Page 82 - DMGT405_FINANCIAL%20MANAGEMENT

Financial Management

Notes Thus, correlation coefficient of securities X an Y can be computed as:

Con variance XY
Correlation XY =
Standard Deviation X ´ Standard Deviation Y

Cov
Or, Cor = xy
xy 6x6y

The variances and standard deviation of X and Y are as follows:
6x = 0.1 (– 8 –5) + 0.2 (10 – 5) + 0.4 (8 – 5) + 0.2 (5 – 5) + 0.1 (– 4 –5) 2
2
2
2
2
2
= 16.9 + 5 + 3.6 + 0 + 8.1 = 33.6
6x = 33.6 = 5.80%
2
6y 2 = 0.1 (14 – 8) + 0.2 (– 4 – 8) + 0.4 (6 – 8) + 0.2 (15 – 8) + 0.1 (20 – 8) 2
2
2
2
= 3.6 + 28.8 + 1.6 + 9.8 + 14.4 = 58.2
6y = 58.2 = 7.63%
The correlation coefficient of securities X and Y is as follows:

- 33 - 33
= - = – 0.746
5.80 7.63 44.25
´
Securities X and Y are negatively correlated. If an investor invests in the combination of these
securities, risk can be reduced.

5.3.3 Variance of a Portfolio
The variance of two security portfolio is given by the following equation:
2
2
2
2
6p 2 = 6 xwx + 6 ywy + 2wx wy 6x 6y Cor xy
where, 6p = Standard deviation of the portfolio
wx and wy are the weightage of securities in value.
If we assume wx and wy in our above example as 50 : 50, then we get

2
6p 2 = 33.6 × (0.5) + 58.2 × (0.5) + 2 × 0.5 × 0.5 × 5.80 × 7.63 ×
2
–0.746
= 8.4 + 14.55 – 16.51 = 6.44
and standard deviation = 6.44 = 2.54%

5.3.4 Minimum Variance Portfolio

A portfolio that has the lowest level of risk is referred as the optimal portfolio. A risk averse
investor will have a trade-off between risk and return.

We can use the following formula for estimating optimal weights of securities X and Y.
2
6y - Covxy
Wx* = 2 2
6x + 6x - 2Covxy

76 LOVELY PROFESSIONAL UNIVERSITY