BKM Chapter 6 - Risk and Risk Aversion

Question 1

Risk Aversion

Answer 1

Investors are risk averse in the sense that they would not accept a risky proposition without some degree of compensation in the form of a risk premium.

Question 2

Utility Function

Answer 2

U = E(r) - 0.5Aσ²

A: represents the investors’ degree of risk aversion

Question 3

Three types of investors

Answer 3

• Risk Averse Investors: Their risk aversion parameter, A, is greater than zero. Their utility score is adjusted downward when there is risk.
• Risk Neutral Investors: Their risk aversion parameter, A, is 0. They judge prospects solely in terms of their expected return and do no modify their utility score when there is risk.
• Risk Seeking Investors: Their risk aversion parameter, A, is less than zero. Their utility score is adjusted upward when there is risk (prefer risk investments to risk-free assets).

Question 4

Mean-Variance Criterion

Answer 4

Portfolio is preferred if its expected return is at least the same or higher { E(r_A) >= E(r_B) } and its risk is at least the same or lower { σ_A<=σ_B }.

Question 5

Certainty Equivalent Rate

Answer 5

The utility value as the rate of return that would cause the investor to be indifferent between the risky investment and the risk-free investment.

They will therefore accept the risky investment if the certainty equivalent rate is above the risk-free rate and will reject it if it falls below the risk-free rate.

Question 6

Indifference Curves

Answer 6

All of the opportunities that the investor will be indifferent to, meaning that all of them have the same utility score. This collection of opportunities can be plotted as an indifference curve, with each point on the curve representing a combination of risk and return that provides the same level of utility.

Question 7

Capital Allocation Across Risky and Risk-Free Portfolios

Expected Return of Complete Portfolio &

Std. Deviation of Complete Portfolio:

Answer 7

Expected Return of Complete Portfolio:

E(r_c) = yE(r_p) + (1 - y) * r_f

= r_f + y[E(r_p)-r_f] ⇒ complete portfolio return is equal to the risk-free rate plus a risk premium

Std. Deviation of Complete Portfolio:

σ_c = y * σ_p

Since the standard deviation of the risk-free asset is zero and its covariance with the risky portfolio is also zero.

Question 8

Capital Allocation Line - Sharp Ratio

Answer 8

If we plot combinations of risk (as the x-axis) and expected return (as the y-axis) for our combination of the risk-free asset and the risky asset based on the capital allocation expected return and std. deviation, we produce what is known as the Capital Allocation Line (CAL).

• One point on this line is the intercept, when y = 0 then E(r_c) = r_f and σ_c = 0.
• Another point on this line is when y = 1 and therefore E(r_c) = E(r_p) and σ_c = σ_p.

we can calculate the slope of the line that connects the two points as:

S = [E(r_p) - r_f]/σ_p

this slope is called the reward-to-variability ratio or the Sharpe ratio.

Question 9

Capital Allocation Line - Formula & Graph

Answer 9

E(r_c) = r_f + (σ_c/σ_p) * {E(r_p) - r_f}

Question 10

Risk Tolerance and Asset Allocation: Optimal Complete Portfolio

Answer 10

The investor would select the portfolio that maximizes his or her utility.

U = E(r) - 0.5Aσ²

= r_f + y*[E(r_p)-r_f] - 0.5Ay²σ_p²

Optimal y is solve by taking the derivative with respect to y and setting the derivative to zero:

y* = [E(r_p) - r_f]/Aσ_p²

The optional complete portfolio is the interception between the CAL and utility function line.

Question 11

Capital Market Line

Answer 11

If we used T-bills as our risk-free asset and a broad index of common stocks such as the S&P 500 as our risky asset, then the Capital Allocation Line we developed here could be referred to as the Capital Market Line (CML).

Question 12

Two main reasons why this passive strategy may make sense

Answer 12

• Reduce cost: It minimizes costs to acquire the information necessary to select an optimal risky portfolio
• Free Rider Benefit: It takes advantage of everyone else’s efforts to do so, resulting in security prices which are by large fair.

Question 13

Common criticisms of the passive strategy (known as indexing):

Answer 13

• They’re Undiversified
  
  • Although the S&P 500 holds 500 stocks, it weights them according to their market capitalization and therefore 25% is invested in the top ten firms.
  • Actively managed funds are similarly undiversified
• They’re Top Heavy
  
  • Although the S&P 500 only includes 500 of the more than 6,000 US stocks, this does represent 77% of the US market’s total market value.

Question 14

What three characteristics do U.S. Treasury Bills have that make them an appropriate choice as the risk-free asset?

Answer 14

• free of default risk,
• short term nature makes them insensitive to interest rate risk and
• minimal exposure to inflation uncertainty