RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS

1
RISK AVERSION AND CAPITAL ALLOCATION TO RISKY
ASSETS

• CHAPTER 6

2
Outline of the Chapter

• Risk
• Difference between speculation and gambling
• Risk averse investors and utility functions
• Indifference curve
• Estimating risk aversion
• Capital Allocation
• Risky versus risk-free portfolios
• Forming portfolios of a risky and a risk-free
  asset
• Expected risk and return of the portfolio
• CAL
• Lending and borrowing
• Risk tolerance and asset allocation
• Finding optimal complete portfolio
• Passive investment strategies

3
Risk and Risk Aversion

• Constructing a portfolio is a two step process
• Selecting the composition of risky assets such
  as stocks and bonds
• Deciding how much to invest in that risky
  portfolio versus in a safe asset such as T-bills
• In order to decide how much to invest in risky
  and risk-free assets the investor should know the
  risk and return trade-off

4
Risk and Risk Aversion (Continued)

• Risk is the concept that denotes the specific
  probabilities of specific events.
• Can be negative or positive
• In this concept risk is volatility of the return
  of the risky asset
• Keep in mind that risk (negative) must be
  accompanied by a reward
• Speculation versus gambling
• Speculation
• The assumption of considerable investment risk to
  obtain commensurate (corresponding,
  proportionate) gain

5
Risk and Risk Aversion (Continued)

• Considerable risk
• Sufficient to affect the decision
• Commensurate gain
• A positive risk premium
• (expected holding period return of a risky
  stock)-(risk-free rate)
• An expected profit greater than the risk-free
  alternative
• Gamble
• Bet on an uncertain outcome
• The difference is the lack of commensurate gain

6
Risk and Risk Aversion (Continued)

• Whereas the gamble is the assumption of risk for
  no purpose but enjoyment of the risk itself, the
  speculation is undertaken in spite of the risk
  involved because one perceives a favorable
  risk-return trade-off
• In order to turn gamble into speculation the
  risk-averse investor has to be offered an
  adequate risk premium to bear the risk
• Risk averse investors are the ones who needs
  positive risk premiums on stocks in order to
  invest
• Risk premium the difference between the expected
  return on risky and risk-free assets
• A risky investment with a risk premium of zero
  called a fair game
• Fair game is usually rejected by the risk-averse
  investors

7
Risk and Risk Aversion (Continued)

• Example (Concept check 1)
• There is a US investor
• The interest rates on UK and US T-Bills are same
  (5)
• The characteristics of these two securities are
  almost same (short-term, default free assets)
• Neither offers a risk premium
• There is an exchange rate risk for the US
  investor since pounds earned on UK T-Bills will
  be exchanged for dollars in the future
• Current exchange rate is 2 per pound
• Is the US investor engaging in speculation or
  gambling?

8
Risk and Risk Aversion (Continued)

• US investor invests 2 in UK T-bills
• Change dollars into pounds
• (with the given exchange rate) 21
• Invest 1 into UK T-bills with 5 rate of return
• At the year end the value of UK T-bills
• 1(10.05)1.05
• One more assumption the end of year exchange
  rate is 2.10 per pound
• Exchange pounds for dollars
• 1.052.102.205

9
Risk and Risk Aversion (Continued)

• The rate of return in dollars
• 2(1r)2.205
• r10.25
• More than available in US T-Bills (positive risk
  premium)
• Thus if US investor is expecting positive
  exchange rate movements, expects dollar to
  depreciate against pound, then the UK T-bills is
  a speculative investment. Otherwise it is a gamble

10
Risk and Risk Aversion (Continued)

• Risk Aversion and Utility Values
• Risk averse investors reject investment
  portfolios that are fair games or worse
• These investors are willing to consider only
  risk-free or speculative prospects with positive
  risk premiums
• Risk averse investors asks for more penalty
  (larger risk premium-return) when the risk is
  greater

11
Risk and Risk Aversion (Continued)

• In order to understand the risk-return relation
  better for the risk-averse investors lets look at
  an example
• An investor is considering three different
  investment opportunities with three different
  risk premiums, expected returns and standard
  deviations (risks)

12
Risk and Risk Aversion (Continued)

• Risk premium (expected return)-(risk-free rate)
• There are three kinds of portflios
• Low risk (L), medium risk (M) and high risk(H)
• The risk and return values given in the table
  shows that
• Riskier portfolios have higher returns

