General Principles

1
General Principles

• Evaluation of a random cash flows
• Directly, using expected value and variance.
• Indirectly, using combination of other known or
  estimated cash flow.

2
Utility Functions

• Utility function provides a procedure for ranking
  random wealth levels.
• Denote by U a utility function and by x,y outcome
  random wealth variables.
• U- utility function
• Defined on real numbers and gives a real value
• Utility for random variable x is better then for
  y if E(U(x)) gt E(U(y))
• U(x) is an increasing continuous function, that
  is
• if x gt y then U(x) gt U(y)

3

• Types of Utilities functions
• Exponential
• U(x) - e-ax, where a gt 0
• Logarithmic
• U(x) ln(x), where x gt 0
• Power
• U(x) bxb, where b lt 1, b ? 0 (if b 1, then
  the
• 
  riskneutral utility)
• Quadratic
• U(x) x bx2, where some b gt 0

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• Example (The Venture Capitalist)
• Two investment alternatives
• (1) Buy T-bill which will give 6M for sure.
• (2) An alternative with random outcomes 10M,
  5M, and 1M with corresponding probabilities of
  .2, .4, and .4 .
• Utility U(x)x1/2 . Expected utilities
• (1) v6 2.45
• (2) .2 x v 10 .4 x v 5 .4 x v 1
• .2 x 3.16 .4 x 2.24 .4 1.93
• 2.45 gt 1.93 gt The first alternative is
  preferred.

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• Equivalent Utility Functions
• Utility functions are used to rank alternatives.
• Equivalent utility functions give identical
  ranking.
• Can be proved that
• V(x) a U(x) b
  ()
• is equivalent to U(x) if agt0.
• Transformation () is the only transformation
  which preserves ranking of all random outcomes.

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• Risk Aversion
• The main purpose of a utility function is to
  provide a systematic way to rank alternatives
  that captures the principle of risk aversion.
• Concave utility and risk aversion.
• A function U defined on am interval a,b of
  real numbers is said to be concave if for any a
  with
• 0 lt a lt 1 and any x and y in a,b there holds
• Ua x (1- a)y gt a U(x) (1 - a)U(y) .
• Risk aversion, usually means strict concavity
  (gt).

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• A utility function U is said to be risk averse on
  a,b if it is concave on a,b. If U is concave
  everywhere, it is said to be (just) risk averse.
• Average utility of outcomes is less the utility
  of average outcomes

9

• Example
• Suppose you face two options.
• (1) Toss of a coin heads, you win 10 tails,
  you win nothing.
• (2) amount M for certain.
• Utility function for money is x - .04x2.
• (1) EU(x) ½(10 - .04 102) ½ 0 3.
• (2) M - .04M2
• If M 5, for example, then the (2) utility value
  is 4, which is greater then value of the (1)
  utility value. This means that you would prefer
  to have 5 for sure rather then 50-50 chance of
  getting 10 or nothing.

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• Example (Contd)
• We solve M - .04M2 3. This gives M 3.49.
  Therefore there are two indifferent choices
• to get 3.49 for sure or 50-50 chance of getting
  10 or 0.

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• Derivatives
• U(x) is increasing if U(x) gt 0.
• U(x) is strictly concave if U(x) lt 0.
• Example
• U(x) -e- a x
• U(x) a e- a x gt 0 gt U is increasing
• U(x) - a 2e- a x lt 0 gt U is concave.

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• Risk Aversion Coefficients
• The degree of risk aversion is defined by
• Arrow-Pratt absolute risk aversion coefficient
• a(x) - U(x) /U(x)
• U (x) normalizes the coefficient a(x) is the
  same for all equivalent utility functions. a(x)
  shows how risk aversion changes with the wealth
  level.

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• Example 1
• U(x) -e- a x ,
• U(x) a e- a x ,
• U(x) - a 2e- a x ,
• a(x) a ,
• aversion coefficient is constant for all x .
• Example 2
• U(x) ln x ,
• a(x) 1/x ,
• risk aversion decreases.

