Portfolio Management formula sheet.

Where $E(r)$ = expected return, $\sigma^2$ = variance of return, $A$ = risk-aversion coefficient (positive if risk averse, zero if risk neutral, negative if risk seeking).

Utility is a risk-adjusted return: the investor picks the choice with the highest $U$. A higher $A$ penalises variance more and gives steeper indifference curves, and a risk-free asset has $U = R_f$ for every investor.

Exam tip Work in decimals ($\sigma = 20\%$ means $\sigma^2 = 0.04$); if you keep percentages, the multiplier becomes $0.005A$, not $0.5A$.

Lesson: Risk aversion and utility →All Portfolio Risk and Return: Part I formulas

Where $R_f$ = risk-free rate, $E(R_i)$, $\sigma_i$ = expected return and standard deviation of the risky portfolio, $E(R_p)$, $\sigma_p$ = those of the combined portfolio.

Every mix of the risk-free asset and one risky portfolio lies on this straight line, whose slope is the risky portfolio's Sharpe ratio. The best CAL is the steepest one, tangent to the efficient frontier; each investor then picks the point where an indifference curve touches it.

Exam tip The choice of risky portfolio does not depend on risk aversion; only the split between it and the risk-free asset does. Points beyond the risky portfolio mean borrowing.

Lesson: Indifference curves and the capital allocation line →All Portfolio Risk and Return: Part I formulas

Where $w_i$ = weight of asset $i$ in the portfolio, $R_i$ = return on asset $i$, $N$ = number of assets.

Portfolio return is always the weighted average of the asset returns. Portfolio risk is not, which is the whole point of diversification: the return is kept while part of the risk disappears.

Exam tip Weights can be negative (a short position) or above 1 (leverage), but they must still add up to 1.

Lesson: Measuring asset and portfolio risk →All Portfolio Risk and Return: Part I formulas

Where $R_{1,t}$, $R_{2,t}$ = returns in period $t$, $\bar R_1$, $\bar R_2$ = mean returns, $n$ = number of observations, $\sigma_1$, $\sigma_2$ = standard deviations, $\rho_{12}$ = correlation.

Covariance shows whether two returns move together, but its size is hard to read. Dividing by both standard deviations gives the correlation, which always lies between $-1$ and $+1$ and is the input the exam uses most.

Exam tip Use $n - 1$ for a sample, and keep units consistent: a covariance from percentage returns is in percent squared, so the standard deviations must also be in percent.

Lesson: Measuring asset and portfolio risk →All Portfolio Risk and Return: Part I formulas

Where $w_1$, $w_2$ = weights, $\sigma_1$, $\sigma_2$ = standard deviations of the two assets, $\rho_{12}$ = their correlation.

The most tested calculation in the topic. Whenever $\rho_{12} < 1$, portfolio risk is below the weighted average of the two standard deviations, and the lower the correlation, the larger the diversification benefit.

Exam tip Do not forget the 2 in the last term or the square root at the end; if a covariance is given, it replaces $\rho_{12}\sigma_1\sigma_2$ directly.

Lesson: Measuring asset and portfolio risk →All Portfolio Risk and Return: Part I formulas

Where $\bar\sigma^2$ = average variance of the assets, $\overline{\text{Cov}}$ = average covariance between pairs of assets, $N$ = number of assets.

As $N$ grows, the first term shrinks towards zero and portfolio variance approaches the average covariance. Diversification removes the risk specific to each asset but cannot remove the risk the assets share.

Exam tip Adding assets can never push variance below the average covariance; only with zero average covariance would risk go to zero.

Lesson: Correlation and the power of diversification →All Portfolio Risk and Return: Part I formulas

Where $E(R_{new})$, $\sigma_{new}$ = expected return and risk of the candidate asset, $E(R_p)$, $\sigma_p$ = those of the current portfolio, $R_f$ = risk-free rate, $\rho_{new,p}$ = their correlation.

Adding the asset improves the portfolio's Sharpe ratio if its own Sharpe ratio beats the portfolio's Sharpe ratio times the correlation. A low correlation makes the hurdle easy to clear.

