Question

1. Explain how the portfolio means are affected by changing the correlation coefficient values.
2. Explain how the portfolio standard deviations are affected by changing the correlation coefficient values.
3. Explain the impact on portfolio diversification of changing the correlation coefficient values only.

Answer 1

Portfolio return will be weighted average return of the various securities making the portfolio and is mathematically given as

Portfolio Return = Rp = W1 XR1 + W2 x R2 + ... ... ... + wn XRn = > W; XR;

Where w_i is the proportion of investment (portfolio size) in security “i” and R_i is the return of the security “i”. A careful observation suggests that portfolio return will be higher than minimum return and lower than the maximum return amongst the securities included in the portfolio.

Portfolio risk is more complicated than the portfolio return. This will be primarily because risks of the individual securities might be correlated.

The individual risks of the securities and nature & correlation amongst the risks of individual securities will determine the portfolio risk. In order to understand portfolio risk we need to under how two securities will move with respect to each other. This movement is captured by two statistical terms “Covariance” and “Correlation”. For the time being, let’s assume there is a portfolio of two securities:

Sl. No.	Parameter	Security 1	Security 2
1.	Investment proportion	w1	w2
2.	Expected Return	E(R1)	E(R2)
3.	Standard Deviation	σ1	σ2

Portfolio Return = Rp = W1 x R1 + W2 x R2

where P12 is coefficient of correlation

between the return of two securities

Part (a)

Please take a minute to examine the equation above. Now following inferences can be easily derived:

• The portfolio means are not affected by changing the correlation coefficient values. The portfolio mean return is independent of the correlation coefficient values.
• It's only dependent over the weight f each security and mean return of each security.

Part (b)

• Covariance describes the co-movement between the returns of two different investments in a portfolio. It can range from negative to positive infinity and are expressed in terms of square units.
• Variance is nothing but covariance of a variable with itself
• Portfolio risk is dependent upon correlation between the return of individual securities
• 
  Since, -1 < P12 <1,w1 x 01 - W2 x 2 < Portfolio risk < W x 01 + W2 x 02
  
• 
  If two securities are perfectly positively correlated then P12 = 1, and hence portfolio risk
  

is maximum at w x 01 + W2 X 02

• 
  If two securities are perfectly negatively correlated then P12 = -1, and hence portfolio risk
  

is minimum at w x 01 - W2 X 02

• 
  If two securities are uncorrelated then P12 = 0, and hence portfolio risk = w? x 02 + w2 x 0,2
  
• Thus by choosing securities properly, portfolio risk can be optimized
• By adding more and more securities negatively correlated with each other, risk can be minimized. This is called diversification. But adding securities beyond 30, incremental benefit of diversification is minimal.

Part (c)

Whatever you do, the risk of the portfolio will not fall below the difference of the weighted risk in this case].

[(w1 x 01 - W2 x 02)

This implies diversification can’t eliminate all the risks. Systematic risks cannot be eliminated by diversification and hence they are called undiversifiable risk.

• If all stocks in the portfolio have positive correlation, then there is no benefit of diversification.
• Benefit of diversification is achieved only if coefficient of correlation is zero or negative.
• Perfectly negatively correlated stocks will lead to lowest possible risk of the portfolio.