13
Risk and Risk Aversion (Continued)

• How does the investor choose among these
  portfolios?
• Of course, every investor wants to choose a
  portfolio with high return and low risk but as
  can be seen the risk increases with the return
• We need to quantify the rate at which investors
  are willing to trade-off return against risk
• The Utility Concept
• Each investor assigns a welfare (utility) score
  to different investment portfolios based on the
  expected return and risk
• Higher utility values are assigned to portfolios
  with more attractive risk and return profile
• The utility scores are higher for higher returns
  and lower for higher volatility (standard
  deviation)

14
Risk and Risk Aversion (Continued)

• The Utility Function
• Positive relation with expected return and
  negative relation with risk

• Where
• U utility
• E ( r ) expected return on the asset or
  portfolio
• A coefficient of risk aversion
• s2 variance of returns
• ½ is a scaling factor
• Rates of returns must be expressed in decimals

15
Risk and Risk Aversion (Continued)

• The Utility score of the risk-free asset is equal
  to its return
• E(r)0.05
• s2 0 (no penalty for risk)
• U E(r)
• A the extent to which the variance of risky
  portfolios lowers utility
• investors degree of risk aversion
• more risk averse investors (larger A) penalize
  risky investments more

16
Risk and Risk Aversion (Continued)

• Now we will go back to the first table and try
  to decide which portfolio to choose based on the
  utility levels
• The one with the highest utility level is going
  to be chosen
• We have three different investor risk aversion
  levels and three differenent portfolios with
  different risk and return values

17
Risk and Risk Aversion (Continued)

• Lets examine one utility score for one
  coefficient of risk aversion
• A 2
• Utility Score of Portfolio L
• E(r)0.07
• s 0.05
• UE(r)-(1/2)As2
• U0.07-(1/2)20.0520.0675

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Risk and Risk Aversion (Continued)

• For each investor with different coefficients of
  risk aversion we choose a single portfolio
• One with the highest utility score
• For A 2.0, we choose portfolio H
• For A 3.5, we choose portfolio M
• For A 5.0, we choose portfolio M
• High-risk-portfolio is chosen by the the lowest
  degree of risk aversion
• All portfolios are better than the risk-free
  asset since the utilities assigned to them are
  higher than the utility assigned to the risk-free
  asset

19
Risk and Risk Aversion (Continued)

• Certainty Equivalent Rate
• Utility score of risky portfolios
• The rate that risk-free investments would need to
  offer to provide the same utility score as the
  risky portfolio
• The rate that if earned with certainty would
  provide a utility score equivalent to that of the
  portfolio in question
• The portfolio is desirable only if its CERgtthat
  of the risk-free alternative
• Risk-neutral
• A0
• The level of risk is irrelevant, no penalty for
  risk
• CERE(r)
• Risk-lover
• Alt0
• In the utility function the expected rate of
  return is adjusted upwards in order to take into
  account the fun of confronting risk

20
Risk and Risk Aversion (Continued)

• A risk-averse investor will always prefer
  portfolio P to any portfolio in the 4th quadrants
• On the other hand portfolios in the 1st quadrant
  are preferred to P
• Mean-Variance (M-V) Criterion
• E(rA) E(rB) and
• sA sB
• rule out the equality (one inequality is strict)

21
Risk and Risk Aversion (Continued)

• The desirability of portfolios in 2nd and 3rd
  quadrants depends on the nature of investors
  risk-aversion
• An investor can identify other portfolios that
  are equally desirable as portfolio P (same
  utility) such as portfolio Q
• Higher risk-higher return
• Lower risk-lower return
• Indifference curve Connects all portfolio
  points with the same utility value
• equally preffered portfolios lie on the same
  indifference curve

22
Risk and Risk Aversion (Continued)

• Estimating Risk Aversion
• Observe individuals decisions when confronted
  with risk
• Observe how much people are willing to pay to
  avoid risk, for example when they buy insurance
  against large losses
• Example
• An investor with risk aversion factor
  (coefficient of risk aversion) A
• Whole wealth is in a piece of real estate
• Probability p whole wealth will be gone
• Rate of return -100

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Risk and Risk Aversion (Continued)

• Probability 1-p whole wealth will stay same
• Rate of return 0
• Expected rate of return
• E(r)p(-1) (1-p)0-p
• Expected loss is a fraction p of the value of
  real estate
• Variance, the deviation from expectation (r-E(r))
• With probability p -1-(-p) p-1
• With probability 1-p 0-(-p)p
• Variance of the rate of return
• s2(r)p(p-1)2(1-p)p2p(1-p)
• Utility
• UE(r)-(1/2)A s2(r)
• U-p-(1/2)Ap(1-p)

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Risk and Risk Aversion (Continued)

