# AFM273 Chapter Notes - Chapter 11: Weighted Arithmetic Mean, Expected Return, Covariance

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7 Dec 2016

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C11: Optimal portfolio choice and the CAPM

The expected return of a portfolio

We can describe a portfolio by its portfolio weights

o How much of each investment makes up the portfolio

Portfolio weights add up to 1

o If xi is the weighted average of portfolio, then Ri would be the weighted average of the

realized returns

o So the expected returns would be

Note: without trading, the weights will increase for stocks in the portfolio whose returns

exceed the portfolios

Volatility of a two stock portfolio

Combining risks

o By combining stocks into a portfolio we can diversify risk

This allows risk in a portfolio to be lower than individual stocks

o Amount of risk that is eliminated in a portfolio depends on the sensitivity of the stock's

price to common risk

If everything in the portfolio is related to a single sector, minimal risk is eliminated

If there is a great degree of diversification then more risk can be eliminated

Determining covariance and correlation

o Covariance

Expected product of the deviations of2 returns from their means

o Formula for estimating covariance from historical data

o If stocks move together, returns will tend to be above or below average at the same

time

Covariance is positive

o If stocks move opposite

Covariance is negative

o Magnitude of covariance is hard to interpret

Correlation

o Value of correlation will be larger the more volatile

Also be larger if it moves more closely relative to other stocks

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2

o

Measures the strength of the relationship between returns while controlling the

stock volatilities

Value output is always between -1 and 1

Closer to -1, perfect negative correlation (moves opposite)

Closer to 1, perfect positive correlation (moves tandem)

When equals 0 no correlation

o Stock returns usually move together if they have they are affected by the same

economic events

Stocks in the same industry usually affected by the same things and may move

together

Computing a portfolio's variance and volatility

o The variance of a portfolio is equal to the weighted average covariance for each stock

with the portfolio

Variance of 2 stock portfolio

Var(Rp)=X12Var(R1)+X22Var(R2)+2X1X2Cov(R1,R2)

Standard deviation

SD(Rp) = Sqrt Var(Rp)

Volatility of a large portfolio

Large portfolio variance

o The variance of a portfolio is equal to the weighted average covariance for each stock

with the portfolio

o

o The risk of a portfolio depends on how each stock's return moves in relation to it

The variability of a portfolio depends on the co-movement of the stocks

Variance of a portfolio is the sum of all the covariance's of the returns of all pairs

of stocks in the portfolio multiplied by each of their portfolio weights

Diversification with an equally weighted portfolio

o Equally weighted portfolio is when each stock has the same amount invested in it

o Variance of an equally weighted portfolio of n Stocks

Var(Rp)=1/n(average variance of the individual stocks) + (1-1/n)(average

covariance between the stocks)

o Variance of a large portfolio mainly determined by the average covariance among the

stocks

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3

With the formula above, as the portfolio gets larger and (n) increases,

First portion will approach 0, while the second portion approaches 1

The standard deviation would then be the square root of the average

covariance between the stocks

o Effects of diversification are largest at the initial change

Effects can be observed with a few stocks

Still must carefully choose stocks that are not correlated to be well diversified

Diversification with general portfolios

o

o Each security contributes to the volatility of the portfolio according to its total risk scaled

by its correlation to the portfolio

The correlation to the portfolio adds a fraction from the risk of the stock to the

portfolio

o Unless all stocks are perfectly correlated, the risk of the portfolio will be lower than the

weighted average volatility of the individual stocks

Risk Versus Return: Choosing an efficient portfolio

Efficient portfolio with two stocks

o Different weights will vary the volatility and expected return

o These varying weights can be turned into a graph

Inefficient portfolios

o Occur when there can be another portfolio that is better in expected returns/ less

volatility

Moving northwest

Efficient portfolios

o No other portfolio of the 2 stocks that will offer less volatility for a higher return

Efficient portfolios are different to people depending how risk averse they may

be

The effect of correlation

o Correlation doesn’t effect on the expected return of a portfolio

o Correlation does affect that volatility

As correlation decreases larger effects to volatility since diversification starts

making a large impact

o No diversification, points connected by a straight line

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