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• Certainty Equivalent
• The certainty equivalent of a random wealth
  variable x is defined to be the amount of a
  certain (that is risk-free) wealth that has a
  utility level equal to the expected utility of x.
  The certainty equivalent C of a random wealth
  variable x is the value C satisfying
• U(C) EU(x)
• For concave utility function the certainty
  equivalent of a random outcome x is less or
  equal to the expected value, i.e., C lt Ex .

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EU(x)
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• Picture illustrates the certainty equivalent for
  two outcomes x1 and x2 is. The certainty
  equivalent is found by moving horizontal
  leftwards from the point where the line between
  U(x1) and U(x2) intersects the vertical line
  drawn at E(x).
• Example The certainty equivalent of the 50-50
  chance of winning 10 or 0 is 3.49 because that
  is the value that, if obtained with certainty,
  would have the same utility as reward based on
  the outcome of the coin toss.

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Specification of Utility Functions

• There are systematic approaches for assigning an
  appropriate utility function to an investor.
  Several approaches will be outlined.
• One way to measure an individuals utility
  function is to ask the individual to assign
  certainty equivalents to various risky
  alternatives.

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• Direct Measurement of Utility
• Let A and B are selected as wealth reference
  points.It is proposed that lottery has outcome A
  with probability p and outcome B with probability
  1-p. For various values of p the investor is
  asked how much certain wealth C he/she would
  accept in in place of the lottery. C will vary as
  p changes.
• A lottery with probability p has an expected
  value of e pA (1 p)B. However, a risk
  averse investor would accept less than this
  amount to avoid the risk of the lottery gt C lt e
  .

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• The values of C reported by investor for various
  ps are plotted. The value of C is placed under
  the corresponding e. A curve is function C(e).
• To define a utility function from this diagram,
  we normalize by setting U(A) A and U(B) B.
  The expected utility is
• pU(A) (1 p)U(B) pA (1 p)B,
• which is the same as the expected value e.
  Since C is defined so that U(C) is expected
  utility of the lottery gt U(C) e. Therefore C
  U-1(e) gt curve define by C(e) is inverse of the
  utility function.

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• Example
• Sybil, a moderately successful venture
  capitalist, is anxious to make her utility
  function explicit. A consultant ask her to
  consider lotteries with outcomes of either 1M or
  9M. She is asked to follow the direct procedure
  as the probability of p of receiving of 1M
  varies. For a 50-50 chance of the two outcomes,
  the expected value is 5M, but she assigns a
  certainty equivalent of 4M. Other values she
  assigns are shown in the table on the next slide.

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Table 9.1. Expected Utility Values and Certainty
Equivalents
Here is shown the utility function (U(C) e).
For example U(4) 5. However, the values of C in
the table are not the whole numbers and a new
table of utility values could be constructed by
interpolation in the table. For example U(2)
3.4(2.00 1.96) 2.6(2.56 2.00) / (2.56
1.96) 2.65
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• Parameter Families
• Another method of assigning a utility function is
  to select a parameterized family of functions and
  then determine a suitable set of parameter
  values.
• Many people prefer to use a logarithmic or power
  utility function, since these functions have the
  property that risk aversion decreases with
  wealth. For the logarithmic utility, the risk
  aversion coefficient is a(x) 1/x, and for the
  power utility function U(x) ?x? the coefficient
  is
• a(x) (1 - ?)/x. We will see further that these
  are appropriate utility functions for investors
  concerned with long-term grows in their wealth.

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• Example
• Suppose that an investor has the exponential
  utility functions
• U(x) - e-xa
  .
• Let us ask how much he/she would accept in
  place of a lottery that offers a 50-50 chance of
  winning 1M or 100,000. Suppose the investor
  felt that this was equivalent to a certain wealth
  of 400,000. We then set
• -e-400,000a - .5e-1,000,000a - .5e-100,000a
• where a 1/623,426.

