BKM Chapter 6

1
Risk Aversion and Capital Allocation

• BKM Chapter 6

2
Risk Aversion Utility

• Investors view of risk
• Risk Averse
• Risk Neutral
• Risk Seeking
• Utility
• A measure of investor welfare based on the risk
  and return of the investments.
• Utility Function
• U E ( r ) - 0.5 A s 2
• Where A is the coefficient of risk aversion, a
  measure of the degree of risk aversion.
• Question Does a large A stand for a great or
  small degree of risk aversion?

3
Utility Values for Investors with Different
Degree of Risk Aversion

• U E ( r ) - .5 A s 2
• 0.22 - .5 A (0.34) 2
• Risk Aversion A Value
• High 5 -0.069
• 3 0.0466
• Low 1 0.1622

T-bill 0.05
4
Comparing Utilities for Different Investments

• Which portfolio should investors choose?
• It depends on A!

5
Dominance Principle
Expected Return
4
2
3
1
Variance or Standard Deviation
Choices under Mean-Variance Criterion 2 v.s. 1
2 v.s. 3 4 v.s. 3 2 v.s. 4
6
Utility and Indifference Curves

• Represent an investors willingness to trade-off
  return and risk.
• An Example Utility Values of Possible Portfolios
  for an Investor with Risk Aversion, A 4

7
Indifference Curves
Expected Return
Increasing Utility
Standard Deviation
8
Top-down Portfolio Construction Process

• Capital Allocation
• Allocation between the risk free asset and a
  risky portfolio (from now on, the proportion on
  risky portfolio is denoted as y).
• Asset Allocation
• Choice of risky asset classes within the risky
  portfolio, i.e., bonds, stocks, and real estates,
  etc.
• Security Selection
• Choice of individual assets within each asset
  class, e.g., MSFT, IBM in the stock portfolio.

9
Allocating Capital Risky Risk Free Assets

• Suppose there is a complete portfolio of
  300,000, with 30 in risk free asset, and the
  remaining in a risky portfolio. The risky
  portfolio invests 54 in equity (E) and 46 in
  bond (B). What are the weights of E and B in the
  complete portfolio?
• wE/C y wE 70 54
• wD/C y wD 70 46

10
Expected Return and Variance for Combinations of
One Risky and a Risk-free Asset
11
Example Using Chapter 6.4 Numbers
12
Combinations Without Leverage

• When y0.75
• E(rc)0.750.150.250.07
• sc0.750.22

• When y1
• E(rc)1.00.1500.07
• sc1.00.22

• When y0
• E(rc)0.00.151.00.07
• sc0.00.22

13
Capital Allocation Line with Leverage

• Borrow at the Risk-Free Rate and invest in stock.
• Using 50 Leverage, i.e., y1.5, 1-y-0.5
• rc (-.5) (.07) (1.5) (.15)
• ?c (1.5) (.22)

14
Figure 6.4 The Investment Opportunity Set with a
RiskyAsset and a Risk-free Asset in the Expected
Return Standard Deviation Plane
15
CAL (Capital Allocation Line)

• Mathematical representation of CAL
• scysp
• y sc/sp
• E(rc) yE(rp) (1 - y)rf
• rf y(E(rp) rf)
• rf y(E(rp) rf)

16
CAL (Capital Allocation Line)

• Mathematical representation of CAL
• Is the slope of the CAL, also know as the
  reward-to-variability ratio.

17
Figure 6.5 The Opportunity Set with Differential
Borrowing and Lending Rates
18
Utility Function and Capital Allocation

• U E ( r ) - .5 A s2
• Where
• U utility
• E ( r ) expected return on the asset or
  portfolio
• A coefficient of risk aversion
• s2 variance of returns
• In the case of allocation, we need to choose a y
  so that U is maximized

19
Utility Function and Capital Allocation

• Recall
• E(rc) rf y(E(rp) rf)
• sc2y2 sp2
• Therefore
• U E (rC) - .5 A sC2
• rf y (E(rp) rf) 0.5 Ay2 sp2
• To maximize U, we need two conditions
• First order derivative w.r.t. y is zero
• Second order derivative w.r.t. y is negative

20
Utility Function and Capital Allocation

• First derivative w.r.t y
• Second derivative w.r.t. y

21
Utility Function and Capital Allocation

• From first condition, we obtain
• Observations
• Greater levels of risk aversion lead to larger
  proportions of the risk free rate.
• Lower levels of risk aversion lead to larger
  proportions of the portfolio of risky assets.

22
Utility Function and Capital Allocation

• Example
• Rf7, E( rP )15, sP22, A4
• What is the optimal allocation?

23
Table 6.5 Utility Levels for Positions in Risky
Assets for an Investor with Risk Aversion A 4
24
Figure 6.6 Utility as a Function of Allocation to
the Risky Asset, y
25
Utility Function and Capital Allocation
26
CAL and Risk Preferences
E(r)
The lender has a larger A when
compared to the borrower
P
Borrower
7
Lender
?
?p 22
27
Assignments

• Chapter 6
• Problems 1-2, 7-11, 13-15, 17, 18, 23, 24, 30-33.