2}
\end{align*}\]

which is the same result we found in Section 3.1. In this case, we see that small means a first order Taylor expansion of k̂ is sufficient, i.e. μ is close to 0.