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Archive for March, 2013

Risk and Return for CFP-6

Posted by Prashant Shah on March 25, 2013

Risk Adjusted Performance Measurement (Important for Examination)

The Sharpe measure

Sharpe (1966) defined this ratio as the reward-to-variability ratio, but it was soon called the Sharpe ratio in articles that mentioned it. It is defined by

Capture

This ratio measures the excess return, or risk premium, of a portfolio compared with the risk-free rate, compared, this time, with the total risk of the portfolio, measured by its standard deviation.

If the portfolio is well diversified, then its Sharpe ratio will be close to that of the market. With this ratio the manager can check whether the expected return on the portfolio is sufficient to compensate for the additional share of total risk that he is taking. Since this measure is based on the total risk, it enables the relative performance of portfolios that are not very diversified to be evaluated, because the unsystematic risk taken by the manager is included in this measure.

The Treynor measure

The Treynor (1965) ratio is defined by

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The term on the left is the Treynor ratio for the portfolio, and the term on the right can be seen as the Treynor ratio for the market portfolio, since the beta of the market portfolio is 1 by definition. Comparing the Treynor ratio for the portfolio with the Treynor ratio for the market portfolio enables us to check whether the portfolio risk is sufficiently rewarded.

The Treynor ratio is particularly appropriate for appreciating the performance of a well diversified portfolio, since it only takes the systematic risk of the portfolio into account.

The Jensen measure

Jensen’s alpha (Jensen, 1968) is defined as the differential between the return on the portfolio in excess of the risk-free rate and the return explained by the market model, or

CaptureIllustration:

You are evaluating the rankings based on Sharpe and Treynor Ratio of three funds A, B and C . The average returns obtained from funds A, B and C have been 16%, 19% and 14%, respectively against the market return of 13%. The standard deviations of fund returns have been 17, 22 and 16, respectively versus the market return standard deviation of 15. If the beta reported of these funds is 1.2, 1.4 and 1.1, respectively and the risk-free rate of return is 5.5%, what are your rankings in the order of best to worst?

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Risk and Return for CFP-5

Posted by Prashant Shah on March 19, 2013

Modern Portfolio Theory and Asset Pricing

The relation between portfolio returns and portfolio risk was recognized by two Nobel Laureates in Economics, Harry Markowitz and William Sharpe. Harry Markowitz tuned us into the idea that investors hold portfolios of assets and therefore their concern is focused upon the portfolio return and the portfolio risk, not on the return and risk of individual assets.

The relevant risk to an investor is the portfolio’s risk, not the risk of an individual asset. If an investor considers buying an additional asset or selling an asset from the portfolio, what must be considered is how this change will affect the risk of the portfolio.

Possible Portfolio

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 The Efficient Frontier: the best combinations of expected return and standard deviation.

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All the assets in each portfolio, even on the frontier, have some risk. Now let’s see what happens when we add an asset with no risk—referred to as the risk-free asset.

If we look at all possible combinations of portfolios along the efficient frontier and the risk-free asset, we see that the best portfolios are no longer those along the entire length of the efficient frontier; rather, the best portfolios are now the combinations of the risk-free asset and one and only one portfolio of risky assets on the frontier.

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Sharpe demonstrates that this one and only one portfolio of risky assets is the market portfolio—a portfolio that consists of all assets, with the weights of these assets being the ratio of their market value to the total market value of all assets.

If investors are all risk averse—they only take on risk if there is adequate compensation—and if they are free to invest in the risky assets as well as the risk-free asset, the best deals lie along the line that is tangent to the efficient frontier. This line is referred to as the capital market line (CML). If the portfolios along the capital market line are the best deals and are available to all investors, it follows that the returns of these risky assets will be priced to compensate investors for the risk they bear relative to that of the market portfolio.

The CML specifies the returns an investor can expect for a given level of risk. The CAPM uses this relationship between expected return and risk to describe how assets are priced.

The CAPM specifies that the return on any asset is a function of the return on a risk-free asset plus a risk premium. The return on the riskfree asset is compensation for the time value of money. The risk premium is the compensation for bearing risk. Putting these components of return together, the CAPM says:

Expected return on an asset = Expected return on a risk-free asset Risk premium

Hence, Required rate or return = Rf + (Rm – Rf) β

Where,  (Rm – Rf) is Market Risk Premium

For each asset there is a beta. If we represent the expected return on each asset and its beta as a point on a graph and connect all the points, the result is the security market line (SML) 

Security Market Line:

Capture

Illustrations:

Find Beta of security X if expected market premium is 15%, risk free return is 7% and expected return of security X is 20%?

  • (a)   0.834
  • (b)   0.900
  • (c)    0.700
  • D)    0.867

Risk free rate of return is 8%, expected market premium is 15% and Beta of security is 0.80. What is the expected rate of return of the security?

  • (a)   13.6%
  • (b)   15.00%
  • (c)    12.00%
  • (d)   20.00%

Assume Rf  is 7%, Km  is 16% for a security X and has a beta factor of 1.4, the required return of the security is_______.

  • 21.6%
  • 20.0%
  • 17.4%
  • 19.6%
  • 23.2%

 Source: Financial Management and Analysis by FRANK J. FABOZZI.

All content belongs to FRANK J. FABOZZI.

Posted in Investment Planning, Risk And Return | 2 Comments »