Sharpe Single Index Model 📊🧬

In Chapter 30, we saw that the Markowitz Model requires thousands of calculations for even a small portfolio. In 1963, William Sharpe proposed a solution: The Single Index Model (SIM).

Instead of measuring how every stock moves with every other stock, Sharpe suggested measuring how every stock moves with a Single Market Index (like the Nifty 50 or S&P 500).

1. The Core Logic

Sharpe's simplification is based on one observation: Most stocks move together because they are all responding to the same "Market Forces" (the economy).

• Markowitz: Stock A moves with Stock B.
• Sharpe: Stock A moves with the Market; Stock B moves with the Market. Therefore, we can find the relationship between A and B via the Market.

2. The SIM Equation

Every stock's return is broken into two parts:

Ri = Alpha_i + (Beta_i * Rm) + Error_i

Where:

• Ri: Return of the individual stock.
• Alpha (a): The stock's unique return (not related to the market).
• Beta (B): The stock's sensitivity to the market.
• Rm: The return of the market index.
• Error: Unpredictable random events (Unique risk).

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3. Huge Reduction in Calculations

Sharpe's model is much easier to use:

• For 100 stocks, Markowitz needs 5,150 inputs.
• For 100 stocks, Sharpe needs only 302 inputs (Alpha, Beta, and Error for each stock, plus Market Return and Variance).

Summary

• The Single Index Model is a practical simplification of the Markowitz model.
• It assumes stock returns are driven by a single common factor (The Market).
• It uses Alpha (Unique return) and Beta (Sensitivity) as key variables.
• It drastically reduces the mathematical work required to diversify.

Quiz Time! 🎯

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