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 other
words, 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 price
appreciation. 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 cash
flows. For Example: The price of a bond is the present value of all its future cash flows (coupon payments
and bond’s “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)
2
3
6
-4
8
Stock-2 (Y)
4
-2
8
-4
4
X = 3
Var(X) = 21
2= Y
24 = Var(Y) and Cov(X,Y) = 17
Rp = X1K1 + X2K2
Rp = X1(3) +X2(2)
Rp = 3X1 + 2(1-X1)
Rp = 3X + 2 - 2X1
Rp = 2 + X1
Since X1+ X2 = 1  X2 = 1-X1
Var (p) = variance of the portfolio
Var (p) = X12 + Var(1) + X22Var(2) + 2X1X2 Cov (1,2)
Var (p) = X12(21) + X22(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 + X12) + 34(X1-X12)
Var (p) = 21X12 + 24 - 48X1 + 24X12 + 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%
= 11 X12 - 14 X1 + 24
= 11 (.64)2 - 14 (.64) + 24
= 19.55
St. Dev(p) = 4.42%
Var(P)
Three Asset portfolio:
Data:
K1 = .05 |
K2 = .07 |
K3 = .3 |
Var(1) = .1
Var(2) = .4
Var(3) = .7
Cov(1,2) = - .1
Cov(1,3) = 0
Cov(2,3) = .3
Rp =
=
=
X1 K1 + X 2 K2 + X 3 K3
.05 X1 +.07 X2 + (1-X1-X2) (.3)
-.25X1 - .23X2 + .3
Var(P)
= X12 (.1) + X22 (.4) + X32 (.7) + 2 X1 X2 (-.1) + 2 X2X3 (.3)
= .1 X12 + .4 X22 + .7 (1-X1-X2)2 - .2 X1 X2 + .6 X2 (1-X1-X2)
= .1 X12 + .4 X22 + .7 (1-(X1+X2))2 - .2 X1 X2 + .6 X2 - .6 X1 X2 - .6 X22
= .1 X12 - .2 X22 + .7 (2(-X1-X2) + (X1+X2)2 +1) - .2 X1 X2 + .6 X2 - .6 X1 X2
* X1 + X2 + X 3 = 1
= .1 X12 - .2 X22 + .7 ((-2 X1 - 2 X2 + X12 + 2 X1 X2 + X22) +1) - .2 X1 X2 + .6 X2 - .6 X1 X2
= .1 X12 - .2 X22 -1.4 X1 - .1.4 X2 + .7 X12 + 1.4 X1 X2 + .7X22 + .7 - .2 X1 X2 + .6 X2 - .6 X1 X2
= .8 X12 + .5 X22 -1.4 X1 -.8 X2 + .6 X1 X2 +.7
Var(P) = .8 X12 + .5 X22 -1.4 X1 -.8 X2 + .6 X1 X2 +.7
Var' (Respect to X2) =
Var' (Respect to X1) =
X2 + .6 X1 - .8 = 0 ==> X2 = .8 - .6 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 X22 - 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