Northspread
Formula sheet/NISM Series X-A: Investment Adviser (Level 1)/Portfolio Construction, Performance Monitoring and Evaluation
[ PORTFOLIO CONSTRUCTION, PERFORMANCE MONITORING AND EVALUATION ]

Modern Portfolio Theory (MPT): Expected Return, Variance & Covariance

Asset Expected Return:E(Ri)=∑[pk×Rik]∣Portfolio Return:E(Rp)=∑[wi×E(Ri)]\text{Asset Expected Return}: E(R_{i}) = \sum [ p_{k} \times R_{\text{ik}} ]\quad\Big|\quad \text{Portfolio Return}: E(R_{p}) = \sum [ w_{i} \times E(R_{i}) ]
2−Asset Portfolio Variance:σp2=w12σ12+w22σ22+2w1w2Cov(1,2)=w12σ12+w22σ22+2w1w2ρ12σ1σ22-\text{Asset Portfolio Variance}: \sigma_{p}^{2} = w_{1}^{2} \sigma_{1}^{2} + w_{2}^{2} \sigma_{2}^{2} + 2 w_{1} w_{2} Cov(1,2) = w_{1}^{2} \sigma_{1}^{2} + w_{2}^{2} \sigma_{2}^{2} + 2 w_{1} w_{2} \rho_{12} \sigma_{1} \sigma_{2}
Portfolio Standard Deviation:σp=σp2\text{Portfolio Standard Deviation}: \sigma_{p} = \sqrt{\sigma_{p}^{2}}
Number of Unique Covariance Terms for n assets:(n2−n)/2\text{Number of Unique Covariance Terms for n assets}: ( n^{2} - n ) / 2
Variables

ρ\rho = -1: Perfect negative correlation eliminates all portfolio risk; ρ\rho = +1: Linear risk, zero diversification benefit. pkp_{k}: Probability of scenario k; RikR_{\text{ik}}: Return of asset i in scenario k; wiw_{i}: Portfolio weight of asset i; σ1\sigma_{1}, σ2\sigma_{2}: Standard deviations of assets 1 and 2; ρ12\rho_{12}: Correlation between assets 1 and 2.

Exam note

MPT demonstrates that portfolio risk depends far more on asset covariances than on individual asset variances.

Worked examples
2-Asset Portfolio Variance and Standard Deviation (MPT)

A portfolio manager allocates 60% of funds to Stock A (Expected Return 18%, Standard Deviation 20%) and 40% to Stock B (Expected Return 12%, Standard Deviation 10%). The correlation coefficient (ρ_AB) between Stock A and B is 0.30. Calculate the portfolio expected return, portfolio variance, and portfolio standard deviation.

Advanced Risk-Adjusted Ratios (Sortino & Modigliani-Modigliani M^2)

A fund delivers an annualized return of 18.0% with total Standard Deviation of 16.0%, downside semi-standard deviation (loss volatility) of 8.0%, and Beta of 1.20. The market index returns 14.0% with market standard deviation of 12.0%, and risk-free rate is 6.0%. Calculate Sharpe Ratio, Sortino Ratio, and the Modigliani-Modigliani (M^2) measure.

Cash-Drag & Foreign Exchange Currency Adjustment

An Indian investor allocates $100,000 to a US Tech Fund. Over 1 year, the US fund gains 15.0% in USD terms. Concurrently, the USD appreciates against INR from Rs 82.00 to Rs 86.10 (+5.0% USD appreciation). If the investor also kept 10% in an INR cash balance earning 6.0% p.a., calculate the Total INR Return in the international fund and the overall Cash-Drag Adjusted Portfolio Return.