Topic 1 Risk Aversion and Capital Allocation to Risky Assets

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Topic 1 Risk Aversion and Capital Allocation to
Risky Assets

• Risk with simple prospects
• Investors 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

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Risk with Simple Prospects

• The presence of risk means that more than one
  outcome is possible.
• A simple prospect is an investment opportunity in
  which a certain initial wealth is placed at risk,
  and there are only two possible outcomes.
• Take as an example initial wealth, W, of
  100,000, and assume two possible results in one
  year.

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? The expected end-of-year wealth

• The expected profit
• 122,000 - 100,000 22,000.

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• The variance of end-of-year wealth
• (the expected value of the squared deviation
  of each possible outcome from the mean)

• The standard deviation of end-of-year wealth
• (the square root of the variance)

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? 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.
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? The expected marginal, or incremental,
profit of the risky investment over investing
in safe T-bills is
22,000 - 5,000 17,000

• One can earn a risk premium of 17,000 as
• compensation for the risk of the investment.

• One of the central concerns of finance theory is
  the measurement of risk and the determination of
  the risk premiums that investors can expect of
  risky assets in well-functioning capital markets.

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Investors View of Risk

• Risk averseConsiders only risk-free or risky
  prospects with positive risk premia.
• Risk neutralFinds the level of risk irrelevant
  and considers only the expected return of risky
  prospects.
• Risk loverAccepts lower expected returns on
  prospects with higher amounts of risk.

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Risk Aversion and Utility

• Assume that each investor can assign a welfare,
  or utility, score to competing investment
  portfolios (collections of assets) based on the
  expected return and risk of those portfolios.
• The utility score may be viewed as a means of
  ranking portfolios. Higher utility values are
  assigned to portfolios with more attractive
  risk-return profiles. Portfolios receive higher
  utility scores for higher expected returns and
  lower scores for higher volatility.

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• One utility function that is commonly used
• where U utility value E(r)
  expected return ?2 variance of
  returns A index of the investors
  risk aversion
• ?Consistent with the notion that utility is
  enhanced by high expected returns and diminished
  by high risk.

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• Example 1
• Choose between
• (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
• 0.0466.
• ? Choose T-bills.

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Example 2
Risk-free rate 5
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• The extent to which variance lowers utility
  depends on A, the investors degree of risk
  aversion. More risk-averse investors (who have
  the larger As) penalize risky investments more
  severely.
• Investors choosing among competing investment
  portfolios will select the one providing the
  highest utility level.

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Trade-off between Risk and Return

• Portfolio P has expected return E(rP) and
  standard deviation ?P.

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• P is preferred by risk-averse investors to any
  portfolio in quadrant IV because it has an
  expected return ? any portfolio in that quadrant
  and a standard deviation ? any portfolio in that
  quadrant.
• Conversely, any portfolio in quadrant I is
  preferable to portfolio P because its expected
  return ? Ps and its standard deviation ? Ps.

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• The mean-standard deviation or mean-variance
  (M-V) criterion
• A dominates B if

and
and at least one inequality is strict (rules out
the equality).
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Expected Return
Increasing Utility
Standard Deviation
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• The indifference curve
• 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.

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• 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

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Asset Risk versus Portfolio Risk
Asset risk

• Best Candy stock has the following possible
  outcomes

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• The expected return of an asset is a
  probability-weighted average of its return in all
  scenarios

where Pr(s) the probability of scenario s
r(s) the return in scenario s ?
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• The variance of an assets returns is the
  expected value of the squared deviations from the
  expected return

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Portfolio risk

• The rate of return on a portfolio is a weighted
  average of the rates of return of each asset
  comprising the portfolio, with portfolio
  proportions as weights.
• The expected rate of return on a portfolio is a
  weighted average of the expected rate of return
  on each component asset.

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• SugarKane stock has the following possible
  outcomes

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• Consider a portfolio when it splits its
  investment evenly between Best Candy and
  SugarKane

• Covariance
• Measures how much the returns on two risky
  assets move in tandem.
• A positive covariance means that asset
  returns move together.
• A negative covariance means that they vary
  inversely.

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• SugarKanes returns move inversely with
• Bests.

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? Correlation coefficient Scales the
covariance to a value between -1 (perfect
negative correlation) and 1 (perfect
positive correlation).

• This large negative correlation (close to -1)
• confirms the strong tendency of Best and
• SugarKane stocks to move inversely.

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? Portfolio variance (2-asset case)
where wi fraction of the portfolio
invested in asset i variance of the
return on asset i

? With equal weights in Best and SugarKane
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• ? 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.
• Hedging involves the purchase of an asset that
  
• is negatively correlated with the existing
  portfolio.
• This negative correlation reduces the overall
  risk of the portfolio.

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• ?p 4.83 is much lower than ?Best or
  ?SugarKane.
• ?p 4.83 is lower than the average of ?Best and
  ?SugarKane (16.82).
• Portfolio provides average expected return but
  lower risk.
• Reason negative correlation.

 
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Capital Allocation Across Risky and Risk-free
Portfolios

• The choice of the proportion of the overall
  portfolio to place in risk-free securities versus
  risky securities.
• Denote the investors portfolio of risky assets
  as P and the risk-free asset as F.
• For now, we take the composition of the risky
  portfolio as given and focus only on the
  allocation between it and risk-free securities.

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• Example

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Risk-free assets

• Treasury bills
• Short-term, highly liquid government securities
  issued at a discount from the face value and
  returning the face amount at maturity.
• Their short-term nature makes their values
  insensitive to interest rate fluctuations.
  Indeed, an investor can lock in a short-term
  nominal return by buying a bill and holding it to
  maturity.
• Inflation uncertainty over the course of a few
  weeks, or even months, is negligible compared
  with the uncertainty of stock market returns.

