NISM Certifications · NISM-Series-X-A: Investment Adviser (Level 1)

Introduction to Modern Portfolio Theory: formula sheet

Full chapter guide

Key formulas

Expected return (with probabilities): E(R) = Σ pᵢ × Rᵢ; Probabilities must add up to 1 (100%). Multiply each return by its probability and sum.
Variance (with probabilities): σ² = Σ pᵢ × (Rᵢ − E(R))²; Use deviations from the expected return, not from zero.
Standard deviation: σ = √σ²; Same units as return. This is the usual measure of total risk of a single asset.
Historical average return: R̄ = Σ Rₜ ÷ n; Simple (arithmetic) average of n past period returns.
Sample variance of historical returns: s² = Σ (Rₜ − R̄)² ÷ (n − 1); Use n − 1 for a sample. If the question says to treat the data as the whole population, divide by n.
Single-period return: R = (P₁ − P₀ + D) ÷ P₀; P₀ is the starting price, P₁ the ending price, D the dividend or income received.
Correlation: ρ12 = Cov(1,2) ÷ (σ1 × σ2); Always between -1 and +1. Same sign as covariance.
Covariance from correlation: Cov(1,2) = ρ12 × σ1 × σ2; Use this when the question gives correlation and standard deviations.
Portfolio expected return: Rp = w1R1 + w2R2; Weighted average. Correlation plays no role.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2w1w2Cov(1,2); Equivalent to using ρσ1σ2 in place of Cov(1,2).
Two-asset portfolio standard deviation: σp = √(w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2); Take the square root of variance as the last step.
Perfect positive correlation case: σp = w1σ1 + w2σ2 (when ρ = +1); The only case where risk is the weighted average of standard deviations.
Total risk: Total risk = Systematic risk + Unsystematic risk; Systematic is market-wide and non-diversifiable. Unsystematic is specific and diversifiable.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Lower ρ12 gives lower portfolio variance, with weights and σ held the same.
Correlation range: -1 ≤ ρ ≤ +1; At +1 there is no diversification benefit. At -1 risk can be eliminated with suitable weights. Portfolio standard deviation is never above the weighted average of the two standard deviations.
Portfolio standard deviation at perfect correlation: σp = w1σ1 + w2σ2 (when ρ = +1); This is the weighted average. For ρ < +1, σp is lower.
Systematic risk measure: Beta (β) measures sensitivity to market movements; Market beta = 1. Beta above 1 means more volatile than the market.
Two-asset portfolio return: E(Rp) = w1 × E(R1) + w2 × E(R2); Weights add up to 1 (100%). Return is a simple weighted average.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2; Standard deviation is the square root of variance. Lower correlation ρ gives lower risk.
Minimum variance weight (two assets): w1 = (σ2² − ρ12σ1σ2) ÷ (σ1² + σ2² − 2ρ12σ1σ2); w2 = 1 − w1. Used for the lowest-risk mix, assuming short sales are not allowed and the weights stay between 0 and 1.
Optimal portfolio rule: Optimal point = tangency of the highest reachable indifference curve with the efficient frontier; Depends on the investor's risk aversion. Frontier alone does not give the optimal portfolio.
Efficiency rule: Efficient if no other portfolio has higher return at the same or lower risk; Portfolios below the minimum variance point are not efficient.
Portfolio expected return: E(Rc) = Rf + w × (E(Rp) − Rf); w is the weight in the risky portfolio. Equivalent to w × E(Rp) + (1 − w) × Rf.
Portfolio risk: σc = w × σp; The risk-free asset has zero standard deviation and zero covariance with the risky portfolio, so risk scales with w.
CAL equation: E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc; Slope is the Sharpe ratio of the chosen risky portfolio.
CML equation: E(Rc) = Rf + [(E(Rm) − Rf) ÷ σm] × σc; Same as CAL but the risky portfolio is the market (tangency) portfolio m.
Slope of CML: (E(Rm) − Rf) ÷ σm; Called the market price of risk per unit of standard deviation.
Weight for target risk: w = σtarget ÷ σp; Use to find the risky weight that gives a desired total risk.
CAPM required return: E(Ri) = Rf + βi × (Rm − Rf); Rf is the risk-free rate, Rm the expected market return. Use the same period for all rates, usually annual.
Market risk premium: MRP = Rm − Rf; This is the slope of the Security Market Line.
Beta: β = Cov(Ri, Rm) ÷ Var(Rm); Also equals correlation × (σi ÷ σm).
Portfolio beta: βp = Σ (wi × βi); Weighted average of individual betas. Weights are market value proportions and sum to 1.
Alpha (Jensen's): α = Actual or expected return − CAPM required return; Positive alpha means above the SML. Negative means below the SML.
SML versus CML: SML: risk = beta. CML: risk = standard deviation; SML prices any security. CML covers only efficient portfolios of the risk-free asset and the market portfolio.
MPT risk measure: Risk = standard deviation (σ) of portfolio returns; MPT treats upside and downside deviations alike. This is a criticism.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Needs estimated σ and correlation inputs. Errors in these feed straight into the result.
CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf]; Valid only under CAPM assumptions such as homogeneous expectations and a common risk-free rate.
Core assumption set: Rational + risk-averse + normal returns + same inputs + frictionless markets; A quick memory line for the assumptions.

