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

Advanced Risk-Adjusted Ratios: Sortino, Information & M^2

Sharpe Ratio:(Rp−Rf)/σp∣Treynor Ratio:(Rp−Rf)/βp∣Jensen’s Alpha:α=Rp−[Rf+βp(Rm−Rf)]\text{Sharpe Ratio}: ( R_{p} - R_{f} ) / \sigma_{p}\quad\Big|\quad \text{Treynor Ratio}: ( R_{p} - R_{f} ) / \beta_{p}\quad\Big|\quad \text{Jensen's Alpha}: \alpha = R_{p} - [ R_{f} + \beta_{p} (R_{m} - R_{f}) ]
Sortino Ratio:(Rp−Rf)/Downside Semi-Standard Deviation(σdown)\text{Sortino Ratio}: ( R_{p} - R_{f} ) / \text{Downside Semi-Standard Deviation} (\sigma_{\text{down}})
Information Ratio(IR):(Rp−Rbenchmark)/Tracking Error\text{Information Ratio} (IR): ( R_{p} - R_{\text{benchmark}} ) / \text{Tracking Error}
Modigliani-Modigliani Measure(M2):Rp∗=(σmarket/σp)×Rp+[1−(σmarket/σp)]×Rf  ⇒  M2=Rp∗−Rmarket\text{Modigliani-Modigliani Measure} (M^{2}): R_{p}^{*} = ( \sigma_{\text{market}} / \sigma_{p} ) \times R_{p} + [ 1 - ( \sigma_{\text{market}} / \sigma_{p} ) ] \times R_{f} \;\Rightarrow\; M^{2} = R_{p}^{*} - R_{\text{market}}
Cash-Drag Adjusted Return:(Weightinvested×Returnassets)+(Weightcash×Returncash)\text{Cash-Drag Adjusted Return}: ( Weight_{\text{invested}} \times Return_{\text{assets}} ) + ( Weight_{\text{cash}} \times Return_{\text{cash}} )
Currency-Adjusted Foreign Return:RINR=(1+Rforeign)×(1+%ΔFX)−1≈Rforeign+%ΔFX\text{Currency-Adjusted Foreign Return}: R_{\text{INR}} = ( 1 + R_{\text{foreign}} ) \times ( 1 + \% \Delta FX ) - 1 \approx R_{\text{foreign}} + \% \Delta FX
Variables

Sortino penalizes only downside volatility (losses), making it superior to Sharpe for asymmetric return distributions. RpR_{p}: Portfolio return; RfR_{f}: Risk-free rate; σp\sigma_{p}: Portfolio standard deviation; βp\beta_{p}: Portfolio beta; RbenchmarkR_{\text{benchmark}}: Benchmark return; σmarket\sigma_{\text{market}}: Market/benchmark standard deviation; σdown\sigma_{\text{down}}: Downside semi-standard deviation; Rp∗R_{p}^{*}: M²-adjusted portfolio return; % Δ\Delta FX: Percentage change in exchange rate.

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.

Time-Weighted Rate of Return (TWRR) vs. MWRR (IRR)

An investor begins with Rs 5,000 in Year 0. In Year 1, the portfolio drops by -5.0% (Value = Rs 4,750). At the start of Year 2, she adds Rs 10,000 (Total = Rs 14,750), and portfolio earns -15.2% (Value = Rs 12,508). In Year 3, she adds Rs 15,000 (Total = Rs 27,508) and earns +8.1% (Value = Rs 29,736). In Year 4, she adds Rs 20,000 and earns +30.75% (Value = Rs 65,030). In Year 5, she adds Rs 25,000 and earns +17.65% (Ending Value = Rs 1,05,920.31). Calculate the 5-year TWRR and compare with MWRR.

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.