Exam tip An asset with a lower Sharpe ratio than the portfolio can still be worth adding if its correlation with the portfolio is low enough.

Lesson: Correlation and the power of diversification →All Portfolio Risk and Return: Part I formulas

Where $w_1$, $w_2$ = weights of assets 1 and 2, $\sigma_1$, $\sigma_2$ = their standard deviations; valid only when $\rho_{12} = -1$.

With perfect negative correlation the two assets can be combined into a portfolio with zero standard deviation. These weights make $w_1\sigma_1 = w_2\sigma_2$, so the two risks cancel exactly.

Exam tip The numerator is the other asset's standard deviation: the riskier asset gets the smaller weight.

Lesson: Integrated cases: a foreign bond with currency, and pairs from a scenario table →All Portfolio Risk and Return: Part I formulas

Where $w_1$ = weight in the risk-free asset, $R_f$ = risk-free rate, $E(R_m)$, $\sigma_m$ = expected return and standard deviation of the market portfolio.

Because the risk-free asset has zero variance and zero covariance with the market, portfolio risk is simply the market weight times $\sigma_m$. Both return and risk are linear in the weight, which is why the combinations form a straight line.

Exam tip Borrowing gives a negative $w_1$: borrowing 25% of wealth means $w_1 = -0.25$, a market weight of 1.25 and $\sigma_p = 1.25\sigma_m$.

Lesson: The risk-free asset, the market portfolio and the CML →All Portfolio Risk and Return: Part II formulas

Where $R_f$ = risk-free rate, $E(R_m)$, $\sigma_m$ = expected return and standard deviation of the market portfolio, $\sigma_p$ = standard deviation of the portfolio.

The CML is the CAL whose risky portfolio is the market portfolio, which is optimal when investors share the same expectations. Its slope is the market price of risk, and only efficient, fully diversified portfolios plot on it.

Exam tip The CML uses total risk $\sigma$ and applies to efficient portfolios; the SML uses $\beta$ and prices any security. A higher borrowing rate makes the CML kink and flatten past the market.

Lesson: The risk-free asset, the market portfolio and the CML →All Portfolio Risk and Return: Part II formulas

Where $\sigma_i^2$ = total variance of asset $i$, $\beta_i$ = its beta, $\sigma_m^2$ = market variance, $\sigma_{e_i}^2$ = nonsystematic (error) variance.

Total risk splits into a systematic part driven by the market and a nonsystematic part specific to the asset. Diversification removes the nonsystematic part, so only systematic risk earns a premium.

Exam tip Variances add, standard deviations do not: with $\sigma_i = 30\%$, $\beta = 1.2$, $\sigma_m = 20\%$, nonsystematic variance is $0.09 - 0.0576 = 0.0324$, so $\sigma_e = 18\%$.

Lesson: Return-generating models, the market model and beta →All Portfolio Risk and Return: Part II formulas

Where $\text{Cov}(R_i, R_m)$ = covariance of the asset with the market, $\sigma_m^2$ = market variance, $\rho_{i,m}$ = correlation with the market, $\sigma_i$, $\sigma_m$ = standard deviations.

Beta measures systematic risk: how much the asset's return moves for a given move in the market. The market has a beta of 1 and the risk-free asset a beta of 0; beta is also the slope of the market model regression.

Exam tip Divide the covariance by the market variance, not the market standard deviation; with the correlation form, the last expression is quicker.

Lesson: Return-generating models, the market model and beta →All Portfolio Risk and Return: Part II formulas

Where $E(R_i)$ = expected (required) return on asset $i$, $R_f$ = risk-free rate, $\beta_i$ = beta, $E(R_m) - R_f$ = market risk premium.

The CAPM gives the return an asset should earn for its systematic risk; plotted against beta it is the SML. A security whose forecast return lies above the SML is undervalued, and one below it is overvalued.

Exam tip Multiply beta by the market risk premium, not by $E(R_m)$; if the question gives the market risk premium directly, do not subtract $R_f$ again.