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• Example
• The Table 9.1 results can be expressed compactly
  by fitting a curve to the results. Assume
• U(x) ax? c.
• From normalization
• ac 1
• a9? c 9

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• Example (Contd)
• Consequently,
• a 8/9? -1
• c (9? - 9)/ (9? - 1)
• Using Excel optimizer we can find the best value
  ?
• matching U(C) to e.
• The best value equals ? ½ . gt U(x) 4vx
  3

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• Questionnaire Method
• Sometimes, the best way to deduce the
  appropriate risk factor and utility function is
  to administer a questionnaire. This must include
  questions related to the investors situation,
  investors investment approach, characteristics
  of the market, and the value of a managed fund.
• The risk tolerance is defined by the internal
  feeling toward risk and by investors financial
  environment.

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Utility and Mean Variance Approach

• Mean-variance criterion used by Markowitz can be
  reconciled with expected utility approach by two
  ways
• (1) using a quadratic utility functions,
• (2) making the assumption that returns are
  normally distributed

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• Quadratic Utility
• Suppose that a portfolio has a random wealth y .
• Using quadratic utility function we get
• EU(y) E(ay - (1/2)by2) aE(y) - (1/2)b E(
  y2)
• aE(y) - (1/2)b E( y)2 - (1/2)b
  var(y),
• Recall that var(y) E( y2) - E( y)2 .

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• Quadratic Utility (Contd)
• Let us consider the problem
• maxx E U (y(x))
• Suppose that at the optimal point x , Ey(x )
  R .
• We can reformulate
• maxx E U (y(x))
• maxx aE(y(x)) - (1/2)b E(y(x))2 - (1/2)b
  var(y(x))
• maxx aR - (1/2)b R2 - (1/2)b var(y(x))

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• Quadratic Utility (Contd)
• The last problem is equivalent to
• minx var(y(x))
• s.t. Ey(x) R
• Different mean-variance efficient points are
  obtained by selecting different values for the
  parameters a and b .

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• Normal Returns
• When all returns are normal random variables, the
  mean-variance criterion is also equivalent to the
  expected utility approach.
• Since the probability distribution is completely
  defined by mean M and s
• EU(y)f(M,s)
• If U is risk averse, then f(M,s) will be
  increasing w.r.t. M and decreasing w.r.t. s .
• The portfolio of normally distributed assets is
  also normally distributed.
• The portfolio optimization problem is therefore
  equivalent to minimizing variance with fixed
  return.

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Linear Pricing

• Fundamental property of security pricing
  linearity.
• Security is formalized as a random payoff
  variable.
• Example. Security pays d10 if it rains
  tomorrow, and d-10 if it is sunny (weather
  derivatives electricity, agriculture).

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• Type A Arbitrage (give an example with strips)
• If an investment produces an immediate positive
  reward with no future payoff (either positive or
  negative), that investment is said to be a TYPE A
  ARBITRAGE.
• Linearity follows from the assumption that there
  is no possibility of a type A arbitrage.
• Example. Suppose d is a security with price P.
  Consider the security 2d (pays always twice more
  than d). Suppose that its price Plt2P. Then we
  can buy the 2d security at a reduced price, brake
  in two parts and sell making profit 2P- P. No
  future obligations. Similar argument can be
  reverted that price of double security can not be
  more than 2P (can buy two d securities, combine
  and sell with profit).

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• Similar to the example, if d1 and d2 are
  securities with prices P1 and P2 , the price of
  the security d1d2 must be P1P2 . If the price
  of d1d2 were Plt P1P2 we can purchase the
  combined security with price P then brake it and
  sell with profit. As before this argument can be
  reverted.
• In general Linear Pricing means
• P(a 1d1 a 2d2) a 1 P(d1) a 2 P( d2)

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• Portfolios
• Suppose that there are n securities d1,,dn .
  Portfolio is a vector ? (? 1,, ? n). Payoff of
  the portfolio is a random variable
• d ? 1 d1 ? ndn ,
• under assumption of no A arbitrage, the price of
  the portfolio ? must be
• P(d) ? 1 P(d1) ? n P( dn) .

