Question

1. Consider an investor with $1 of wealth. He has to compose a portfolio with the following two risky assets. The rates of returns from those assets are specified by expectations, variances and correlations.
var[r 1] = 0.02 E[r 1] = 0.05 var[r 2] = 0.03 E[r 2] = 0.08 corr[r 1, r 2] = 0.7
(a) Draw an efficient frontier of the investor.
(b) How will the mean-variance efficient frontier in (a) change if corr[r 1, r 2] = 1?
(c) How will the mean-variance efficient frontier in (a) change if corr[r 1, r 2] = −1?
(d) In (c), what would happen in the financial market if the risk-free rate is 0.06? Does the investor has an incentive to buy the risk-free asset?
(e) Return to the assumptions of (a). In addition, suppose that the risk-free asset is available at the rate of 0.03. Then, how the efficient frontier will change? In other words, find a CML(Capital Market Line).

Answer 1

Part a:

We will determine different values of expected return E(R) and Variance of the portfolio by increasing the weight of asset 1 from 0 to 1 in increments of 0.1. Since the total weight is 1, the weight of the other asset is (1-w)

And then draw the E(R) v/s Variance graph. This is the efficient frontier.

E(R1)	0.05
V(R1)	0.02
E(R2)	0.08
V(R2)	0.03
Corr(R1,R2)	0.7
Cov(R1,R2)	0.017146
W1	E(R )	V (R )	Std. Dev. (R )
0	0.08	0.0300	0.1732
0.1	0.077	0.0276	0.1661
0.2	0.074	0.0255	0.1596
0.3	0.071	0.0237	0.1540
0.4	0.068	0.0222	0.1491
0.5	0.065	0.0211	0.1452
0.6	0.062	0.0202	0.1422
0.7	0.059	0.0197	0.1404
0.8	0.056	0.0195	0.1396
0.9	0.053	0.0196	0.1400
1	0.05	0.0200	0.1414

Copying the same excel table with formulae:

E(R1)	0.05
V(R1)	0.02
E(R2)	0.08
V(R2)	0.03
Corr(R1,R2)	0.7
Cov(R1,R2)	=B5*SQRT(B2*B4)
W1	E(R )	V (R )	Std. Dev. (R )
0	=A9*$B$1+(1-A9)*$B$3	=(A9^2*$B$2 +(1-A9)^2*$B$4+2*A9*(1-A9)*$B$6)	=C9^0.5
=A9+0.1	=A10*$B$1+(1-A10)*$B$3	=(A10^2*$B$2 +(1-A10)^2*$B$4+2*A10*(1-A10)*$B$6)	=C10^0.5
=A10+0.1	=A11*$B$1+(1-A11)*$B$3	=(A11^2*$B$2 +(1-A11)^2*$B$4+2*A11*(1-A11)*$B$6)	=C11^0.5
=A11+0.1	=A12*$B$1+(1-A12)*$B$3	=(A12^2*$B$2 +(1-A12)^2*$B$4+2*A12*(1-A12)*$B$6)	=C12^0.5
=A12+0.1	=A13*$B$1+(1-A13)*$B$3	=(A13^2*$B$2 +(1-A13)^2*$B$4+2*A13*(1-A13)*$B$6)	=C13^0.5
=A13+0.1	=A14*$B$1+(1-A14)*$B$3	=(A14^2*$B$2 +(1-A14)^2*$B$4+2*A14*(1-A14)*$B$6)	=C14^0.5
=A14+0.1	=A15*$B$1+(1-A15)*$B$3	=(A15^2*$B$2 +(1-A15)^2*$B$4+2*A15*(1-A15)*$B$6)	=C15^0.5
=A15+0.1	=A16*$B$1+(1-A16)*$B$3	=(A16^2*$B$2 +(1-A16)^2*$B$4+2*A16*(1-A16)*$B$6)	=C16^0.5
=A16+0.1	=A17*$B$1+(1-A17)*$B$3	=(A17^2*$B$2 +(1-A17)^2*$B$4+2*A17*(1-A17)*$B$6)	=C17^0.5
=A17+0.1	=A18*$B$1+(1-A18)*$B$3	=(A18^2*$B$2 +(1-A18)^2*$B$4+2*A18*(1-A18)*$B$6)	=C18^0.5
=A18+0.1	=A19*$B$1+(1-A19)*$B$3	=(A19^2*$B$2 +(1-A19)^2*$B$4+2*A19*(1-A19)*$B$6)	=C19^0.5

The efficient frontier drawn in excel looks like the following:

Chart Title 0.0000 0.0500 0.1000 0.1500 0.2000

Sub part b:

When correlation is changed to 1, the efficient frontier looks like below:

E(R1)	0.05
V(R1)	0.02
E(R2)	0.08
V(R2)	0.03
Corr(R1,R2)	1
Cov(R1,R2)	0.024495
W1	E(R )	V (R )	Std. Dev. (R )
0	0.08	0.0300	0.1732
0.1	0.077	0.0289	0.1700
0.2	0.074	0.0278	0.1668
0.3	0.071	0.0268	0.1637
0.4	0.068	0.0258	0.1605
0.5	0.065	0.0247	0.1573
0.6	0.062	0.0238	0.1541
0.7	0.059	0.0228	0.1510
0.8	0.056	0.0218	0.1478
0.9	0.053	0.0209	0.1446
1	0.05	0.0200	0.1414

Chart Title 0.0000 0.0500 0.1000 0.1500 0.2000

Sub part c:

When correlation is changed to -1, the efficient frontier looks like below:

E(R1)	0.05
V(R1)	0.02
E(R2)	0.08
V(R2)	0.03
Corr(R1,R2)	-1
Cov(R1,R2)	-0.02449
W1	E(R )	V (R )	Std. Dev. (R )
0	0.08	0.0300	0.1732
0.1	0.077	0.0201	0.1417
0.2	0.074	0.0122	0.1103
0.3	0.071	0.0062	0.0788
0.4	0.068	0.0022	0.0474
0.5	0.065	0.0003	0.0159
0.6	0.062	0.0002	0.0156
0.7	0.059	0.0022	0.0470
0.8	0.056	0.0062	0.0785
0.9	0.053	0.0121	0.1100
1	0.05	0.0200	0.1414

Chart Title 0.0000 0.0500 0.1000 0.1500 0.2000

Sub part d:

In the absence of risk free asset, the efficient frontier provides the combination of all the risky assets. This is the opportunity set for the investor to invest.

|f a risk free asset is introduced. then the efficient frontier is a straight line that starts from y-axis at the risk free rate and is tangent to the original efficient frontier of risky assets. This line is called the Capital Market Line and this is the basis of CAPM (Capital Asset Pricing Model). The point where the tangent insects the efficient frontier is called the tangency portfolio and this is the most efficient portfolio. With the introduction of the risk free asset, the opportunity set now increases as the investor can buy a combination of risk free and risky assets.

Hence the investor has an incentive to buy the risk free asset.