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2. Capital Asset Pricing Model Capm Notes

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Capital Asset Pricing Model (CAPM)
Mean-variance Analysis and CAPM
Expected Rate of Return
Probability
P(X) = Probability that X happens
P(Y) = Probability that Y happens
Note: P(X) + P(Y) = 1
Change in Price (P) = (P1 - P0) / P0
P0 = Purchase Price (Period 0)
P1 = Stock Price at Period 1
Rate of Return (R) = Change in Price (%)
Stock A

Stock B

R under P(X)

RAX = (PAX1 - PA0) / PA0

RB = (PBX1 - PB0) / PB0

R under P(Y)

RAY = (PAY1 - PA0) / PA0

RB = (PBY1 - PB0) / PB0

Expected Rate of Return on Stock A
E(RA) = P(X) x RAX + P(Y) x RAY
Expected Rate of Return on Stock B
E(RB) = P(X) x RBX + P(Y) x RBY
Expected Rate of Return on the Portfolio
E(RP) = aE(RA) + bE(RB)
Weighting: a + b = 1 or b = 1 - a a = (No. of Stock A x PA0) / (No. of Stock A x PA0 + No. of Stock B x PB0)
b = (No. of Stock B x PB0) / (No. of Stock A x PA0 + No. of Stock B x PB0)
Standard Deviation () = Risk
A = {[P(X) x RAX - E(RA)]2 + [P(Y) x RAY - E(RA)]2}0.5
B = {[P(X) x RBX - E(RB)]2 + [P(Y) x RBY - E(RB)]2}0.5
Variances (2)

1 Variance of return (risk) on the portfolio: P2 = a2A2 + b2B2 + 2abAB
Covariances (AB)
Relationship between the rate of return on Stock A and Stock B
AB = E(RAB) - E(RA) x E(RB)
E(RAB) = [E(RA) + E(RB)] / 2
Correlation Coefficient (AB)
Relationship between the rate of return on Stock A and Stock B
Value between -1 and +1
AB = AB / AB

2 2

2 A

or 2

AB =ABAB

2 B

 P = a  + b  + 2abABAB
Perfectly positively correlated: AB = 1
Positively correlated: 0 < AB < 1
Uncorrelated: AB = 0
Negatively correlated: -1 < AB < 0
Perfectly negatively correlated: AB = -1
Portfolio with 3 Stocks (N = 3)
Expected Rate of Return on Stock A
E(RA) = P(X) x RAX + P(Y) x PAY
Expected Rate of Return on Stock B
E(RB) = P(X) x RBX + P(Y) x RBY
Expected Rate of Return on Stock C
E(RC) = P(X) x RCX + P(Y) x RCY
Expected Rate of Return on the Portfolio
E(RP) = aE(RA) + bE(RB) + cE(RC)
Weighting: a + b + c = 1

a=

b=

No. of Stock A x PA0
No. of Stock A x PA0 + No. of Stock B x PB0 + No. of Stock C x PC0
No. of Stock B x PB0
No. of Stock A x PA0 + No. of Stock B x PB0 + No. of Stock C x PC0 2 c=

No. of Stock B x PC0
No. of Stock A x PA0 + No. of Stock B x PB0 + No. of Stock C x PC0

Standard Deviation () = Risk
A = {[P(X) x RAX - E(RA)]2 + [P(Y) x RAY - E(RA)]2}0.5
B = {[P(X) x RBX - E(RB)]2 + [P(Y) x RBY - E(RB)]2}0.5
C = {[P(X) x RCX - E(RC)]2 + [P(Y) x RCY - E(RC)]2}0.5
Variances (2)
Variance of return (risk) on the portfolio:
P2 = a2A2 + b2B2 + c2C2 + abAB + baBA + acAC + caCA + bcBC + cbCB
P2 = a2A2 + b2B2 + c2C2 + 2abAB + 2acAC + 2bcBC
N(N - 1) = 6
N = Number of Stocks in the Portfolio
Diversification
 To eliminate risk derived from individual stocks by reducing the portfolio variance
 The portfolio variance (P ) falls as the number of the assets held increases (N  )
P2 = (N / N2) VarP + [N (N - 1) / N2] CovP
P = (1 / N) VarP + (1 - 1 / N) CovP
VarP = Average Variance
CovP = Average Covariance
Two Stocks

Three Stocks

VarP = (A + B) / 2

VarP = (A + B + C) / 3

CovP = (AB + BA) / 2

CovP = (AB + BA + AC + CA + BC + CB) / 3

CovP = 2AB / 2

CovP = (2AB + 2AC + 2BC) / 3

CovP = AB

CovP = 2 (AB + AC + BC) / 3

Note: 1 / N + (1 - 1 / N) = 1

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