FINS2624 Lecture Notes - Lecture 9: Sharpe Ratio, Idiosyncrasy, Standard Deviation

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18 May 2018

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9 – Performance Measures

Need to take risk into account – CAPM returns too high based on:

• Loading high on systematic risks

• High alpha – normally attributable to greater idiosyncratic risk

Main difference lies in adjustment of returns for risks & how risks are defined

• Risks defined & measured differently depending on context and objective

Evaluating portfolios for their entire investment in risky assets: Sharpe Ratio

• Numerator is excess return (i.e. attributable to risk taking)

• Denominator is standard deviation of returns (i.e. total risk)

• Iterpreted as a assets prie of total risk easure

• Can leverage up or down a risky portfolio by borrowing or lending

• Utility generated from entire complete portfolio:

• Higher the Sharpe ratio of a risky portfolio, higher utility given = more desirable

• In this context, return is compared with the total risk (standard deviation of returns)

Evaluating portfolios for their entire investment in risky assets: The M2

• Difference between two Sharpe ratios interpreted as difference in slope of

respective CALs (or prices of risk)

• Forming hypothetical portfolio P’ for evaluated portfolio P, which has same

standard deviation as market portfolio

• M2 measure is difference between expected return on the hypothetical portfolio,

E(rP), and that of the market portfolio, E(rM)

• Essentially a standardized Sharpe ratio

• Method:

 Step 1: For P y oiig the ealuated portfolio ith risk-free asset

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Document Summary

Need to take risk into account capm returns too high based on: loading high on systematic risks, high alpha normally attributable to greater idiosyncratic risk. Main difference lies in adjustment of returns for risks & how risks are defined: risks defined & measured differently depending on context and objective. I(cid:374)terpreted as a(cid:374) asset(cid:859)s (cid:858)pri(cid:272)e of total risk(cid:859) (cid:373)easure. In this context, return is compared with the total risk (standard deviation of returns: higher the sharpe ratio of a risky portfolio, higher utility given = more desirable. E(rp), and that of the market portfolio, e(rm: essentially a standardized sharpe ratio, method: Step 1: for(cid:373) p(cid:859) (cid:271)y (cid:272)o(cid:373)(cid:271)i(cid:374)i(cid:374)g the e(cid:448)aluated portfolio (cid:449)ith risk-free asset. Step 2: choose the weight in the evaluated portfolio to set the resulting standard deviation equal to the standard deviation of the market. Step 3: calculate implied expected return & compare to that of the market: ethical considerations for the m2.

Related Questions

Hi guys could someone help me and explain in detail the two fundamental principles of investments â the risk/return tradeoff and diversification. Also, describe how asset allocation and the correlation of various securities within asset classes impact the riskiness of an investorâs portfolio.

I found an answer but it seems to me very long and not sure about it so can you help me out guys

ANS:

Risk /Return Tradeoff

Risk, return and liquidity are three dimensions of investment decision. Risk/Return trade off considers two dimension ie risk and return. As risk of an investment increases , its expected return will decrease. Risk is measured by volatility of return which can be statistically obtained by standard deviation of return. Higher the standard deviation, higher will be volatility of return and higher the risk

Risk can be also measured as a relative risk compared to the market. This is measured by stock beta with reference to a market index. Higher the beta, higher the value of stock Beta, higher will be the risk .

Investors seeking higher return will need to take higher risks.

Treasury bonds are almost risk free assets. This has the lowest rate of return. Stocks have higher risk , hence have higher expected return. High beta stocks will have higher risk and higher returns.

An investor should understand his risk taking ability and will have to make a trade off between risk and return depending on his risk apetite.

Diversification:

There is a saying, do not put all your eggs in one basket. The reason is that it has higher risk if all eggs are in one basket. If the eggs are distributed in different basket, risk of loss decreases.

Same principle is applied in investment. Diversification reduces risk.

If one invests in different classes of assets like bonds, stock, commodities and properties âhis risk is likely to reduce. Reduction in return in one asset is likely to be compensated by return in other asset.

Within one class of asset also, there should be diversification to reduce risk.

For example one stock âA âmay have expected return of 10% and standard deviation of 5%.This risk can be reduced by diversification in number of assets and the ratio of Risk/return can be decreased.

If we diversify and instead of investing 100% in stock A, we invest 60% in âAâ and 40% in another stock âBâ having expected return15% and standard deviation of 8%.assume the correlation between the two assets are 0.5

Portfolio return=wa*Ra+wb*Rb

Wa=Weight of asset A=0.6

Wb=Weight of asset B=0.4

Ra=Return of A=10%

Rb=Return of B=15%

Sa=Standard deviation of A=5%

Sb=Standard deviation of B=8%

Corr(a,b)=Correlation of A & B=0.2

Portfolio return=0.6*10+0.4*15=12%

Portfolio Variance=(wa^2)*(Sa^2)+(wb^2)*(Sb^2)+2wa*wb*Cov(a,b)

Cov(a,b)=Corr(a,b)*Sa*Sb=0.1*5*8=4

Portfolio Variance=(0.6^2)*(5^2)+(0.4^2)*(8^2)+2*0.6*0.8*4=23.08

Portfolio standard deviation=Square root of Variance=(23.08^0.5)=4.8%

The diversification has increased return and reduced risk

Stock A was giving 10% return with 5% standard deviation. With diversification of portfolio we have 12% return and 4.8% standard deviation. Risk return ratio is decreased.

It may be noted that, higher the correlation between assets in the portfolio, higher will be risk in terms of standard deviation. Hence it is better to diversify with uncorrelated assets.

Diversification reduces unsystematic risk,but it cannot reduce systematic risk of overall market or economy

FINS2624 Lecture Notes - Lecture 9: Sharpe Ratio, Idiosyncrasy, Standard Deviation

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