• The question how much an individual would be
  willing to pay for insurance against potential
  loss?
• Insurance company offers to cover the loss for a
  fee of v dollars
• Certain negative rate of return of the insurance
  policy v
• U-v (since s2(r) is 0)
• In order to decide how much will our investor pay
  for the policy at maximum.
• U-p-(1/2)Ap(1-p)-v
• v p1(1/2)A(1-p)

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Risk and Risk Aversion (Continued)

• Risk-neutral investor (A0)
• vp
• Prefers to pay an amount equal to expected loss
  to the insurance company
• As risk aversion increases (A increases) the
  investor is willing to pay more as insurance
  premium

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Capital Allocation Across Risky and Risk-Free
Portfolios

• Investors construct their portfolios using
  securities from all asset classes, such as
  T-bills, long-term bonds and stocks
• The risk of these assets are different
• Investor can control the risk of their portfolio
  by investing portion of it into safe money market
  securities such as T-bills
• Asset allocation choice
• The most basic asset allocation choice is between
  the risk-free money market securities and other
  risky asset classes

27
Capital Allocation Across Risky and Risk-Free
Portfolios (Continued)

• P Investors portfolio of risky assets
• Stocks and Long-term Bonds
• F Investors portfolio of risk-free assets
• Investors Total PortfolioPF
• Example
• Total portfolio value 300,000
• Risk-free value 90,000
• Risky (Equities (E) and Long-term bonds (B))
  210,000
• B96,600
• E113,400

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Capital Allocation Across Risky and Risk-Free
Portfolios (Continued)

• Weight of equity and long-term bond in the risky
  portfolio
• wE113,400/210,0000.54
• wB96,600/210,0000.46
• Weight of the risky portfolio P in the complete
  portfolio
• y210,000/300,0000.70 (risky assets)
• 1-y90,000/300,0000.30 (risk-free assets)
• Weight of each risky asset class (E and B) in the
  complete portfolio
• E113,400/300,0000.378
• B96,600/300,0000.322
• EB0.70

29
Capital Allocation Across Risky and Risk-Free
Portfolios (Continued)

• Example
• Suppose the investor wants to decrease the risk
  of the portfolio by decreasing the proportion of
  the risky assets in the complete portfolio
  (y0.70 to y0.56)
• Total value of risky portfolio in the new case
• 0.56300,000168,000
• Have to sell 42,000 (210,000-168,000) worth of
  risky assets to buy more risk-free assets

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Capital Allocation Across Risky and Risk-Free
Portfolios (Continued)

• Total value of risk-free asset in the new case
• 300,000(1-0.56)132,000
• 90,00042,000132,000
• The proportions of risky assets are unchanged
• In order to keep the weights same
• 0.5442,00022,680 worth of E
• 0.4642,00019,320 worth of B was sold

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The Risk-Free Asset

• Only the government can issue default-free bonds
• Guaranteed real rate only if the duration of the
  bond is identical to the investors desire
  holding period
• If you want to sell it before it matures the
  price you get is not guaranteed
• secondary market
• T-bills viewed as the risk-free asset
• Less sensitive to interest rate fluctuations
  because of their short-term maturity
  characteristic

32
Portfolios of One Risky Asset and a Risk-Free
Asset

• We are now interested in the risk-return
  combination available to investors
• The opportunities available to investors given
  the features of the broad asset markets
• Its possible to split investment funds between
  safe and risky assets
• The composition of risky assets is already
  decided

33
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• Question What portion of the complete portfolio
  is going to be allocated to the risky portfolio
  P?
• What is y?
• Risky rate of return of P rP
• Expected rate of return of P E(rP)
• Standard deviation of P sP
• Risk-free rate of return rf

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Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• Example
• E(rP)15
• sP22
• rf7
• The risk premium on the risky asset
• E(rP)-rf8
• The return on the complete portfolio (C) with a
  proportion y in the risky portfolio and (1-y) in
  the risk-free asset
• rCyrP(1-y)rf

35
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• Take the expectation
• E(rC) yE(rP)(1-y)rf
• rfyE(rP)-rf7y(8)
• The base rate of return for any portfolio is the
  risk-free rate
• Investors are assumed to be risk averse so they
  take on a risky position when there is a positive
  risk premium
• So, they also expect to earn a risk premium that
  depends on the risk premium of the risky
  portfolio and the investors position in that
  risky asset, y

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Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• The standard deviation of the complete portfolio
• sCysP22y
• The standard deviation of a complete portfolio
  with a risky and a risk-free asset is the
  standard deviation of the risky asset multiplied
  by the weight of the risky asset in that
  portfolio
• Note
• s2Cy2 s2P(1-y)2 s2F2y(1-y)cov(P,F)
• y2 s2P
• Since s2F 0 and cov(P,F)0
• Cov(P,F)E(rP-E(rP) (rf-E(rf)0
• Since E(rf)rf