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• Type B Arbitrage
• If an investment has nonpositive cost but has a
  positive probability yielding a positive payoff
  and no probability of yielding a negative payoff,
  that investment is a TYPE B ARBITRAGE.
• Type B arbitrage pay nothing but have a chance
  to get something.
• Example a free lottery ticket is a type B
  arbitrage.
• Usually, it is assumed that there is no arbitrage
  (both A and B).

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Portfolio Choice

• This section combines results of the previous
  sections and consider a portfolio with the
  expected utility performance function.
• Let x be a random variable. We write xgt0 if the
  variable is never less than 0 and it is strictly
  positive with some positive probability.
• Suppose that U is a strictly increasing utility
  function and initial wealth W is available. There
  are n securities d1,,dn . The investor wishes to
  form a portfolio ? (? 1,, ? n) maximizing the
  expected utility .

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• Mathematical programming problem
• max EU(x)
• s.t. ?? idi x
• xgt0
• ? ? iPi W
• Note summation ? is over n securities in the
  portfolio.

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• Portfolio choice theorem.
• Suppose that U(x) is continuous and increases
  toward infinity as x tends to infinity. Suppose
  also that there is a portfolio ?0 such that
• ? ?0i di gt 0.
• Then, the optimal portfolio problem (on the
  previous transparence) has a solution if and only
  if there is no arbitrage possibility.

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• Proof. We prove only one direction statement.
• (1) Suppose that there is a type A
  arbitrage. Using this portfolio it is possible to
  infinitely increase wealth W. This implies that
  EU(x) does not have a maximum.
• (2) If there is type B arbitrage, it is possible
  to obtain (at zero or negative cost) an asset
  that has payoff xgt0 (with nonzero probability of
  being positive). We can acquire arbitrarily large
  amounts of this asset to increase EU(x)
  arbitrary.
• Hence if there is a solution, there can be no
  type A or B arbitrage.

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• We can characterize the solution. Suppose that
  there is no arbitrage and hence there is an
  optimal portfolio ? . We also assume that the
  corresponding payoff x ? ?i di satisfies
  x gt0.
• We can deduce that the constraint ? ?iPi W is
  active (equality at the solution) otherwise we
  can improve the performance function.
• To derive the equation we substitute x ? ?i di
  in the objective and ignore constraint x gt0 . The
  problem becomes
• max EU(? ?i di )
• s.t. ? ?iPi W

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• Lagrange function for this problem is (see
  appendix B in the book)
• EU(? ?i di ) - ?(? ?iPi - W )
• Differentiation with respect to each ?i and
  equating to zero (at optimal point x ? ?i di
  ) gives
• d( EU(? ?i di ) ) / d(?i) EU( x )
  di ?Pi
• for i 1,,n.

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• We have n equations
• EU(x) di ? Pi ()
• and one more equation
• ? ?iPi W .
• Altogether there are n1 equations for n1
  variables ?1,, ?n and ? .
• If there is a risk-free asset with the return R,
  then () must apply when di R and Pi 1 . Thus
• ? EU(x) R ()

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• Substituting () into () gives
• EU(x) di / (R EU(x) ) Pi
• Summary Portfolio pricing equation.
• If x ? ?i di is the solution of the
  optimal portfolio problem on page 7, then
• EU(x) di ? Pi
• for i 1,2,,n where ? gt0 . If there is a
  risk-free asset with the return R, then
• EU(x) di / (R EU(x) ) Pi
• for i 1,2,,n.

