Lesson 1 · Portfolio ManagementCFA L1

Portfolio Risk and Return

Modern portfolio theory takes a deliberately macroscopic view: an investment is summarised by just two numbers — its expected return (mean) and its risk (standard deviation). This lesson builds the machinery from there: a utility function that trades return against risk, the capital allocation line mixing a risky and a riskfree asset, the optimal split, then two risky assets, the efficient frontier, the optimal risky portfolio, and the mutual-fund theorem.

On this page

Return, risk & utility Capital allocation line Optimal risky/riskfree split Two risky assets The efficient frontier The optimal risky portfolio The mutual-fund theorem Practical issues

The entire prospect of an investment is summarised by its return — the expected value (mean) $\mu=E(X)$ — and its risk — the standard deviation $\sigma=\sqrt{\operatorname{var}(X)}$, a measure of variability. Both are estimated from history (Excel's average, stdev), and plotting investments on a $(\mu,\sigma)$ plane lets us compare them directly.

Learning outcomes

Quantify return and risk and rank investments with a mean-variance utility function.
Derive the capital allocation line and interpret its slope as the Sharpe ratio.
Solve for the optimal allocation between a risky and a riskfree asset.
Compute the mean and variance of a two-asset portfolio and the minimum-variance and efficient sets.
Find the optimal risky portfolio and apply the two-step allocation and mutual-fund theorem.

Return, risk and the utility function

An investor loves return but dislikes risk. On the $(\mu,\sigma)$ plane, an investment that offers higher return and lower risk dominates — but trade-offs (more return for more risk) need a rule. We score each investment with a mean-variance utility function:

Mean-variance utility

$$ U(\mu,\sigma)=\mu-\tfrac{1}{2}A\,\sigma^2 $$

$U$ rises with return and falls with risk; $A$ is the investor's risk-aversion parameter (larger $A$ penalises risk more heavily). Investments giving the same $U$ trace an indifference curve — upward-sloping (more risk must be paid for with more return), with higher curves carrying higher utility.

Worked example

Aggressive vs conservative fund

An investor has $A=4.5$. Fund 1: $\mu=20\%,\ \sigma=18\%$. Fund 2: $\mu=15\%,\ \sigma=10\%$. Which is preferred?

Show solution

$U_1=0.20-\tfrac{1}{2}(4.5)(0.18)^2=0.1271$; $U_2=0.15-\tfrac{1}{2}(4.5)(0.10)^2=0.1275$. The conservative Fund 2 wins — for this risk-averse investor, its much lower risk outweighs the 5-point return shortfall.

Fund 2 (U = 0.1275 > 0.1271)

The capital allocation line

Mix a riskfree asset (return $r_f$) with a risky asset (mean $\mu_p$, s.d. $\sigma_p$), putting fraction $y$ in the risky asset. Using $E(a+bX)=a+bE(X)$ and $\operatorname{var}(a+bX)=b^2\operatorname{var}(X)$:

Complete portfolio and the CAL

$$ \mu_c=r_f+(\mu_p-r_f)\,y, \qquad \sigma_c=y\,\sigma_p $$

$$ \mu_c=r_f+\underbrace{\frac{\mu_p-r_f}{\sigma_p}}_{\text{Sharpe ratio}}\,\sigma_c $$

All attainable risk-return profiles lie on this straight capital allocation line (CAL), whose slope is the Sharpe ratio. $y=100\%$ is full investment in the risky asset; $y>100\%$ means borrowing at $r_f$ to buy more (margin); $y<0$ means short-selling the risky asset (which flips the slope's sign — and is never optimal when $\mu_p>r_f$).

The optimal risky/riskfree split

Maximising $U=r_f+(\mu_p-r_f)y-\tfrac{1}{2}A\sigma_p^2 y^2$ over $y$ (a quadratic) gives:

Optimal weight on the risky asset

$$ y^{*}=\frac{\mu_p-r_f}{A\,\sigma_p^2} $$

Graphically, the optimum is where the CAL is tangent to the highest reachable indifference curve. $y^{*}$ falls as risk aversion $A$ rises — a more cautious investor holds less of the risky asset.

Worked example

Optimal investment in an S&P 500 ETF

$r_f=1\%$, ETF $\mu_p=9\%$, $\sigma_p=15\%$. (1) Risk/return of a 50-50 split? (2) Optimal $y$ if $A=4$? (3) Another investor picks $y=80\%$ — infer his $A$; who is more aggressive?

Show solution

(1) $\mu_c=0.5(1\%)+0.5(9\%)=5\%$; $\sigma_c=0.5(15\%)=7.5\%$. (2) $y^{*}=\frac{0.09-0.01}{4(0.15)^2}=0.889$ → 88.9% ETF, 11.1% riskfree. (3) $0.8=\frac{0.08}{A(0.15)^2}\Rightarrow A=4.44$. Since $4.44>4$, the second investor is more risk averse — the first is more aggressive.

