Question

These are a conceptual question.

1. Discuss the meaning of the following statements: the standard deviation of any portfolio of stocks can never be higher than the highest individual stock standard deviation. However, a portfolio standard deviation can be lower than the lowest individual stock standard deviation. (The corollary to this is a portfolio stand-alone risk could be zero even when individual stocks each have a lot of stand-alone risks.)

2. If given a choice to invest in a single mutual fund ( a diverse group of assets) or invest in your own stock/ bonds/ asset selection, which would you rather do and why?

3. Describe one scenario that you could employ to reduce stand-alone risk in your all portfolio if you had constructed the portfolio three years ago. That is if you had to invest in each of the five random stocks (any), how might your portfolio construction be different to maximize risk reduction.

Answer 1

The standard deviation of a portfolio is a measure of the variability of its expected returns. It helps us measure the consistency of the returns earned on a portfolio. Since standard deviation measures how volatile or different the actual returns are from the expected returns, it is also an important measure of the risk of the portfolio.

Mathematically, we express the formula of portfolio variation for two assets as follows:

Where,

W₁ : weight of asset

W₂ : weight of asset 2

σ_1: Standard deviation of asset 1

σ_2: Standard deviation of asset 2

ρ₁₂: Coefficient of correlation between the two assets.

In the formula above, the third term is known as the covariance of the two assets. This term measures the degree to which the change in the return on one asset affects the return on the other asset in the portfolio.

The coefficient of correlation within the covariance term tells us the direction and indicates how strongly the returns vary with each other. The coefficient of correlation can take on values between -1 and +1.

When the correlation coefficient is +1, it means that the returns of the two assets move in the same direction. We can measure the standard deviation of various portfolios of two assets with a correlation co-efficient is +1.

Supposing we have two stocks A and B. Their expected returns and standard deviations are as follows:

	Expected Return	Standard Deviation
Stock A	10%	15%
Stock B	5%	10%
Correlation Coeeficient	1

The partial list of portfolios with different weights of these assets along with standard deviation calculations in excel gives us the following result:

Portfolio No	Stock A (w1)	Stock B (w2)				Portfolio Variance	Std Deviation
1	0%	100%	0	0.01	0	0.01	10.00%
2	10%	90%	0.000225	0.0081	0.0027	0.011025	10.50%
3	20%	80%	0.0009	0.0064	0.0048	0.0121	11.00%
4	30%	70%	0.002025	0.0049	0.0063	0.013225	11.50%
5	40%	60%	0.0036	0.0036	0.0072	0.0144	12.00%
6	50%	50%	0.005625	0.0025	0.0075	0.015625	12.50%
7	60%	40%	0.0081	0.0016	0.0072	0.0169	13.00%
8	70%	30%	0.011025	0.0009	0.0063	0.018225	13.50%
9	80%	20%	0.0144	0.0004	0.0048	0.0196	14.00%
10	90%	10%	0.018225	0.0001	0.0027	0.021025	14.50%
11	100%	0%	0.0225	0	0	0.0225	15.00%

We observe that the portfolio with the highest standard deviation is when the portfolio consists entirely of the stock A, the stock with the higher standard deviation. We also observe that in all other portfolios, the standard deviation is smaller since the proportion of stock A is less than 100%.

If the correlation between the assets is negative, the covariance becomes negative, and the sum total of all the three terms in the portfolio standard deviation reduces. A negative correlation implies that the returns of the two assets move in opposite directions. Hence it is entirely possible to have a portfolio that has standard deviation as zero i.e. riskless.

We can observe this when we change the correlation coeeficient to -1 in the same case as above.

	Expected Return	Standard Deviation
Stock A	10%	15%
Stock B	5%	10%
Correlation Coeeficient	-1

The partial list of portfolios with different weights of these assets along with standard deviation calculations in excel changes as follows:

Portfolio No	Stock A (w1)	Stock B (w2)				Portfolio Variance	Std Deviation
1	0%	100%	0	0.01	0	0.01	10.00%
2	10%	90%	0.000225	0.0081	-0.0027	0.005625	7.50%
3	20%	80%	0.0009	0.0064	-0.0048	0.0025	5.00%
4	30%	70%	0.002025	0.0049	-0.0063	0.000625	2.50%
5	40%	60%	0.0036	0.0036	-0.0072	0	0.00%
6	50%	50%	0.005625	0.0025	-0.0075	0.000625	2.50%
7	60%	40%	0.0081	0.0016	-0.0072	0.0025	5.00%
8	70%	30%	0.011025	0.0009	-0.0063	0.005625	7.50%
9	80%	20%	0.0144	0.0004	-0.0048	0.01	10.00%
10	90%	10%	0.018225	0.0001	-0.0027	0.015625	12.50%
11	100%	0%	0.0225	0	0	0.0225	15.00%

From the above we observe that Stock B has a lower standard deviaiton of than stock A. A porfolio with 40% invested in stock A and 60% invested in stock B has a standard deviation of 0 i.e it is a riskless portfolio, which is lower than the standard deviation of stock B. Hence we can conclude that the portfolio standard deviation can be lower than the lowest individual standard deviation.