Question

Suppose that the standard deviation of returns from a typical share is about 0.40 (or 40%) a year. The correlation between the returns of each pair of shares is about 0.3.

a. Calculate the variance and standard deviation of the returns on a portfolio that has equal investments in 2 shares, 3 shares, and so on, up to 10 shares.

b. Use your estimates to draw a graph of the variance on the y-axis and the number of shares on the X-axis. How large is the underlying market variance that cannot be diversified away?

c. Now repeat the problem, assuming that the correlation between each pair of stocks is zero.

This question has been posted before, but can someone explain it in more simple terms for all the parts for the answers? Like for example, are there certain set equations where we just stick the figures in for example part a? --> 1/N · 0.4 2 + (N2−N)/ N2 · 0.3 · 0.4 · 0.4.

What does diversified away means?

Thank you, I'm just struggling to get my head around the answer someone else has posted.

Answer 1

Portfolio Variance=(w1^2)*(S1^2)+(w2^2)(S2^2)+………….(wn^2)*(Sn^2)+2w1w2*Cov(1,2)+2w1w3*Cov(1,3)+………

Cov(1,2)=Covariance of returns of asset1 and asset2 Portfolio Standard Deviation =Square Root (Portfolio Variance)

Standard Deviation=0.4 Variance =Square of Standard Deviation=0.4^2=0.16

0.4

Correlation(1,2)=0.3

0.3

Covariance(1,2) =Correlation(1,2)* (Standard Deviation1)*(StandardDeviation2)

Covariance(1,2) =0.3*0.4*0.4=0.048 .(a)If There are 2 Shares Portfolio Variance =((1/2)^2)(0.16+0.16)+2*(1/2)*(1/2)*0.048 Portfolio Variance =0.104 Portfolio Standard Deviation =Square Root(0.104)=0.3225

0.048

If there are n shares with equal weight

Weight of each Shares=(1/n) Portfolio Variance=((1/n)^2)*n*0.16+2*((1/n)^2))*(nC2)*0.048 Portfolio Variance=((1/n)^2)*n*0.16+((1/n)^2))*(nC2)*0.096 A B C D E=C+D F=E^2 n nC2 ((1/n)^2)*n*0.16 ((1/n)^2))*(nC2)*0.096 Portfolio Portfolio No. of shares Variance Standard Deviation 2 1 0.0800 0.0240 0.1040 0.3225 3 3 0.0533 0.0320 0.0853 0.2921 4 6 0.0400 0.0360 0.0760 0.2757 5 10 0.0320 0.0384 0.0704 0.2653 6 15 0.0267 0.0400 0.0667 0.2582 7 21 0.0229 0.0411 0.0640 0.2530 8 28 0.0200 0.0420 0.0620 0.2490 9 36 0.0178 0.0427 0.0604 0.2459 10 45 0.0160 0.0432 0.0592 0.2433

Market Variance, that cannot be diversified away=0.0592

If Correlation between each pair is ZERO

Covariance between each pair =0

Portfolio Variance =((1/n)^2)*n*0.16

A	B	C=B^2
n	((1/n)^2)*n*0.16	Portfolio
No. of Shares	Portfolio Variance	Standard Deviation
2	0.0800	0.2828
3	0.0533	0.2309
4	0.0400	0.2000
5	0.0320	0.1789
6	0.0267	0.1633
7	0.0229	0.1512
8	0.0200	0.1414
9	0.0178	0.1333
10	0.0160	0.1265

MEANING OF DIVERSIFICATION GIVEN BELOW:

There is an old saying “Do not put all eggs in one basket”

Why?

Because risk is high.By keeping eggs in different baskets, we reduce risk

We try to do same thing through diversification.

By investing our capital in different assets , we try to reduce risk.

Diversified folios reduce the risk and also the ratio of Risk to reward.

If w1, w2 , w3 …wn are weight in the portfolio for assets 1, 2,3 ….n

Then,w1+w2+w3+……………………+wn=1

R1, R2,R3,…….Rn are the return of the assets 1, 2 , 3 ….n

S1, S2, S3……Sn are the standard deviation of the assets 1, 2, 3 …n

Portfolio Return=w1R1+w2R2+w3R3+…….+wnRn

Portfolio Variance=(w1^2)*(S1^2)+(w2^2)(S2^2)+………….(wn^2)*(Sn^2)+2w1w2*Cov(1,2)+2w1w3*Cov(1,3)+………+w(n-1)wn*Cov(n,(n-1)

Cov(1,2)=Covariance of returns of asset1 and asset2

Portfolio Standard Deviation =Square root of Portfolio variance

Risk of a stock is measured by standard deviation.

Hence reduction of standard deviation through diversification means reduction of risk.

We can take a simple example of two assets 1 and 2

Return of asset1=R1=15%

Return of asset2=R2=12%

Standard deviation of asset 1=S1=10%

Standard deviation of asset 2=S2=8%

Correlation of asset 1 and 2=Corr(1,2)=0.1

Covariance(1,2)=Corr(1,2)*S1*S2=0.1*10*8=8

Assume for simplicity, equal amount is invested in asset 1 and asset 2

Hence, w1=w2=0.5

Portfolio Return;

0.5*15+0.5*12=13.5%

Portfolio Variance=(0.5^2)*(10^2)+(0.5^2)*(8^2)+2*0.5*0.5*8=45

Portfolio Standard Deviation=Square root of Variance=(45^0.5)= 6.708204

We can see, the risk of portfolio as measured by Standard Deviation has reduced significantly to 6,7 whereas the assets in the portfolio had standard deviation of 10 and 8

Risk / Return ratio of the portfolio=6.7/13.5=0.496

Risk/Return ratio of asset1=10/15= 0.666667

Risk/Return ratio of asset2=8/12= 0.666667

Risk return ratio of the portfolio is lower

WHY INVEST IN DIFFERENT TYPES OF ASSETS?

Return of same types of assets are highly correlated.Suppose, if you invest in 10 different Technology Companies, you may not be able to reduce risk, because all the companies will have similar return and highly correlated.

We have seen in the equation of Portfolio Variance, one factor=2 w1*w2*Cov(1,2)

If there is high correlation between return of asset1 and 2, the Covariance(1,2) will be high . Hence this factor 2w1w2Cov(1,2) will be high. As a result portfolio standard deviation will be high and diversification will not serve any purpose.

Hence, in order to get benefit of diversification, we need to invest in different classes of assets such that the correlation between different assets are low, and consequently, the portfolio risk (measured through standard deviation) is low.