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Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• Now we have
• E(rC)rfyE(rP)-rf7y(8)
• sC22y
• Plot the portfolio characteristics (for different
  ys) in the expected return-standard deviation
  plane
• The risk-free asset is on the vertical axis since
  its standard deviation is 0.
• The portfolio P is also plotted with expected
  return of 15 and standard deviation of 22

38
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• If y1 then the complete portfolio is P
• If y0 then the complete portfolio is the
  risk-free portfolio F
• The solid, straight line called capital
  allocation line (CAL)
• CAL (up to point P) connects the mid-points
  where y is between 0 and 1

39
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• The equation for the straight line between points
  F and P
• E(rC)rfyE(rP)-rf
• where ysC/sP
• E(rC)rf (sC/sP)E(rP)-rf
• In this example
• E(rC)7(8/22)(sC)
• The expected return of the complete portfolio is
  a straight line with intercept rf and slope S
• SE(rP)-rf/ sP 8/220.36
• The expected return of the complete portfolio is
  a linear function of its standard deviation

40
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• The figure 6.4 shows the investment opportunity
  set
• Set of feasible expected return and standard
  deviation pairs of all portfolios resulting from
  different values of y

41
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• The Capital Allocation Line (CAL) is a straight
  line originating at rf and going through the
  point P
• Shows all the risk-return combinations available
  to the investor with given risk-free rate and
  expected return and standard deviation
  combinations of a risky portfolio
• Slope of CAL (S) shows the increase in the
  expected return of the complete portfolio per
  unit of additional standard deviation
• Incremental return per incremental risk
• It is also called reward-to-volatility ratio

42
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• Example
• An equally divided portfolio between risk-free
  and risky asset
• y0.5
• E(rC)rfyE(rP)-rf7y(8)
• 7(0.5)811
• sC22y22(0.5)11
• Midway between F and P

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Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• Example (Capital Allocation Line with Leverage)
• We are interested in the points on the CAL to the
  right of point P
• If investors can borrow at the risk-free rate
  then they can construct portfolios that may be
  plotted on the CAL to the right of point P
• Investment budget300,000
• Investor borrows120,000
• Invest total available funds in the risky assets
• y420,000/300,0001.4

44
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• (1-y)1-1.4-0.4
• Short (borrowing) position in the risk-free asset
• Investor borrows at the risk-free rate (7)
  instead of lending
• E(rC)rfyE(rP)-rf
• 71.4(15-7)
• 18.2
• sC22y22(1.4)30.8
• SE(rP)-rf/ sP(18.2-7) /30.80.36
• Note that in this example not only the expected
  return but also the standard deviation (risk) of
  the portfolio increases

45
Portfolios of One Risky Asset and a Risk-Free
Asset (Continued)

• In reality the lending and borrowing rates are
  different
• Usually the borrowing rate for investors are
  higher than the lending rate
• This situation makes the CAL kinked as shown in
  the figure
• The portion from F to P is same as before but the
  part to the right of P has a different slope now.
  Slope to the left of P is still 0.36, where slope
  to the right of P is 0.27 (6/22)

46
Risk Tolerance and Asset Allocation

• The investor must choose one optimal portfolio,
  C, from the set of feasible choices on CAL
• Trade-off between risk and return
• Individual investors have different risk aversion
  
• Given the same opportunity set (risk-free rate
  and reward-to-volatility ratio) (CAL) they will
  choose different positions in the risky asset.
• The more risk averse investor will prefer to have
  more risk-free asset and less risky asset

47
Risk Tolerance and Asset Allocation (Continued)

• Expected return of the complete portfolio is
  given by
• E(rC) rfyE(rP)-rf
• Variance is
• s2C y2 s2P
• An investor tries to maximize his/her utility by
  choosing the best allocation of risky asset
• The utility function is
• UE(r)-(1/2)As2
• As the y increases the return increases
• However the risk also increases
• The Utility can increase or decrease

48
Risk Tolerance and Asset Allocation (Continued)
49
Risk Tolerance and Asset Allocation (Continued)
50
Risk Tolerance and Asset Allocation (Continued)

• We are trying to maximize the utility
• Max U with respect to y
• UE(rC)-(1/2)As2C
• rfyE(rP)-rf-(1/2)Ay2s2P
• We take the derivative of this equation with
  respect to y and set it equal to 0
• y E(rP)-rf/As2P
• Optimal proportion of risky asset is directly
  proportional to the risk premium of risky asset
  and inversely proportional to the level of risk
  aversion and level of risk.