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• Money market instruments
• Commercial paper (CP)
• Short-term unsecured debt note issued by large,
  well-known companies.
• Certificate of deposit (CD)
• Time deposit with a bank.
• Virtually free of interest rate risk because of
  their short maturities and are fairly safe in
  terms of default or credit risk.

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Capital Allocation Line

• Suppose the investor has already decided on the
  composition of the risky portfolio.
• Now the concern is with the proportion of the
  investment budget, y, to be allocated to the
  risky portfolio, P.
• The remaining proportion, 1 - y, is to be
  invested in the risk-free asset, F.

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• Let rP risky rate of return on P
• 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
• With y in the risky portfolio and 1 - y in the
  risk-free asset, the rate of return on the
  complete portfolio C

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Interpretation 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
investors position in the risky asset, y.
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• Recall Portfolio variance (2-asset case)

?
?

• The standard deviation of the portfolio is
• proportional to both the standard deviation
  of the
• risky asset and the proportion invested in
  it.

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• The capital allocation line (CAL)

• - shows all feasible risk-return combinations
• of a risky and risk-free asset to investors.
  

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• The slope of the CAL

• equals the increase in the expected return of
• the complete portfolio per unit of additional
  standard deviation (i.e. incremental return per
  incremental risk).
• - also called the reward-to-variability ratio.

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• y 1
• E(rC) rf yE(rP) rf 7 1 ? 8 15
  
• ?C y?P 1 ? 22 22.
• y 0
• E(rC) 7 0 ? 8 7 ?C y?P 0.
• y 0.5
• E(rC) 7 0.5 ? 8 11
• ?C y?P 0.5 ? 22 11
• Will plot on the line FP midway between F P.
  
• The reward-to-variability ratio is S 4/11
  .36
• (precisely the same as that of portfolio P,
  8/22).

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What about points on the CAL to the right of
portfolio P?

• If investors can borrow at the risk-free rate of
  rf 7, they can construct portfolios that may
  be plotted on the CAL to the right of P.
• Suppose the investment budget is 300,000 and our
  investor borrows an additional 120,000,
  investing the total available funds in the risky
  asset.
• This is a leveraged position in the risky asset
  it is financed in part by borrowing.

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? y (420,000/300,000) 1.4. 1 y 1
1.4 -0.4 (short or borrowing position
in the risk-free assets).

• The leveraged portfolio has a higher expected
  return and standard deviation than does an
  unleveraged position in the risky asset.
• Exhibits the same reward-to-variability ratio.

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• Nongovernment investors cannot borrow at the
  risk-free rate.
• The risk of a borrowers default causes lenders
  to demand higher interest rates on loans.
• Therefore, the nongovernment investors borrowing
  cost will exceed the lending rate of rf 7.
• Suppose the borrowing rate is

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• In the borrowing range, the reward-to-variability
  ratio (the slope of the CAL) will be
• The CAL will therefore be kinked at point P.
• To the left of P the investor is lending at
  7, and the slope of the CAL is 0.36.
• To the right of P, where y gt 1, the investor
  is borrowing at 9 to finance extra investments
  in the risky asset, and the slope is 0.27.

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Risk Tolerance and Asset Allocation

• The investor confronting the CAL now must choose
  one optimal portfolio, C, from the set of
  feasible choices.
• This choice entails a trade-off between risk and
  return.
• The more risk-averse investors will choose to
  hold less of the risky asset and more of the
  risk-free asset.

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Recall 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 investors
risk aversion
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Recall 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
yE(rP) rf. The variance of the complete
portfolio is
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• The investor attempts to maximize utility U by
  choosing the best allocation to the risky asset,
  y.
• e.g.

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• To solve the utility maximization problem more
  generally

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• This particular investor will invest 41 of the
  investment budget in the risky asset and 59 in
  the risk-free asset.
• The rate of return of the complete portfolio will
  have an expected return standard deviation
• The risk premium of the complete portfolio

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• Another graphical way of presenting this decision
  problem is to use indifference curve analysis.
• Recall
• The indifference curve is a graph in the expected
  return-standard deviation plane of all points
  that result in a given level of utility.
• The curve displays the investors required
  trade-off between expected return and standard
  deviation.

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e.g. 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.
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• ?

• ?

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.
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• Because the utility value of a risk-free
  portfolio is simply the expected rate of return
  of that portfolio, the intercept of each
  indifference curve (at which ? 0) is called the
  certainty equivalent of the portfolios on that
  curve and in fact is the utility value of that
  curve.
• Notice that the intercepts of the indifference
  curves are at 0.05 and 0.09, exactly the level of
  utility corresponding to the two curves.

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• The more risk-averse investor has steeper
  indifference curves than the less risk-averse
  investor.
• Steeper curves mean that the investor requires a
  greater increase in expected return to compensate
  for an increase in portfolio risk.
• Given the choice, any investor would prefer a
  portfolio on the higher indifference curve, the
  one with a higher certainty equivalent (utility).
• Portfolios on higher indifference curves offer
  higher expected return for any given level of
  risk.

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• The investor thus attempts to find the complete
  portfolio on the highest possible indifference
  curve.
• When we superimpose plots of indifference curves
  on the investment opportunity set represented by
  the capital allocation line, we can identify the
  highest possible indifference curve that touches
  the CAL.
• That indifference curve is tangent to the CAL,
  and the tangency point corresponds to the
  standard deviation and expected return of the
  optimal complete portfolio.

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e.g. A 4.
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• The indifference curve with U .08653 is tangent
  to the CAL.
• The tangency point corresponds to the complete
  portfolio that maximizes utility.
• The tangency point occurs at ?C 9.02 and E(rc)
  10.28, the risk/return parameters of the
  optimal complete portfolio with y 0.41.