Sharpe Ratio

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yazena

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I ran through this. Didnt make much sense to me. Any help would be helpful:
‘assuming all investors agree on all asset return, variance, and correlation expectations, then the market portfolio has the highest sharpe ratio”
Any idea?
 
PM is one of my weakest points, but the above seems to say that in an efficient market, the best possible return is the market’s return. Any other return is not possible, so the market portfolio has the highest sharpe ratio by definition….weird, but that’s how I can interpret the above statement.
 
Ya.. I guess it’s the only way! I hate these theory impractical questions. As if we will ever use them in real world!
 
Sharpe ratio is return/risk, since the market portfolio is the most superior portfolio in an efficent market i.e. it has the lowest risk for a given return, all other portfolios have either a lower return for the same risk, or a higher risk for the same return, so the market portfolio will have the highest sharpe, hope this helps
 
With all the assumptions the market portfolio is the best portfolio (highest sharpe ratio, highest additional return over risk free rate for each unit of additional risk taken by means of leverage). No other combination can beat it.
 
While everybody seems to ‘interpret’ sharpe ratio, nobody has answered the fundamental question - why market portfolio has the highest sharpe ratio!
Here are some additional thoughts:
1. There are infinite number of portfolios with the same Sharpe ratio as the market porfolio.
2. These portfolios all have market portfolio in them
What are these portfolios? To fundamentally understand this - ask yourself, what is ‘market’ portfolio?
Anybody wanna try?
 
The market portfolio is every security in the world. If every asset is fairly priced (sits on SML line), then each one provides the correct amount of return for the correct amount of risk. Plus, each security added provides incremental diversification benefit (lowers the volatility).
1) Removing assets with returns higher than the market portfolio would lower the return & increase the volatility by decreasing diversification (lower Sharpe).
2) Removing assets with returns lower than the market portfolio would increase returns, but increase the volatility of the portfolio by a great amount (again, lower Sharpe).
If you don’t believe this, then rework the table on page 154 of Schweser, Book 5 by hand. Or look at figure 1 on page 150. Both illustrate the math involved, on a smaller scale.
“1. There are infinite number of portfolios with the same Sharpe ratio as the market porfolio.”
Just reference page 162 of book five. Again, it may not be intuitive, but that’s how the math works out. As you change the mix of the market portfolio and the rf asset, the volatility of the portfolio and the expected return both increase and decrease linearly. For each additional % of return added, volatility increases proportionally, thus the Sharpe ratio will remain constant.
“2. These portfolios all have market portfolio in them”
You are essentially dialing the risk and return of the market portfolio up and down by mixing in the rf asset.
 
There may be number of portfolios that may give bigger return than market portfolio. However, we have to ask the question at what cost(risk). Considering standard deviation of returns as risk, sharpe ratio gives the excess return per unit of additional risk.
For example you have 4% return on risk free asset and 10% return on market portfolio (A theoretical portfolio with all different classes of asset available for investing across the globe) and with the assumption of perfect markets and risk/return based investor expectations, all the systematic risk is diversified and only risk left is unsystematic risk due to business cycles. So this portfolio gives the best return per additional unit of risk taken. Any other portfolio, irrespective of there returns (high/low), return per unit of risk taken (sharpe ratio) is less than that of market portfolio.
Now with this combination of risk free asset and market portfolio taking additional risk, i.e., using leverage (borrowing at risk free rate) and investing in market portfolio you can generate any rate of returns. You can double or triple your returns.
The CAPM evaluates every asset with two assets, 1. risk free asset and 2. market portfolio (theoretically the best portfolio that contains all risky assets in the world in a combination that gives highest return for every unit of risk taken, sharpe ratio).
Now if you consider S&P as market portfolio with 500 assets (not perfect, relaxing assumptions) with certain return and risk, you get SML. You could build a portfolio with 40 assets with the same return and risk characteristics or may be 50 assets or so on … infinite # of combinations of assets and weights. SML plots above CML because SML is theoretical perfect portfolio, so in theory all systematic risk is not diversified, so it must return higher rate than CML.
Relaxing the assumptions further if you find set of assets that are undervalued you could build a portfolio with those under valued assets and generate alpha. Eventually your portfolio line will be above CML and SML. This way you can have infinite # lines with portfolios with combination of different assets with different weights.
A portfolio is simply a combination of risk free asset (for a high return portfolio this could be -ve, i.e., leverage) and different risk assets at certain weights. Logically speaking every asset or set of assets or portfolio or set of portfolios must return more than market portfolio, because it is exposed to non systematic risk in addition to systematic risk. Now if it does not then we say this portfolio lies below CML/SML and not worthy to invest in.
 
FTW, great job. Janar, you can’t compare SML to CML - they use different metric for risk. You may want to look this up once more.
 
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