51
Risk Tolerance and Asset Allocation (Continued)

• Example
• rf7
• E(rP)15
• sP22
• A4
• In order to find the optimal risky asset
  proportion (y), use the percentages as decimals
  in the formula derived
• y(0.15-0.07)/(40.222)0.41

52
Risk Tolerance and Asset Allocation (Continued)

• Thus the investor with a coefficient of risk
  aversion equals to 4 will maximize his/her
  utility if he/she invests 41 of his/her complete
  portfolio in risky assets and 59 of his/her
  complete portfolio in risk-free assets
• Given the risk-free rate, expected return and
  standard deviation of the risky portfolio
• E(rC) rfyE(rP)-rf70.41(15-7)10.28
• sC y sP0.41229.02

53
Risk Tolerance and Asset Allocation (Continued)

• Next we try to find the combination of expected
  returns and standard deviations that gives the
  same utility level to an investor with a given
  coefficient of risk aversion
• Plotting the expected return and standard
  deviation combinations that give the same utility
  (with a given coefficient of risk aversion) gives
  the indifference curve
• On this curve the investors are indifferernt
  between different risk-return combinations

54
Risk Tolerance and Asset Allocation (Continued)

• How to construct an indifference curve
• Consider an investor with A4
• Holds all wealth in risk-free asset (rf5, sf0)
• UE(rf)-(1/2)As2f0.05
• Keep U constant and try to find the expected
  returns for different levels of risk (standard
  deviation)
• UE(r)-(1/2)As2
• E(r) U(1/2)As2

55
Risk Tolerance and Asset Allocation (Continued)
56
Risk Tolerance and Asset Allocation (Continued)

• We have two utility curves for each investor
  with different coefficients of risk aversion
• The intercepts are same for each investor but
  different for utility levels
• Each intercept corresponds to a different
  utility level (risk-free rate)
• Any investor prefers a portfolio on a higher
  indifference curve, the one with higher utility
• Portfolios on higher indifference curves offers
  higher expected return for any given level of risk

57
Risk Tolerance and Asset Allocation (Continued)

• The Figure 6.7 in the book shows that more
  risk-averse investors have steeper indifference
  curves than less risk-averse investors
• More risk-averse investors require a greater
  increase in their expected return to compensate
  for an increase in portfolio risk
• Since higher indifference curves correspond to
  higher utility any investor wants to find the
  complete portfolio on the highest possible
  indifference curve
• Constraint is the CAL

58
Risk Tolerance and Asset Allocation (Continued)

• In order to find the complete portfolio that has
  the highest utility the indifference curves of an
  investor with given A, are moved to the
  investment opportunity set represented by CAL.
• The optimal complete portfolio is where the
  highest indifference curve is tangent to CAL.
• At C, we have the standard deviation and
  expected return of the optimal complete portfolio

59
Risk Tolerance and Asset Allocation (Continued)

• For the expected returns on different utility
  functions
• E(r)U(1/2)As2
• where A4
• For the expected returns on CAL
• E(rC)rf (sC/sP)E(rP)-rf
• 7(8/22)(sC)

60
Risk Tolerance and Asset Allocation (Continued)
61
Passive Strategies The Capital Market Line

• The Capital allocation Line (CAL) is derived with
  a risk-free and risky portfolio, P
• Determination of which assets to include in P is
  a result of passive and/or active strategy
• Passive strategy involves a decision that avoids
  any direct or indirect security analysis
• It is holding highly diversified portfolios
• efficient market hypothesis

62
Passive Strategies The Capital Market Line
(Continued)

• A natural candidate for a passively held risky
  asset would be a well-diversified portfolio of
  common stocks
• Because a passive strategy requires devoting no
  resources to acquiring information on any
  individual stock or group we must follow a
  neutral diversification strategy
• The most popular value-weighted index of US
  stocks is the StandardPoors Composite Index of
  500 large capitalization US corporations (SP
  500)

63
Passive Strategies The Capital Market Line
(Continued)

• The advantages of passive strategy
• Cheaper than active strategy because of costs to
  acquire information needed to generate optimal
  active portfolio or the fees you pay to the
  professionals
• Free-rider benefit If EMH holds then most assets
  will be fairly priced, so the well-diversified
  portfolio is going to be a fair buy

64
Passive Strategies The Capital Market Line
(Continued)

• Capital Market Line (CML) The Capital Allocation
  Line provided by 1-month T-Bills and a broad
  index of common stocks
• A passive strategy involves investment in two
  passive portfolios risk-free short-term T-bills
  and a fund of common stocks that mimics a broad
  market index
• This strategy generates an investment opportunity
  set that is represented by the CML