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Covariance. And portfolio variance. The states of nature model. Time zero is now. Time one is the future. At time one the possible states of the world are s = 1,2,…,S. Mutually exclusive, collectively exhaustive states. This IS the population. No sampling. The states of nature model.

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Covariance


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Presentation Transcript
covariance

Covariance

And portfolio variance

the states of nature model
The states of nature model
  • Time zero is now.
  • Time one is the future.
  • At time one the possible states of the world are s = 1,2,…,S.
  • Mutually exclusive, collectively exhaustive states.
  • This IS the population. No sampling.
the states of nature model3
The states of nature model
  • States s = 1,2,…,S.
  • Probabilities ps
  • Asset j
  • Payoffs Rj,s
  • Expected rate of return
slide4

= rate of return on j in state s

= probability of state s

= expectation

of rate on j

variance and standard deviation
Variance and standard deviation
  • Form deviations
  • Take their expectation.
covariance8
Covariance
  • Form the product of the deviations
  • (positive if they both go in the same direction)
  • and take the expectation of that.
covariance10
Covariance
  • It measures the tendency of two assets to move together.
  • Variance is a special case -- the two assets are the same.
  • Variance = expectation of the square of the deviation of one asset.
  • Covariance = expectation of the product of the deviations of two assets.
correlation coefficient
Correlation coefficient
  • Like covariance, it measures the tendency of two assets to move together.
  • It is scaled between -1 and +1.
correlation coefficient12
Correlation coefficient
  • = covariance divided by the product of the standard deviations.
  • Size of deviations is lost.
intuition from correlation coefficients
Intuition from correlation coefficients
  • = 1, always move the same way and in proportion.
  • = -1, always move in opposite directions and in proportion.
  • = 0, no tendency either way.
portfolio risk and return
Portfolio Risk and Return
  • Portfolio weights x and 1-x on assets A and B.
an amazing fact
An amazing fact
  • Mixing a risky asset with a safe asset
  • is often safer than the safe asset.
variance of portfolio return
Variance of portfolio return
  • Diversification effects
slide20

Portfolio deviation

Deviation squared

Remember

slide23

Portfolio variance depends on

covariance of the assets.

Positive covariance raises the

variance of the portfolio.

historical data
Historical data
  • Holding period return
  • Equivalent annual return
  • Not the same
holding period return 1926 1929
Holding period return1926-1929
  • Rhp is the holding period return
  • 1+Rhp = (1+r26)(1+r27)(1+r28)(1+r29)
  • = 1.1162*1.3749*1.4362*.9158
  • = 2.0183592
  • Rhp = 101.83592%.
question
Question:
  • Is Rhp the return from holding 4 years at the sample average rate?
  • No.
  • 4 years at 21.075% would yield (1.21075)^4-1 =1.1489084
  • i.e. 114.89 %, instead of Rhp = 101.83592%
review question
Review question
  • Define the internal rate of return.
answer
Answer:
  • The internal rate of return of a project is r such that, given the cash flows CFt of the project,
review item
Review item
  • In the first year a portfolio has a rate of return of -30%.
  • In the second year it has a rate of return of +30%.
  • What is the holding period return?
answer32
Answer:
  • Solve 1+Rhp=(.7)(1.3).
  • Then Rhp = .91 - 1 = -.09.
equivalent annual rate is the geometric average
Equivalent annual rate is the geometric average
  • Solve for x in (1+x)^4 =2.0183592
  • Solution 19.19269%.
  • approximately.
  • It answers the question: what is the equivalent rate over 4 years?
  • Population mean answers the question:
  • What is the average for next year?