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Measuring Security & Portfolio Return and Risk (SIM) 📐📉

In the Single Index Model (SIM), we use a different set of formulas compared to the Markowitz model to calculate the vital statistics of our investments.

1. Individual Security Measurement

Under SIM, a security's performance depends on its unique characteristics (Alpha) and its relationship with the market (Beta).

  • Expected Return (Ei): 
    Ei = Alpha_i + (Beta_i * Rm)
  • Total Risk (Variance): 
    Total Variance = (Beta_i^2 * Var_m) + Var(Error_i)
    • Systematic Risk: (Beta_i^2 * Var_m)
    • Unsystematic Risk: Var(Error_i)

2. Portfolio Measurement

The beauty of SIM is that the portfolio's stats are just weighted averages of the individual stock's Alpha and Beta.

  • Portfolio Alpha (A_p): Σ (wi * Alpha_i)
  • Portfolio Beta (B_p): Σ (wi * Beta_i)
  • Portfolio Expected Return (E_p): 
    E_p = A_p + (B_p * Rm)

3. Portfolio Risk (Variance)

This is where Sharpe's model shines. Even for a complex portfolio, the risk calculation remains simple:

Var_p = (B_p^2 * Var_m) + Σ (wi^2 * Var_ei)
  1. Systematic Component: B_p^2 * Var_m (The portion of risk you can't diversify away).
  2. Unsystematic Component: Σ (wi^2 * Var_ei) (The portion that drops as you add more stocks).
Important

As the number of stocks in the portfolio increases (n -> infinity), the Unsystematic Component approach zero, leaving only the Market Risk. This is the mathematical proof of why diversification works!

Summary

  • SIM breaks risk and return into Market-related and Company-specific parts.
  • Portfolio Beta is the weighted average of individual betas.
  • The Variance of a portfolio is greatly simplified compared to the Markowitz formula.
  • Diversification effectively kills the "Sum of weighted individual variances" part of the risk.

Quiz Time! 🎯

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