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Quantitative MethodsModule 5 of 11

Portfolio Mathematics

5

Concepts

4

Formulas

1

Decisions

4

Quiz Questions

Key Concepts

5 concepts covered in this module.

Portfolio Expected Return

E(Rp) = Σ wi E(Ri). Weighted average of individual expected returns.

Portfolio Variance (2 assets)

σ²p = w²1σ²1 + w²2σ²2 + 2w1w2Cov(1,2). Key: correlation drives diversification benefit.

Diversification Effect

Unless ρ = +1, portfolio SD < weighted average of individual SDs. Lower correlation = more diversification benefit.

Normal Distribution Properties

68% within ±1σ, 95% within ±1.96σ, 99% within ±2.58σ. Skewness = 0, kurtosis = 3.

Safety-First Ratio (SFRatio)

SFRatio = [E(Rp) - Rthreshold] / σp. Choose portfolio with highest SFRatio (fewest SDs above disaster level).

Formulas

4 essential formulas for this module.

Portfolio Variance (2 assets)

σ²p = w²Aσ²A + w²Bσ²B + 2wAwBρABσAσB

Where: w = weights, ρ = correlation, σ = standard deviation

Covariance from Correlation

Cov(A,B) = ρAB × σA × σB

Where: ρ = correlation coefficient

Safety-First Ratio

SFRatio = [E(Rp) - RL] / σp

Where: RL = minimum acceptable (threshold) return

Z-score

z = (X - μ) / σ

Where: Converts value to standard normal units

Decision Frameworks

1 decision frameworks to guide your analysis.

Effect of correlation on portfolio risk?

  • ρ = +1: no diversification, portfolio SD = weighted average
  • ρ = 0: significant diversification
  • ρ = -1: maximum diversification, can create zero-variance portfolio

Mind Map

Visual overview of how concepts connect in this module.

Portfolio Mathematics
Portfolio Return
Weighted average of asset returns
E(Rp) = Σ wi×E(Ri)
Linear in weights
Portfolio Variance
2 assets: need ρ or Cov
n assets: n variances + n(n-1)/2 covariances
Correlation drives diversification
Diversification
ρ < 1: portfolio SD < weighted avg SD
ρ = -1: can eliminate all risk
Only reduces unsystematic risk
Normal Distribution
±1σ = 68%
±1.96σ = 95%
±2.58σ = 99%
Symmetric, bell-shaped
Safety-First
SFRatio = (E(R)-RL)/σ
Choose highest SFRatio
Similar to Sharpe ratio
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Portfolio Expected Return

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Answer
E(Rp) = Σ wi E(Ri). Weighted average of individual expected returns.
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