8/3/2019 Portfolio MGMT Theory

1/3

CALIFORNIA STATE UNIVERSITY, SACRAMENTO

College of Business Administration

REVIEW OF THE BASICS IN

PORTFOLIO THEORY

Portfolio manger is expected to form a portfolio of stocks and bonds and other assets that

provides the highest possible rate of return at the lowest posible level of risk. In otherwords, to form a portfolio that allocates funds in different assets in such a way that the

portfolio is well diversified and provides the highest return at the lowest possible risk.

Definitions:Assets by definition are expected to have positive cash flow(s). For Instance: The cash

flow of a bond is coupon payments, and the cash flow of a tock is dividends or priceappreciation. A portfolio is a combination of 2 or more assets (stocks, bonds, real estate,contracts, currency, etc)Price of any assets by definition is the present value of all its future cashflows. For Example: The price of a bond is the present value of all its future cash flows (coupon payments

and bonds Face Value)

Two Asset portfolio:

Notations:Rp = return of the portfolio

Var (p) = variance of the portfolio

X1

= proportion of the portfolio invested in stock 1

K1 = average return of stock 1

Rp = X1K1 + X2K2

Stock-1 (X) Stock-2 (Y)

2 4

3 -2

6 8

-4 -4

8 4

X = 3 2 = YVar(X) = 21 24 = Var(Y) and Cov(X,Y) = 17

Rp = X1K1 + X2K2Rp = X1(3) +X2(2)

Rp = 3X1 + 2(1-X1) Since X1+ X2 = 1 X2 = 1-X1

Rp = 3X + 2 - 2X1

8/3/2019 Portfolio MGMT Theory

2/3

Rp = 2 + X1

Var (p) = variance of the portfolio

Var (p) = X12 + Var(1) + X22Var(2) + 2X1X2 Cov (1,2)

Var (p) = X12(21) + X2

2(24) + 2X1X2(17)

Var (p) = X12 (21) + (1-X1)

2(24) + 2X1(1-X1)(17)

Var (p) = 21X12 = 24(1-X1)

2 + 34(X1)(1-X1)

Var (p) = 21X12 + 24(1 - 2X + X1

2) + 34(X1-X12)

Var (p) = 21X12 + 24 - 48X1 + 24X1

2 + 34X1 - 34X12

Var (p) = 11X12 - 14X1 + 24

Y = 11 X12 - 14 X1 + 24

Minimum Varaince Portfolio:

In order to find the proportions invested in assets to guarantee the minimum risk (variance), we take the

first derivative of the above function and make it equal to zero.

Derivative (Var(p) = 22 X1 - 14 = 0 X1 =14/22 = .64 and X2 = .36

This means if we invest 36% of our fund in stock1 and the rest in stock.2 then we will get lowest possible

risk or variance.

X1 = 64% X2 = 36%

Return of the MVP is Rp = 2 +.64 = 2.64%

Var(P) = 11 X12 - 14 X1 + 24

= 11 (.64)2 - 14 (.64) + 24

= 19.55

St. Dev(p) = 4.42%

Three Asset portfolio:

Data: K1 = .05 | Var (1) = .1 Cov(1,2) = - .1K2 = .07 | Var (2) = .4 Cov(1,3) = 0

K3 = .3 | Var (3) = .7 Cov(2,3) = .3

Rp = X1 K1 + X2 K2 + X3 K3 * X1 + X2 + X3 = 1= .05 X1 +.07 X2 + (1-X1-X2) (.3)

= -.25X1 - .23X2 + .3

Var(P) = X12 (.1) + X2

2 (.4) + X32 (.7) + 2 X1 X2 (-.1) + 2 X2X3 (.3)

= .1 X12 + .4 X2

2 + .7 (1-X1-X2)2 - .2 X1 X2 + .6 X2 (1-X1-X2)

8/3/2019 Portfolio MGMT Theory

3/3

= .1 X12 + .4 X2

2 + .7 (1-(X1+X2))2 - .2 X1 X2 + .6 X2 - .6 X1 X2 - .6 X2

2

= .1 X12 - .2 X2

2 + .7 (2(-X1-X2) + (X1+X2)2 +1) - .2 X1 X2 + .6 X2 - .6 X1 X2

= .1 X12 - .2 X2

2 + .7 ((-2 X1 - 2 X2 + X12 + 2 X1 X2 + X2

2) +1) - .2 X1 X2 + .6 X2 - .6 X1 X2

= .1 X12 - .2 X2

2 -1.4 X1 - .1.4 X2 + .7 X12 + 1.4 X1 X2 + .7X2

2 + .7 - .2 X1 X2 + .6 X2 - .6

X1 X2= .8 X1

2 + .5 X22 -1.4 X1 -.8 X2 + .6 X1 X2 +.7

Var(P) = .8 X12 + .5 X2

2 -1.4 X1 -.8 X2 + .6 X1 X2 +.7

Var' (Respect to X2) = X2 + .6 X1 - .8 = 0 ==> X2 = .8 - .6 X1Var' (Respect to X1) = 1.6 X1 + .6 X2 -1.4 = 0

1.6 X1 + .6(.8-.6X1 ) - 1.4 = 0 ==> X1 =74.2%X2 = .8 - .6(.742) = 35.5%

X3 = -9.7%

Characteristics of MinimumVariance Portfolio (MVP):

X1 = 74.2%

X2 = 35.5%

X3 = - 9.7%

Rp = -.25 X1 - .23 X2 + .3

= -.25 (74.2%) - .23 (35.5%) + .3= 3.29%

Var(P) = .8 X12 + .5 X2

2 - 1.4 X1 - .8 X2 + .6 X1 X2 + .7

= .8 (74.2%)2 + .5 (35.5%)2 - 1.4 (74.2%) - .8 (35.5%) + .6 (74.2%) (35.5%) + .7

= .0287