Topic 1 (Ch. 6) Capital Allocation to Risky Assets. Risk with simple prospects Investor’s view of risk Risk aversion and utility Trade-off between risk and return Asset risk versus portfolio risk Capital allocation across risky and risk-free portfolios. Risk with Simple Prospects.
$122,000 - $100,000 =$22,000.
(the expected value of the squared deviation of each possible outcome from the mean)
(the square root of the variance)
Suppose that at the time of the decision, a one- year T-bill offers a risk-free rate of return of 5%; $100,000 can be invested to yield a sure profit of $5,000.
profit of the risky investment over investing
in safe T-bills is:
$22,000 - $5,000 = $17,000
compensation for the risk of the investment.
Consistent with the notion that utility is enhanced by high expected returns and diminished by high risk.
(1) T-bills providing a risk-free return of 5%.
(2) A risky portfolio with E(r)= 22% and
= 34% .
A = 3:
T-bills: U = 0.05 – 0 = 0.05.
Risky portfolio: U = 0.22 – 0.5 3 (0.34)2
Risk-free rate = 5%
and at least one inequality is strict (rules out the equality).
A curve connecting all portfolios that are equally desirable to the investor (i.e. with the same utility) according to their means and standard deviations.
To determine some of the points that appear on the indifference curve, examine the utility values of several possible portfolios for an investor with A = 4:
where Pr(s): the probability of scenario s
r(s): the return in scenario s
Consider a portfolio when it splits its investment evenly between Best Candy and SugarKane:
Scales the covariance to a value between -1
(perfect negative correlation) and +1 (perfect
confirms the strong tendency of Best and
SugarKane stocks to move inversely.
wi: fraction of the portfolio invested in asset i
: variance of the return on asset i
With equal weights in Best and SugarKane:
A positive covariance increases portfolio variance, and a negative covariance acts to reduce portfolio variance.
This makes sense because returns on negatively correlated assets tend to be offsetting, which stabilizes portfolio returns.
is negatively correlated with the existing portfolio.
This negative correlation reduces the overall risk of the portfolio.
E(rP) (= 15%): expected rate of return on P
P(= 22%): standard deviation of P
rf(= 7%): risk-free rate of return on F
E(rP) - rf(= 8%): risk premium on P
The base rate of return for any portfolio is the risk-free rate.
In addition, the portfolio is expected to earn a risk premium that depends on the risk premium of the risky portfolio, E(rP) - rf, and the investor’s position in the risky asset, y.
proportional to both the standard deviation of the
risky asset and the proportion invested in it.
- shows all feasible risk-return combinations
of a risky and risk-free asset to investors.
1 – y = 1 – 1.4 = -0.4 (short or borrowing
position in the risk-free assets).
In the borrowing range, the reward-to-variability ratio (the slope of the CAL) will be:
The utility that an investor derives from a portfolio with a given expected return and standard deviation can be described by the following utility function:
where U: utility value
E(r): expected return
2: variance of returns
A: index of the investor’s risk aversion
An investor who faces a risk-free rate, rf, and a risky portfolio with expected return E(rP)and standard deviation p will find that, for any choice of y, the expected return of the complete portfolio is:
E(rC) = rf + y[E(rP) – rf].
The variance of the complete portfolio is:
The investor attempts to maximize utility U by choosing the best allocation to the risky asset, y.
This particular investor will invest 41% of the investment budget in the risky asset and 59% in the risk-free asset.
Consider an investor with risk aversion A = 4 who currently holds all her wealth in a risk-free portfolio yielding rf = 5%.
Because the variance of such a portfolio is zero, its utility value is U = 0.05.
Now we find the expected return the investor would require to maintain the same level of utility when holding a risky portfolio, say with = 1%.
We can repeat this calculation for many other levels of , each time finding the value of E(r)necessary to maintain U = 0.05.
This process will yield all combinations of expected return and volatility with utility level of .05; plotting these combinations gives us the indifference curve.