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• Example 9.5 (A film venture)
• Three possible incomes and possibility to make
  20 risk free. Investor want to know how much to
  invest . Initial prices of both instruments 1.
• The expected return is
• 0.330.410.30 1.3 gt 1.2

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• Example 9.5 (Contd)
• Utility U(x) ln (x) . Let (?1, ?2) be weights
  of risky and risk free instruments. The
  optimization problem
• max EU(x) .3 ln(3?11.2 ?2) .4 ln(?11.2 ?2)
• .3 ln(1.2 ?2)
• s.t. ?1 ?2 W

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• The portfolio pricing equation EU(x) di ?
  Pi gives
• .9/(3?11.2 ?2) .4/(?11.2 ?2) ?
• .36 /(3?11.2 ?2) .48/(?11.2 ?2) .36 ln(1.2
  ?2) ?
• By solving these equations together with
• ?1 ?2 W
• we obtain ?10.089W , ?20.911W , and ? 1/ W .

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Finite State Models

• Finite number of possible states (outcomes)
  1,2,,S
• Example. Two states it rains tomorrow, or it is
  sunny.
• Initial point States
  

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• States define uncertainty in a vary basic manner.
• Security is a vector of the form dd1,,dS ,
• where di is a payoff at state i .
• P is security price .
• Example. Security pays d110 if it rains
  tomorrow, and d1-10 if it is sunny. This
  security is represented by the vector d10,-10
  .

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• State Prices
• S elementary state securities es0,,0,1,0,,0).
  Payoff is obtained in only one state.
• ?s denotes the price of elementary security es .
• The security dd1,,dS can be expressed as a
  combination of elementary state securities
• d ? ds es
• By linearity of pricing price P of d equals
• P ? ds ?s

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• Positive State Prices
• If a complete set of elementary securities exists
  or can be constructed as a combination of
  existing securities, it is important that their
  prices be positive. Otherwise there would be an
  arbitrage opportunity.
• The condition of no arbitrage is equivalent to
  the existence of positive state prices.
• Positive State Prices Theorem.
• A set of positive state prices exists if and
  only if there are no arbitrage opportunities.

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• Proof of Positive State Prices Theorem
• 1. (Positive prices gt no arbitrage)
• Suppose the there are positive state prices and
  that the security with d0 can be constructed
• (ds 0 for s1,2,,S).
• Since P ? ds ?s , ?s gt0, and ds 0 for
  s1,2,,S, we have P 0 . Also, Pgt0 if at least
  one ds gt0, i.e., there is no B type arbitrage.

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• Proof of Positive State Prices Theorem (Contd)
• 2. (No arbitrage gt positive prices)
• Assume that there is no arbitrage.
• Assume that there is a portfolio ?0 (?01 ,,
  ?0s ) such that ? ?0s ds gt 0 .
• Assign state probabilities ps 0 for s1,2,,S
• such that ? ps 1 .
• Select a strictly increasing utility function
  U(x) .
• Since there is no arbitrage there is an optimal
  portfolio (see section 9.8). Assume that optimal
  portfolio payoff x gt 0.

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• Proof of Positive State Prices Theorem (Contd)
• Portfolio pricing equation (Pportfolio price,
  ?Lagrange multiplier)
• EU(x) d ? P gt
• P 1/ ? ? ps U(x)s ds
• where U(x)s is value of U(x) in state s.
• Define ?s (ps U(x)s ) / ? .
• Consequently, ?s gt0, because, psgt0, U(x)s
  gt0, and ? gt0 .

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• Example 9.8 (Film venture)
• Three possible incomes and possibility to make
  20 risk free. Investor want to know how much to
  invest. Initial prices of both instruments 1.
• Optimal values are found in Example 9.5
  ?10.089W , ?20.911W , and ?1/ W . Assuming W1
  we have ?10.089, ?20.911 , and ?1 .