Quick revision

Expected return is the probability-weighted average of possible returns.
Standard deviation is the square root of variance and measures total risk.
Correlation ranges from −1 to +1; covariance = correlation × σA × σB.
Portfolio risk is not the weighted average of risks unless correlation is +1.
Correlation of −1 gives the most risk reduction; it can allow zero risk with the right weights.
Diversification removes unsystematic risk but not systematic (market) risk.
Beta measures systematic risk; market beta is 1, and beta above 1 means more volatile than the market.
The efficient frontier shows portfolios with the highest return for each level of risk.
CAL slope is the Sharpe ratio: (portfolio return − Rf) ÷ portfolio standard deviation.
The CML uses the market portfolio and total risk; the SML uses beta and applies to individual securities.
CAPM: Expected return = Rf + β × (Rm − Rf); (Rm − Rf) is the market risk premium.
A security plotting above the SML is undervalued; below it is overvalued.

Common mistakes

Giving the variance when the question asks for standard deviation. Fix: Underline the word asked. Always finish with the square root if standard deviation is asked.
Measuring deviations from zero instead of from the expected return. Fix: Always compute E(R) first, then subtract it from every return.
Reporting the weighted average of standard deviations as portfolio risk. Fix: Use the full formula with correlation. The weighted average is right only when ρ = +1.
Forgetting the factor 2 in the cross term. Fix: Always write 2 × w1 × w2 × ρ × σ1 × σ2.
Saying diversification removes all risk. Fix: Diversification reduces unsystematic risk only. Systematic risk remains.
Classing a RBI rate hike or an inflation surge as unsystematic risk. Fix: Ask whether it affects the broad economy or market. If yes, it is systematic.
Calling the whole bullet-shaped curve the efficient frontier. Fix: Only the part from the minimum variance portfolio upward is efficient. The lower part is dominated.
Thinking the optimal portfolio is the same for every investor. Fix: The frontier is common to all. The optimal point differs by indifference curves, which reflect risk aversion.
Treating CAL and CML as the same thing. Fix: Remember that CML is a CAL that passes through the tangency (market) portfolio. A CAL can use any risky portfolio.
Using E(Rp) instead of the excess return (E(Rp) − Rf) in the slope. Fix: Always subtract Rf before dividing by σ.

Exam tips

Read whether the question wants expected return, variance or standard deviation. Options often include all three.
Use the 'probabilities add up to 1' check to catch typo-style trap data.
Keep deviations in whole % points to avoid decimals, and convert at the end.
Know the conceptual difference: historical is ex-post and actual; expected is ex-ante and probability-based. Theory questions test this directly.
Remember that on 2-mark caselet questions a wrong answer costs twice as much, so finish the calculation before answering rather than guessing.
Memorise the three special cases (ρ = +1, 0, -1). Many options can be eliminated or solved in seconds.
Check whether the question gives covariance or correlation. Using one in place of the other is a favourite trap.
Expect conceptual questions: which correlation gives the greatest risk reduction? The answer is -1.