Capital allocation line

The capital allocation line is a straight line that has the following equation:

:$\backslash mathrm\{CAL\}\; :\; E(r\_\{C\})\; =\; r\_F\; +\; \backslash sigma\_C\; \backslash frac\{E(r\_P)\; -\; r\_F\}\{\backslash sigma\_P\}$

In this formula P is the risky portfolio, F is riskless portfolio, and C is a combination of portfolios P and F.

The slope of the capital allocation line is equal to the incremental return of the portfolio to the incremental increase of risk. Hence, the slope of the capital allocation line is called the reward-to-variability ratio because the expected return increases continually with the increase of risk as measured by the standard deviation.{{Cite web|url=http://thismatter.com/money/investments/capital-allocation.htm|title = Capital Allocation Between a Risk-Free Asset and a Risky Asset; Capital Allocation Line (CAL)}}{{better source needed|date=July 2023|reason=This is just some guy's blog}}

If investors can purchase a risk free asset with some return r_F, then all correctly priced risky assets or portfolios will have expected return of the form

:$\{E(R\_P)\}\; =\; r\_F\; +\; b\; \backslash sigma\_P$

where b is some incremental return to offset the risk (sometimes known as a risk premium), and σ_P is the risk itself expressed as the standard deviation. By rearranging, we can see the risk premium has the following value

:$b\; =\; \backslash frac\{E(R\_P)\; -\; r\_F\}\{\backslash sigma\_P\}$

Now consider the case of another portfolio that is a combination of a risk free asset and the correctly priced portfolio we considered above (which is itself just another risky asset). If it is correctly priced, it will have exactly the same form:

:$E(R\_C)\; =\; r\_F\; +\; \backslash sigma\_C\; b$

Substituting in our derivation for the risk premium above:

:$E(R\_C)\; =\; r\_F\; +\; \backslash sigma\_C\; \backslash frac\{E(R\_P)\; -\; r\_F\}\{\backslash sigma\_P\}$

This yields the Capital Allocation Line.{{cite journal|last1=Sharpe|first1=William| journal=The Journal of Business | title = Mutual Fund Performance |year=1966| volume=39| issue = 1 , pt 2 | pages=119–138 |doi=10.1086/294846| url=http://finance.martinsewell.com/fund-performance/Sharpe1966.pdf }}

• Capital market line
• Security market line
• Security characteristic line
• Market portfolio
• Sharpe ratio, which equals the slope of the Capital Allocation Line

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Category:Investment management

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