Phase 14 - Lesson 14.5

Portfolio Theory, Utility, and Optimal Investment

Mean–variance efficiency, expected-utility choice, and a first look at the Merton problem.

⏱ 60 min● Advanced🔗 Prereqs: 14.1–14.2, linear algebra (Phase 3), probability (Phase 7)

Builds on: Pricing (14.1–14.4) used \(\mathbb{Q}\); investment is a real-world \(\Prob\) problem about preferences and risk.

Learning Objectives

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Formulate the mean–variance problem and derive the tangency (maximum-Sharpe) portfolio.
Explain expected-utility theory and the role of risk aversion.
Derive the Merton optimal risky fraction for CRRA utility in the single-period / constant-investment case.
Distinguish investment (real-world \(\Prob\), preferences) from pricing (risk-neutral \(\mathbb{Q}\)).
Compute an efficient portfolio and its Sharpe ratio from expected returns and a covariance matrix.

Key Vocabulary

Mean–variance efficiency: A portfolio with the least variance for its expected return; the set of such portfolios is the efficient frontier.

Tangency portfolio: The risky portfolio maximizing the Sharpe ratio; with a risk-free asset, all investors hold it plus cash (two-fund separation).

Sharpe ratio: Excess return per unit of volatility, \((\mu_p-r)/\sigma_p\); the slope of the capital market line.

Expected utility: Ranking gambles by \(\E[u(W)]\) for an increasing concave \(u\); concavity encodes risk aversion.

CRRA utility: Constant relative risk aversion \(u(W)=W^{1-\gamma}/(1-\gamma)\) with risk-aversion parameter \(\gamma\gt 0\).

Merton problem: Continuous-time optimal consumption/investment; for CRRA the optimal risky fraction is the constant \(w^*=(\mu-r)/(\gamma\sigma^2)\).

Intuition & Motivation

Intuition

Pricing asks ‘what is this claim worth?’ and answers under \(\mathbb{Q}\). Investment asks a different question - ‘how should I allocate my wealth?’ - and it lives squarely in the real world \(\Prob\), because it depends on actual expected returns and on your preferences. Markowitz reduced the problem to a trade-off between mean and variance, producing an efficient frontier and, with a risk-free asset, a single best risky mix (the tangency portfolio) that everyone scales up or down. Expected-utility theory generalizes this: rank outcomes by \(\E[u(W)]\) with concave \(u\) for risk aversion. Merton then made it dynamic - and for constant-relative-risk-aversion preferences the answer is beautifully simple: hold a constant fraction \(w^*=(\mu-r)/(\gamma\sigma^2)\) of wealth in the risky asset.

Mean–variance and the efficient frontier

With risky assets of expected returns \(\mu\) and covariance \(\Sigma\), a portfolio \(w\) (weights summing to 1) has mean \(\mu_p=w^\top\mu\) and variance \(\sigma_p^2=w^\top\Sigma w\). The efficient frontier minimizes \(\tfrac12 w^\top\Sigma w\) for each target mean - a quadratic program whose solution traces a hyperbola in \((\sigma_p,\mu_p)\) space.

Add a risk-free asset \(r\). Maximizing the Sharpe ratio \((w^\top\mu-r)/\sqrt{w^\top\Sigma w})\) gives the tangency portfolio, proportional to

\[w_{\text{tan}}\ \propto\ \Sigma^{-1}(\mu-r\mathbf 1),\qquad \text{normalized so }\ \mathbf 1^\top w=1.\] (14.12)

Two-fund separation: every mean–variance investor holds only the risk-free asset and the tangency portfolio, in a ratio set by risk appetite. The line from \(r\) through the tangency point is the capital market line; its slope is the maximal Sharpe ratio.

Expected utility and risk aversion

Mean–variance is exact only for quadratic utility or normal returns; the general theory ranks random wealth \(W\) by expected utility \(\E[u(W)]\) with \(u'\gt 0\) (more is better) and \(u''\lt 0\) (risk aversion). Concavity means a certain amount is preferred to a fair gamble with the same mean - the gap is the risk premium the investor demands.

Definition - Absolute and relative risk aversion

Arrow–Pratt coefficients \(A(W)=-u''(W)/u'(W)\) (absolute) and \(R(W)=-Wu''(W)/u'(W)\) (relative). CRRA utility \(u(W)=W^{1-\gamma}/(1-\gamma)\) has constant \(R(W)=\gamma\); log utility is the \(\gamma\to1\) limit.

The Merton problem (introduction)

Merton studied an investor allocating between a risk-free bond (rate \(r\)) and a stock \(dS/S=\mu\,dt+\sigma\,dW\) to maximize expected utility of terminal (and/or consumed) wealth. For CRRA utility the striking result is a constant optimal fraction of wealth in the stock:

\[w^*=\frac{\mu-r}{\gamma\,\sigma^2}.\] (14.13)

This is the continuous-time analogue of the tangency intuition: numerator = risk premium, denominator = risk-aversion times variance. Higher expected excess return \(\mu-r\) raises the allocation; more volatility \(\sigma^2\) or more risk aversion \(\gamma\) lowers it. Note this is a real-world \(\Prob\) statement using the physical drift \(\mu\) - unlike pricing, the drift genuinely matters here.

