Lecture 8

• Expected value (Ex)


  x is the probability that different outcomes will occur multiplied by resulting payoffs

  Ex = q1x1 + q2x2 + … qnxn

  Sum all probabilities = 1
• Variance (risk)


  Sum of probabilties that different outcomes will occur multiplied by the squared deviations from mean of random variable

  If possibilities of random variable x are x1, x2 …. xn and their corresponding values (payoffs) are q1, q2 and qn and the expected value of x is Ex then variance of x is:

  σ^2 = q1(x1 - Ex)^2 + q2(x2 - Ex)^2 + … + qn(xn - Ex)^2
• Attitude toward risk


  U = W^a

  • Utility (U) is a function of wealth (W); parameter (a) defines nature of function
  • Marginal utility of wealth;
  • Assume consumer has initial level of wealth (W1);
  • They farce 50:50 chance of losing certain amount of wealth (W1 - W0)
  • Ex = (1/2)U(W0) + (1/2)U(W1)
  • Risk aversion implies that uncertainty of a risky outcome reduces utility. Risk averse consumer prefers certain payoff to same expected payoff with risky outcome - diminishing marginal utility of wealth
• Attitude towards risk graph

  • Consumer faces choice between a 50:50 change of W0 and W1 or the certainty of W2
  • Expected wealth in both is the same, but expected utility of certain alternative is higher by:
  • U(W2) - [1/2 U(W1) + 1/2 U(W0)]
  • Distance W2 - W3 is a measure of consumer surplus gained by insuring the certain alternative

  
• Attitude towards risk: risk neutral

  Consumer is indifferent between all alternatives offering the same expected value

  • Risk itself does not affect utility directly;
  • As long as expected values are the same, a risk neutral person is indifferent towards risk; constant marginal utility of wealth
• Marginal utility of wealth

  Marginal utility of wealth is constant

  • Not only are expected values of wealth involved in certain and uncertain situations the same, but so too is expected utility;
  • Linear utility function indicated that consumer is indifferent between 50:50 chance of W0 or W1 or W2 with certainty

  
• Attitude towards risk: risk loving


  Consumer is indifferent between all alternatives offering the same expected value;

  • Increasing marginal utility of wealth makes risky alternative preferable
  • Gain of W1 - W2 would add more utility than an equal sized loss (W2 - W0) would subtract from it;
  • 50:50 chance of such a gain or loss has higher expected utility than certainty of W2
  • (W3-W2) measures the money value of the extent to which the uncertain alternatives is preferred

  
• Application to firm’s investment decision: Decision tree analysis 


  Sequential decisions are made and probabilities of different outcomes may be conditional on previous events

  • Objective is to calculate expected monetary values (EMV)
• Decision tree example


  Company deciding whether to test market new product

  • Cost test marketing 3m
  • Probability good result 0.6 (bad 0.4)
  • If results good, prob Hugh sales 0.8 (low 0.2)
  • High sales post test marketing represents NPV of 20m (-10m if low)
  • No test marketing conducted, 50:50 of high or low sales
  • NPV high sales 23m (-7m if low)

  
• How decision tree works

  Nodes are shown as numbered squares;

  • State of nature nodes shown as lettered circles
  • Use backward induction to analyse payoffs
  • Calculate expected NPV at each state of nature node
  • At C: NPV = 0.8 (20) + 0.2(-10) - 3 = 11
  • At D: NPV = 0.3 (20) + 0.7(-10) - 3 = -4
  • Two decision paths, if results are good = national launch, if bad = drop product
  • At A: NPV = 0.6(11) + 0.4(-3) = 5.4m
  • At D: NPV = 0.5(23) + 0.5(-7) = 8m
  • At decision node 1 should go straight for nation launch with no test marketing, decision node 2&3 don’t raise

  
• Problems with the foregoing method

  • It simplifies decision making by restricting decision and state of nature variables to discrete values: market test, not market test; high sales (low sales) anticipated
  • This methodology (as well as those involving sensitivity analysis), do not provide a definitive decision rule
  • They only provide a measure of 'stand-alone' risk, but market risk is crucial
• High stand alone risk does not necessarily lead to high market risk
• Relationship between stand-alone risk and market risk

  Depends on the correlation between the project's returns and the returns on other assets
• Risk-adjusted cost of capital (RACC)


  Addresses the problem of estimating the correlation between the project's returns and the returns on other assets so that the effect on market risk can be estimated, and reflected in the cost of capital that is used to discount cash flows (NPV)
• Basic valuation model


  Where: k, the risk adjusted interest rate, is the sum of the risk less rate of return (eg gilts), and risk premium - a function of the variability of the firm’s returns

  If standard deviation of profits increases, discount rate also increases

  
• Basic valuation model example

  Company considering 2 products

  • Product A can only be used by company
  • Product B can be used by company and other companies
  • Total estimates investment outlay = £100,000
  • Expected cash flow for product A = £20,000 for 8 years; product B = £23,000 for 8 years
  • SD of annual returns from A = 1.0(1.5 for project B)
  • Rate of discount for A = 10% (15% for B)

  
• Certainty equivalence 


  How much money must agent receive to make them indifferent between a certain sum and the expected value of a risky sum?

  Firm can purchase franchise for £100,000 with 50:50 chance of success

  • If successful, firm receives £1m, otherwise loses £100,000
  • OR can decide not to purchase franchise (retain £100,000)
  • Expected monetary value of franchise: 0.5 x 1,000,000 + 0.5 x (-) 100,000 = 450,000
  • Therefore, if firm was indifferent between two options, expected monetary valye of £450,000 from risky alternative has a certainty equivalent of £100,000
• Risk-adjusted (RAD) model Vs Certainty equivalence (CE)

  RAD is considered ‘superior’ as:

  • Includes risk
  • Easy to apply (risk premium + risk less rate of return)
  • CE requires analysis of decision maker to discover their preferences toward risk