8-1 Portfolio Theory and CAPM

Portfolio Theory

• reduce the standard deviation of a portfolio by choosing stocks that do not move together.

when measured over a short interval, the past rates of return on any stock conform closely to a normal distribution.

normal distributions can be completely defined by two numbers:

• the mean or expected value.
• the variance or standard deviation.

In this chart:

• A and B
  • have the same expected return of 10%
  • A has the greater spread of possible returns ergo it's more risky than B.
  • the spread can be measured by the standard deviation std(A) = 15% std(B) = 7.5%
  • most investors would prefer B to A.
• B and C
  • have the same standard deviation
  • C offers a higher expected return.
  • most investors would prefer C to B.

Efficient Portfolios

Efficient Portfolios

Efficient portfolios are combinations of investments that maximize their overall returns within an acceptable level of risk.

Finding the best efficient portfolio

With a graph of the efficient portfolios, draw a line starting at the risk free return $r_f$ and tangent to the portfolio line. The efficient portfolio at the tangency point is better than all the others. it offers the highest ratio of risk premium to standard deviation.

Borrowing and Lending

If investors have access to borrowing and lending at the risk-free rate, then the investor can obtain any combination of risk and return along the tangent line by either borrowing money which is then invested in the best efficient portfolio(=more risk) or lending money (=less risk).

example

The best efficient portfolio S has std=16%, r=15%.

• Less risk less return strategy:
  • invest half in S
  • lend the rest at the risk-free rate.

$r=\frac{1}{2}\cdot r_S+\frac{1}{2}\cdot r_{bills}$

$=10\mathrm{\%}$

$\sigma =\frac{1}{2}\cdot st{d}_S+\frac{1}{2}\cdot st{d}_{bills}$

$=8\mathrm{\%}$

• More risk more return strategy:
  • borrow initial amount.
  • invest all in S

$r=2\cdot r_S-1\cdot \text{interest rate}$

$=25\mathrm{\%}$

$\sigma =2\cdot st{d}_S-1\cdot st{d}_{bills}$

$=8\mathrm{\%}$

Sharpe Ratio

The ratio of the risk premium to standard deviation. the best efficient portfolio has the highest sharpe ratio.

$$\text{Sharpe ratio}=\frac{\text{Risk premium}}{\text{standard deviation}}=\frac{r-r_f}{\sigma }$$