(1) 5%, 7.5%; (2) y*=0.889; (3) A=4.44, investor 1 more aggressive

Combining two risky assets

Now two risky assets A and B with weight $w_A$ in A. Covariance $\sigma_{AB}=\operatorname{Cov}(r_A,r_B)$ and correlation $\rho=\sigma_{AB}/(\sigma_A\sigma_B)\in[-1,1]$ capture co-movement. Using $\operatorname{var}(aX+bY)=a^2\operatorname{var}X+2ab\operatorname{Cov}(X,Y)+b^2\operatorname{var}Y$:

Two-asset mean and variance

$$ \mu_p=w_A\mu_A+(1-w_A)\mu_B $$

$$ \sigma_p^2=w_A^2\sigma_A^2+2w_A(1-w_A)\sigma_{AB}+(1-w_A)^2\sigma_B^2 $$

The mean is linear in $w_A$; the variance is quadratic. Two telling special cases: at $\rho=+1$, $\sigma_p=w_A\sigma_A+(1-w_A)\sigma_B$ (just a weighted average — no diversification); at $\rho=-1$, $\sigma_p=|w_A\sigma_A-(1-w_A)\sigma_B|$, and setting $w_A=\sigma_B/(\sigma_A+\sigma_B)$ gives $\sigma_p=0$ — risk is perfectly removed. Anything below $\rho=+1$ provides a diversification benefit.

Minimum-variance portfolio

$$ w_A^{\min}=\frac{\sigma_B^2-\sigma_{AB}}{\sigma_A^2-2\sigma_{AB}+\sigma_B^2} $$

The opportunity set and efficient frontier

As $w_A$ varies, the attainable $(\mu_p,\sigma_p)$ combinations trace a curve — the portfolio opportunity set. A portfolio is efficient if it offers the highest return for its level of risk, so only the upper half of the curve — the efficient frontier — is of interest. (With many risky assets the opportunity set becomes a solid region whose upper boundary is the frontier.)

The optimal risky portfolio

Add a riskfree asset. Each risky portfolio spawns a CAL; the best is the steepest — the one tangent to the efficient frontier, maximising the Sharpe ratio. With $R_{Af}=\mu_A-r_f$ and $R_{Bf}=\mu_B-r_f$:

Weight of A in the optimal risky portfolio

$$ w_A^{*}=\frac{R_{Af}\sigma_B^2-R_{Bf}\sigma_{AB}}{R_{Af}\sigma_B^2+R_{Bf}\sigma_A^2-(R_{Af}+R_{Bf})\sigma_{AB}} $$

Two-step allocation → the complete portfolio

Step 1: find the optimal risky portfolio (the $w_A^{*}$ above) and its $\mu_p,\sigma_p$. Step 2: using the investor's $A$, split between that risky portfolio and the riskfree asset with $y^{*}=(\mu_p-r_f)/(A\sigma_p^2)$. Final weights: riskfree $1-y^{*}$, Asset A $y^{*}w_A^{*}$, Asset B $y^{*}(1-w_A^{*})$. This is the optimal complete portfolio.

Worked example

Pension-fund allocation

Riskfree 1%; bond fund $\mu=6\%,\sigma=8\%$; equity fund $\mu=10\%,\sigma=20\%$; $\rho=0.4$. Find the optimal risky portfolio, then the complete allocation for $A=8$.

Show solution

$R_{Af}=5\%,\ R_{Bf}=9\%,\ \sigma_{AB}=0.4(0.08)(0.20)=0.0064$. $w_A^{*}=\dfrac{(0.05)(0.2)^2-(0.09)(0.0064)}{(0.05)(0.2)^2+(0.09)(0.08)^2-(0.14)(0.0064)}=0.8476$. Then $\mu_p=0.8476(6\%)+0.1524(10\%)=6.6\%$, $\sigma_p=8.47\%$. With $A=8$: $y^{*}=\frac{0.066-0.01}{8(0.0847)^2}=0.9766$. Final: riskfree 2.34%, bond $0.9766\times0.8476=82.78\%$, equity $0.9766\times0.1524=14.88\%$.

Risky portfolio: 84.8% bond / 15.2% equity (μ 6.6%, σ 8.47%); complete: 2.3% rf, 82.8% bond, 14.9% equity

The mutual-fund theorem

One fund for everyone

The optimal risky portfolio does not depend on the investor's risk aversion — only the split between it and the riskfree asset does. So a single risky portfolio (the maximum-Sharpe "mutual fund") serves all investors; they merely choose different points along the same best CAL by mixing that fund with the riskfree asset. This separation of the risky-portfolio choice from the risk-preference choice is the mutual-fund theorem.

Practical issues

The theory is only as good as its inputs. Optimising the Sharpe ratio over many assets is computationally demanding (Excel's Solver, or more) and, more importantly, requires estimates of every mean, variance and pairwise correlation — "junk in, junk out." Volatilities and correlations are time-varying and tend to spike together in market distress (just when diversification is most needed), so advanced statistical models are used to capture these dynamics.

Key takeaway

Summarise investments by mean and standard deviation and rank them with $U=\mu-\tfrac{1}{2}A\sigma^2$. Mixing a riskfree and a risky asset traces the CAL, whose slope is the Sharpe ratio, with optimal risky weight $y^{*}=(\mu_p-r_f)/(A\sigma_p^2)$. Two risky assets diversify whenever $\rho<1$, generating an efficient frontier; adding a riskfree asset singles out the maximum-Sharpe optimal risky portfolio, after which a simple risk-preference split completes the allocation. The mutual-fund theorem means everyone holds the same risky fund — but it all rests on input estimates that are noisy and unstable.

Saved in your browser — your progress bars update automatically.