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• Example 9.8 (Contd)
• By construction ?s (ps U(x)s ) / ?
• Utility in state 1 ln(3?11.2 ?2)
• Utility in state 2 ln(?11.2 ?2)
• Utility in state 3 ln(1.2 ?2)
• ?1 .3/(3?11.2 ?2) .221
• ?2 .4 /(?11.2 ?2) .338
• ?3 .3 /(3?11.2 ?2) .338

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• Example 9.69.9 (Residual Rights)
• Comparing to the previous example, there is one
  more instrument residual rights, which gives
  return 60 if the film is successful.
• Utility U(x) ln (x) .
• Initial prices of all instruments 1 .
• Expected utility
• EU(x) .3 ln(3?11.2 ?26 ?3) .4 ln(?11.2
  ?2)
• .3 ln(1.2 ?2)

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• Example 9.69.9 (Contd)
• Portfolio pricing equation EU(x) d ? P
• .9/(3?11.2 ?2 6 ?3) .4/(?11.2 ?2) ?
• .36 /(3?11.2 ?2 6 ?3).48/(?11.2 ?2)
  .36ln(1.2 ?2) ?
• 1.8/(3?11.2 ?26 ?3) ?
• ?1 ?2 ?3 W
• Solution ?1-1.0W , ?21.5W , ?30.5W, and
• ?1/ W . When W1, ?1-1, ?21.5, ?30.5, and
• ?1.

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• Example 9.69.9 (Contd)
• Instrument 1 (film venture) d1 3,1,0
• Instrument 2 (risk free) d2 1.2, 1.2, 1.2
• Instrument 3 (residual rights) d3 6, 0, 0
• Linearity of pricing Pi ? dsi ?s
• 3 ?1 ?2 1
• 1.2 ?1 1.2 ?2 1.2 ?3 1
• 6?1
  1
• This system has the solution
• ?1 1/6, ?2 1/2, ?3 1/6.
• Therefore
• P 1/6d1 1/2d2 1/6d3
• E.g., the price of basic venture P 3/6 1/2
  1.

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Risk-Neutral Pricing

• Suppose the there are positive state prices ?s gt0
• for s1,2,,S. Then the price of security
  dd1,,dS equals
• P ? ds ?s
• Normalize state prices so their sum equals 1.
• Let ?0 ? ?s and q s ?s / ?0 .
• Pricing formula
• P ? ds ?s ?0 ? ds ?s / ?0 ?0 ? qsds

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• Quantities qs , s1,2,,S can be thought as
  (artificial) probabilities since they are
  positive and sum to 1.
• Pricing equation
• P ?0 ? qsds ?0 E (ds)
• where E? is an expectation with respect to
  artificial probabilities.
• The value ?0 has a useful interpretation.
  Since ?0 ? ?s , ?0 is the price of security
  1,,1 that pays 1 in every state a risk free
  bond. By definition its price is 1/R, where R is
  a risk free return.

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• Thus we can write a pricing formula which we term
  neutral pricing (price is linear with respect to
  probabilities)
• P ?0 E (ds) (1/R) E (ds)
• qs , s1,2,,S are called risk neutral
  probabilities

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• Three ways to calculate neutral probabilities.
• Can be found by multiplying prices by risk free
  rate
• q s ?s / ?0 ?s / (1/R) ?s R
• If the positive state prices were found from a
  portfolio problem
• ?s (ps U(x)s ) / ?
• ?0 ? ?s ? (ps U(x)s ) / ?
• qs ps U(x)s / ? (ps U(x)s )
• If there are n states, and n securities, then the
  risk-neutral probabilities can be found by
  solving system of equations
• Pi(1/R) ? qs dsi
• The expected return is
• 0.330.410.30 1.3 gt 1.2

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• Example 9.10 (Film venture)
• We found
• ?1 1/6, ?2 1/2, ?3 1/6
• Multiplying by risk free rate 1.2 we get
• q1 0.2, q2 0.6, q3 0.2
• Hence the price of a security with payoff
• d1, d2, d3 is
• P 0.2 d1 0.6 d2 0.2 d3 /1.2