1Mathematical and Economic Tools

1
1Mathematical and Economic Tools

• Fundamental to this course is the problem that
  real-world prices can not be forecast with
  absolute certainty. The best one can do is try to
  optimize ones situation given limits on
  information. Therefore, it should be no surprise
  that statistics and calculus are very helpful in
  commodity market analysis.
• The live cattle price for next October is not
  known, but we might be able to ascribe discrete
  probabilities, e.g.,
• State i 1 2 3 4
  5 6 7
• Probability (pi) 0.1 0.1 0.2 0.2 0.2
  0.1 0.1
• Price (xi) 60 62 64 66 68
  70 72

2
2Statistical Moments, and Continuous Distributions

• The mean is ? ?17 pi xi 66c/lb
• The variance is ?2 ?17 pi (xi - ?)2 12(c/lb)2
• The standard deviation is ? 2(?3)c/lb
• Of course, there is no reason why the
  distribution need be discrete. It could be
  continuous whereby the probability of being at a
  particular value is zero, but there are positive
  probabilities of being within certain intervals
  of values.

3
3Continuous Distributions

• Suppose that F(x) Probs ? x is the
  distribution function for s. Then one can
  usually differentiate F(x) to obtain the
  probability density function (p.d.f.) dF(x)/dx
  f(x).
• In this case, f(x) measures the likelihood, in a
  sense, that s has the value x. Of course, the
  probability that it has the value x is zero.
• The mean is given by ? Ex ?A x f(x) dx where
  A is the domain where f(x) is strictly positive.
  Variance is ?2 ?A (x - ?)2 f(x) dx E(x -
  ?)2 .

4
4Normal and Lognormal Distributions

• Discussion. A problem with the normal
  distribution is that random variables can assume
  negative values.
• Defn A random variable is lognormally
  distributed if its natural logarithm is normally
  distributed.
• The lognormal distribution is confined to the
  positive domain (graph of lognormal).

Ln(x)
x
5
5Covariance

• This is a measure of how random variables covary
  with one another. From our discussion on
  hedging, it is clear that covariation is a
  central concept in commodity market analysis.
• Let x represent live cattle prices next October,
  with mean ?x. Let z represent the October
  maturity futures price of cattle, with mean ?z.
  We have Cov(x, z)
  E(x - ?x)(z - ?z).
• It is positive (negative) if x tends to increase
  (decrease) with z. Give integration
  representation.

6
6Correlation

• Covariance is not a unit neutral measure of
  interdependence. Correlation is. It is arrived
  at by dividing through by the standard
  deviations.
• ?x,z Cov(x, z)/?x?z.

7
7Present Value of Money

• See handout 1.

8
8Timelines for a Deferred Income Stream

• a

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0
?
t

)
0
?
9
9Risk Preferences

• See handout 2.
• Graphs of concave functions.

10
10Preference among Lotteries A and B
U(1 m)
U(0.5 m) V(?A)
V(?B)
U(0)
?
0
1 mill
0.5 mill
11
11Mean-Standard Deviation Iso-Utility Curve

• a

?
h(?, ?) k
?
12
12Portfolio Analysis

• See Handout 3

13
13Security Market Line

• a

Security Market Line
?mp
rf
?j
?j 1