Key Idea

Two measures, two jobs. Pricing uses \(\mathbb{Q}\) and the risk-free drift \(r\); the real drift \(\mu\) is irrelevant. Investment uses \(\Prob\) and depends critically on \(\mu\) and on preferences \(\gamma\). Confusing the two is a deep conceptual error.

Worked Example - Merton fraction for a concrete investor

Stock: \(\mu=0.10,\ \sigma=0.20\); risk-free \(r=0.03\); CRRA with \(\gamma=3\).

Risk premium: \(\mu-r=0.07\). Variance: \(\sigma^2=0.04\).

Optimal fraction: \(w^*=0.07/(3\cdot0.04)=0.07/0.12=0.583\).

So invest about 58% of wealth in the stock, 42% in the bond. A more risk-averse investor (\(\gamma=6\)) would hold half as much, \(29\%\).

Interactive: compute the tangency portfolio and Sharpe ratio

Common Mistakes to Avoid

Using the risk-neutral drift \(r\) in an investment problem; allocation depends on the real expected return \(\mu\) and preferences.
Applying mean–variance blindly to non-normal, fat-tailed returns where variance understates tail risk.
Treating estimated \(\mu\) and \(\Sigma\) as exact; sample means are noisy and optimizers amplify the error (the ‘error-maximization’ problem).
Reading \(w^*\) or a high Sharpe ratio as a promise of profit; these are model-dependent, estimate-sensitive, and carry real risk of loss.

Quant Practitioner Tips

Two-fund separation is a huge simplification: pick the tangency portfolio once, then dial risk with the cash weight.
Regularize portfolio optimization (shrinkage of \(\Sigma\), constraints, Bayesian priors) or the weights explode on noisy inputs.
The Merton fraction \((\mu-r)/(\gamma\sigma^2)\) is the mental model for sizing any single risky bet by edge, variance, and risk appetite.
Always separate the pricing question (use \(\mathbb{Q}\), ignore \(\mu\)) from the investment question (use \(\Prob\), \(\mu\) is central).

Knowledge Check

Q1 Medium

With a risk-free asset, two-fund separation says every mean–variance investor holds:

A different risky portfolio each

The risk-free asset plus the same tangency portfolio, scaled by risk appetite

Only the risk-free asset

Only the highest-return asset

Q2 Medium

The Merton optimal risky fraction for CRRA utility is:

\(w^*=(\mu-r)/\sigma\)

\(w^*=(\mu-r)/(\gamma\sigma^2)\)

\(w^*=\gamma(\mu-r)\)

\(w^*=r/\sigma^2\)

Q3 Hard

Which measure and drift govern the optimal-investment problem?

Risk-neutral \(\mathbb{Q}\) with drift \(r\)

Real-world \(\Prob\) with the physical drift \(\mu\)

Neither; it is preference-free

Both give the same answer

Practical Exercise

An investor with CRRA utility (\(\gamma=4\)) faces a stock with \(\mu=0.12,\ \sigma=0.25\) and risk-free \(r=0.02\). (a) Compute the Merton risky fraction. (b) Interpret the sign and size. (c) If leverage above 100% is disallowed, how does the constraint bind here?

▶ Show full solution

(a) \(w^*=(\mu-r)/(\gamma\sigma^2)=(0.12-0.02)/(4\cdot0.0625)=0.10/0.25=0.40\).

(b) Positive and below 1: the investor puts 40% in the stock and 60% in the bond. The positive sign reflects a positive risk premium; the moderate size reflects fairly high risk aversion (\(\gamma=4\)) and substantial volatility.

(c) The unconstrained optimum (40%) is already below 100%, so a no-leverage constraint does not bind here. It would bind only if \((\mu-r)/(\gamma\sigma^2)\gt 1\), e.g. for a low-\(\gamma\), high-premium, low-vol asset.

After the reveal, answer for yourself: How would a sudden doubling of estimated volatility change the allocation, and why does that make robust estimation of \(\sigma\) so important?

Lesson Summary

Investment is a real-world \(\Prob\) problem about expected returns and preferences - distinct from pricing under \(\mathbb{Q}\). Mean–variance analysis yields an efficient frontier and, with a risk-free asset, a tangency portfolio held by all investors (two-fund separation) with tangency weights \(\propto\Sigma^{-1}(\mu-r\mathbf1)\). Expected-utility theory encodes risk aversion via concave \(u\); for CRRA the Merton problem gives the constant optimal risky fraction \(w^*=(\mu-r)/(\gamma\sigma^2)\). All such results depend on noisy estimates and imply no guarantee of profit.

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: Give the tangency-portfolio weights and the Merton risky fraction.
A: Tangency: \(w\propto\Sigma^{-1}(\mu-r\mathbf1)\) normalized to sum 1. Merton (CRRA, single asset): \(w^*=(\mu-r)/(\gamma\sigma^2)\).

Q: Which measure governs pricing versus investment?
A: Pricing uses the risk-neutral \(\mathbb{Q}\) and the risk-free drift \(r\) (the real drift \(\mu\) cancels); investment uses the real-world \(\Prob\) and depends on \(\mu\) and preferences.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

Methods of Mathematical Finance (Karatzas & Shreve, 1998) foundational - Ch. 3-6 - Ch. 3–6: utility maximization, optimal portfolios, and the martingale/duality approach to the Merton problem.
Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 3,5 - Ch. 3,5: geometric Brownian motion dynamics and the real-world vs risk-neutral distinction underlying investment.

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