Portfolio Management — Master Formula Sheet

1. Expected Return (Building Blocks)

$1+E(R)=(1+r_{rf})(1+E(\pi ))(1+E(RP))$

where:

• $r_{rf}$ = real risk-free rate
• $E(\pi )$ = expected inflation premium
• $E(RP)$ = expected risk premium

2. Real Return

$R_{\text{real}}=\frac{1+R_{\text{nominal}}}{1+\pi }-1$

3. Utility Function

$U=E(R)-\frac{1}{2}A{\sigma }^2$

where $A$ = risk aversion coefficient ($A>0$ for risk-averse investors).

4. Capital Allocation Line (CAL)

$E(R_p)=R_f+\left[\frac{E(R_i)-R_f}{{\sigma }_i}\right]{\sigma }_p$

where the slope $\frac{E(R_i)-R_f}{{\sigma }_i}$ is the Sharpe ratio of the risky portfolio.

5. Capital Market Line (CML)

$E(R_p)=R_f+\left[\frac{E(R_m)-R_f}{{\sigma }_m}\right]{\sigma }_p$

Applies only to efficient portfolios (combinations of the risk-free asset and the market portfolio).

6. Portfolio Variance (Two Assets)

${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\rho }_{12}{\sigma }_1{\sigma }_2$

Equivalently: ${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\text{Cov}}_{12}$

7. Covariance

${\text{Cov}}_{12}=\frac{{\sum }_{t=1}^n(R_{1t}-{\bar{R}}_1)(R_{2t}-{\bar{R}}_2)}{n-1}$

8. Correlation

${\rho }_{12}=\frac{{\text{Cov}}_{12}}{{\sigma }_1{\sigma }_2}$

where $-1\le {\rho }_{12}\le +1$.

9. Beta

${\beta }_i=\frac{\text{Cov}(R_i,R_m)}{{\sigma }_m^2}={\rho }_{im}\times \frac{{\sigma }_i}{{\sigma }_m}$

10. CAPM

$E(R_i)=R_f+{\beta }_i[E(R_m)-R_f]$

where $E(R_m)-R_f$ = market risk premium.

11. Market Model

$R_i={\alpha }_i+{\beta }_i{R}_m+e_i$

where:

• ${\alpha }_i$ = abnormal return (intercept)
• $e_i$ = firm-specific error term

12. Portfolio Beta

${\beta }_p={\sum }_{i=1}^n{w}_i\times {\beta }_i$

13. Sharpe Ratio

$\text{Sharpe}=\frac{R_p-R_f}{{\sigma }_p}$

Uses total risk. Appropriate for evaluating the investor’s entire portfolio.

14. Treynor Ratio

$\text{Treynor}=\frac{R_p-R_f}{{\beta }_p}$

Uses systematic risk. Appropriate for evaluating a sub-portfolio.

15. M-Squared ($M^2$)

$M^2=(R_p-R_f)\frac{{\sigma }_m}{{\sigma }_p}-(R_m-R_f)$

Equivalently:

$M^2=({\text{Sharpe}}_p-{\text{Sharpe}}_m)\times {\sigma }_m$

Returns a percentage — the excess return of a leveraged/de-leveraged portfolio matched to market risk.

16. Jensen’s Alpha

${\alpha }_p=R_p-[R_f+{\beta }_p(R_m-R_f)]$

Positive $\alpha$ indicates the manager outperformed the CAPM prediction.

17. Diversification Ratio

$\text{Diversification ratio}=\frac{\sigma (\text{equally-weighted portfolio})}{\sigma (\text{randomly selected security})}$

A ratio closer to zero indicates greater diversification benefit.

See Also

• Portfolio Risk and Return
• CAPM and SML
• Performance Measurement