To Show : How Diversification Reduces Risk



|To Show : How | | | | | | | | | | | | | | | | | |

|Diversification | | | | | | | | | | | | | | | | | |

|Reduces Risk | | | | | | | | | | | | | | | | | |

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|Diversification: | | | | | | | | | | | | | | | | | |

|Holding many stocks | | | | | | | | | | | | | | | | | |

|in one's portfolio. | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | |

|For simplicity, we | | | | | | | | | | | | | | | | | |

|consider a portfolio| | | | | | | | | | | | | | | | | |

|of two stocks. | | | | | | | | | | | | | | | | | |

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|How do we measure | | | | | | | | | | | | | | | | | |

|portfolio risk? If | | | | | | | | | | | | | | | | | |

|stock returns are | | | | | | | | | | | | | | | | | |

|normally | | | | | | | | | | | | | | | | | |

|distributed, | | | | | | | | | | | | | | | | | |

|then a portfolio | | | | | | | | | | | | | | | | | |

|constructed from | | | | | | | | | | | | | | | | | |

|such stocks will | | | | | | | | | | | | | | | | | |

|also be normally | | | | | | | | | | | | | | | | | |

|distributed. | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | |

|The risk of such a | | | | | | | | | | | | | | | | | |

|portfolio can be | | | | | | | | | | | | | | | | | |

|measured by the | | | | | | | | | | | | | | | | | |

|variance of the | | | | | | | | | | | | | | | | | |

|returns | | | | | | | | | | | | | | | | | |

|of the portfolio. | | | | | | | | | | | | | | | | | |

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|Consider stocks A | | | | | | | | | | | | | | | | | |

|and B in portfolio | | | | | | | | | | | | | | | | | |

|P: | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | |

|Expected Return of | | | | | | | | | | | | | | | | | |

|portfolio = (Weight | | | | | | | | | | | | | | | | | |

|on A)* (Expected | | | | | | | | | | | | | | | | | |

|Return of A) | | | | | | | | | | | | | | | | | |

|+ (Weight on | | | | | | | | | | | | | | | | | |

|B)*(Expected Return | | | | | | | | | | | | | | | | | |

|of B) | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | |

|[Weight on A + | | | | | | | | | | | | | | | | | |

|weight on B = 1.0; | | | | | | | | | | | | | | | | | |

|If you have $1000 to| | | | | | | | | | | | | | | | | |

|invest in GM and ATT| | | | | | | | | | | | | | | | | |

|and you invest $300 | | | | | | | | | | | | | | | | | |

|in GM, then weight | | | | | | | | | | | | | | | | | |

|on GM = 300/1000 = | | | | | | | | | | | | | | | | | |

|.3] | | | | | | | | | | | | | | | | | |

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|Variance of | | | | | | | | | | | | | | | | | |

|Portfolio = (Weight | | | | | | | | | | | | | | | | | |

|on A)^2 * (Variance | | | | | | | | | | | | | | | | | |

|of A) | | | | | | | | | | | | | | | | | |

|+ (Weight on B)^2 * | | | | | | | | | | | | | | | | | |

|(Variance of B) | | | | | | | | | | | | | | | | | |

|+ 2 * (Weight on A) | | | | | | | | | | | | | | | | | |

|* (Weight on B) * | | | | | | | | | | | | | | | | | |

|Covariance of A and | | | | | | | | | | | | | | | | | |

|B | | | | | | | | | | | | | | | | | |

| | | | | | | | | |F | | | | | | | | |

|Covariance of A and | | | | | | | | | | | | | | | | | |

|B = Correlation of A| | | | | | | | | | | | | | | | | |

|and B * Std. | | | | | | | | | | | | | | | | | |

|Deviation of A * | | | | | | | | | | | | | | | | | |

|Std. Deviation of B | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | |

|A Numerical Example:| | | | | | | | | | | | | | | | | |

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|Expected Return for | | |μa = |0.05 | |Expected | | |μb = |0.08 | | | | | | | |

|Stock A = | | | | | |Return for | | | | | | | | | | | |

| | | | | | |Stock B = | | | | | | | | | | | |

|Std. Deviation for | | |σa = |0.04 | |Std. Deviation| | |σb = |0.1 | | | | | | | |

|Stock A = | | | | | |for Stock B = | | | | | | | | | | | |

For Correlation of A and B: -1.0, -0.2, 0.0, 0.2, 0.5, and 1.0:

For each correlation compute expected return and standard deviation of return of a portfolio with different weights in stocks A and B. Example: Correlation = -1.0; see RISK2.xls

| | | |ρ = |-1.00 |

|Percent of |Percent of | | | |

|portfolio in |portfolio in | |μp |σ p |

|Stock A |Stock B | | | |

|Wa |Wb=1-Wa | | | |

| | | | | |

|1 |0 | |5.00% |4.00% |

|0.9 |0.1 | |5.30% |2.60% |

|0.8 |0.2 | |5.60% |1.20% |

|0.714285226 |0.285714774 | |5.86% |0.00% |

|0.7 |0.3 | |5.90% |0.20% |

|0.6 |0.4 | |6.20% |1.60% |

|0.5 |0.5 | |6.50% |3.00% |

|0.4 |0.6 | |6.80% |4.40% |

|0.3 |0.7 | |7.10% |5.80% |

|0.2 |0.8 | |7.40% |7.20% |

|0.1 |0.9 | |7.70% |8.60% |

|0 |1 | |8.00% |10.00% |

Return Minimum Variance Frontier

Risk

Efficient Frontier: Upward sloping part of the Minimum Variance Frontier. A portfolio is said to be efficient if it has the least risk for a given level of return, or the most return for a given level of risk.

Rational investors (that is, investors that attempt to maximize their returns and minimize their risk) attempt to hold efficient portfolios.

Return The Efficient Frontier

Risk

Return The Efficient Frontier

The Capital Market Line

The Value-weighted MARKET Portfolio

Risk-free

asset

Risk (Standard Deviation)

Equation of the Capital Market Line:

Expected Return of an efficient portfolio = Risk-free rate

+{[Expected return on the market portfolio - Risk-free rate] / [Std. Dev. of the Market Portfolio] } * Std. Deviation of the efficient portfolio

Tobin’s Separation Theorem

From among all the risky assets and portfolios of risky assets (the efficient set and points to the right and below it), each (and all ) investors would hold the same portfolio of risky asset ( the value-weighted market portfolio), regardless of his/her tastes with respect to risk.

An individual investors’s tastes with respect to risk comes into play in the following way: A risk-averse investor would invest more of his/her wealth in the risk-free asset and less in the market portfolio. A less risk-averse investor would invest less in the risk-free asset and more in the market portfolio.

Risk of an Individual Stock when held in a well-diversified portfolio: BETA.

Rational investors only hold well-diversified portfolios. The risk of a well-diversified portfolio is measured by its variance. (Assuming portfolio returns are normally distributed).

How should investors measure the risk of an individual stock, say ABC, (when held in a well-diversified portfolio)? This question is equivalent to asking:

How does adding one additional share of the stock of company ABC change the variance of their portfolio?

Mathematically,

Risk of stock ABC = Change in variance of well-diversified portfolio when one share of ABC is added to this portfolio

Risk of stock ABC = ( Δ Vp ) / ( Δ wABC )

( d Vp / d wABC ) = 2 * Covariance ( Stock ABC, Market)

Beta is a standardized measure of covariance.

βABC = [ Covariance ( Stock ABC, Market) ] / [ Variance of Market]

Portfolio Betas and Expected Returns

Consider stocks A and B in portfolio P:

Expected Return of portfolio = (Weight on A)* (Expected Return of A) + (Weight on B)*(Expected Return of B)

Beta of portfolio = (Weight on A)* (Beta of A) + (Weight on B)* (Beta of B)

Please go to CAPM-EMH.ppt

Equation of the Capital Market Line:

Expected Return of an efficient portfolio = Risk-free rate

+{[Expected return on the market portfolio - Risk-free rate] / [Std. Dev. of the Market Portfolio] } * Std. Deviation of the efficient portfolio

Numerical example:

Risk-free rate = 3%

Expected return on the market = 15%

Standard deviation of the market’s returns =18%

Standard deviation of mutual fund XYZ =13%; assume this fund is on the efficient frontier.

What is the expected return on mutual fund XYZ?

.03 + {[.15 - .03] / .18} * (.13) = .1167 = 11.67%

Equation of the Security Market Line (CAPM):

Expected Return of any asset = Risk-free rate + (Expected return on the market portfolio - Risk-free rate) * Beta of that asset

Numerical example:

Risk-free rate = 3%

Expected return on the market = 15%

Beta of Netscape = 1.6

What is the expected return for Netscape?

.03 + (.15 - .03) * 1.6 = .222 = 22.2

BETAS OF EFFICIENT PORTFOLIOS

The Capital Market Line applies only to efficient portfolios. The Security Market Line applies to ALL assets (including efficient portfolios, individual stocks, real estate, etc.)

Hence, for an efficient portfolio, both the Capital Market Line and the Security Market Line will apply. From the equations of the Capital Market Line and the Security Market Line we can show that

Beta of an efficient portfolio = Std. Dev.of the efficient portfolio / Market Std. Dev.

Standard deviation of the market’s returns =18%

Standard deviation of mutual fund XYZ =13%; assume this fund is on the efficient frontier.

Hence, Beta of fund XYZ = .13 / .18

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