Lesson: The CAPM: assumptions, the SML and its limits →All Portfolio Risk and Return: Part II formulas

Where $w_i$ = weight of asset $i$, $\beta_i$ = its beta, $\beta_p$ = portfolio beta, $R_f$ = risk-free rate, $E(R_m)$ = expected market return.

Unlike standard deviation, beta is a simple weighted average, so a portfolio's systematic risk and CAPM return follow directly from its holdings.

Exam tip Include any cash or risk-free holding with a beta of 0; leaving it out overstates the portfolio beta.

Lesson: The CAPM: assumptions, the SML and its limits →All Portfolio Risk and Return: Part II formulas

Where $E(R_p)$ or $\bar R_p$ = expected or average portfolio return, $R_f$ = risk-free rate, $\sigma_p$ = standard deviation of portfolio return.

Excess return per unit of total risk, and the slope of the CAL. It suits a portfolio that is the investor's whole wealth or is not fully diversified, because it charges for all volatility.

Exam tip Lending or borrowing at $R_f$ does not change the Sharpe ratio, which is why every point on one CAL has the same Sharpe ratio.

Lesson: Sharpe ratio, Treynor ratio, M² and Jensen's alpha →All Portfolio Risk and Return: Part II formulas

Where $E(R_p)$ or $\bar R_p$ = expected or average portfolio return, $R_f$ = risk-free rate, $\beta_p$ = portfolio beta.

Excess return per unit of systematic risk. It is appropriate for a well-diversified portfolio, or one that is a small part of a larger diversified portfolio, where nonsystematic risk no longer matters.

Exam tip Treynor ignores nonsystematic risk, so it can rank a poorly diversified portfolio above where the Sharpe ratio would place it.

Lesson: Sharpe ratio, Treynor ratio, M² and Jensen's alpha →All Portfolio Risk and Return: Part II formulas

Where $R_p$ = portfolio return, $R_f$ = risk-free rate, $\sigma_p$, $\sigma_m$ = standard deviations of the portfolio and the market, $SR$ = portfolio Sharpe ratio, $R_m$ = market return.

The return the portfolio would have earned if leveraged or de-leveraged to the market's total risk. Being a percentage return, it compares directly with the market: a positive $M^2$ alpha means the portfolio beat the market after adjusting for total risk.

Exam tip $M^2$ always ranks portfolios in the same order as the Sharpe ratio; it just restates it as a return.

Lesson: Sharpe ratio, Treynor ratio, M² and Jensen's alpha →All Portfolio Risk and Return: Part II formulas

Where $R_p$ = realised portfolio return, $R_f$ = risk-free rate, $\beta_p$ = portfolio beta, $R_m$ = realised market return.

Actual return minus the return the CAPM predicts for the portfolio's beta. A positive alpha means the portfolio plots above the SML and outperformed on a systematic-risk basis; it is also the intercept of the security characteristic line.

Exam tip Use the realised market return for the same period, not an expected one, and the portfolio's beta, not the market's beta of 1.

Lesson: Sharpe ratio, Treynor ratio, M² and Jensen's alpha →All Portfolio Risk and Return: Part II formulas

Where $\sigma$ = standard deviation of return of the equally weighted portfolio and of one randomly chosen security from the same set.

Both choices have the same expected return, so the ratio isolates how much risk the portfolio approach removes. The lower the ratio, the greater the benefit of diversification.

Exam tip A ratio of 0.60 means the portfolio carries 60% of a single security's risk; a ratio near 1 means almost no benefit, as happens when correlations rise in a crisis.

Lesson: The portfolio approach and diversification →All Portfolio Management: An Overview formulas

Where $FV_{\text{nominal}}$ = goal in future (nominal) money, $PV$ = assets invested today, $N$ = years to the goal; assumes no further contributions.

The annual return the portfolio must earn to reach a goal, which then feeds the return objective in the IPS. First inflate a goal stated in today's money: $FV_{\text{nominal}} = FV_{\text{today's money}} \times (1 + \pi)^{N}$.

Exam tip Using the goal in today's money instead of nominal money gives the real required return, not the nominal one.

Lesson: Return objectives and the required return →All Basics of Portfolio Planning and Construction formulas