Measurement of Risk and Calculation of Portfolio Risk

Made By:Dharti Shah 46

Dhrumil Shah 47Kavisha Shah 48

Param Shah 49Shairavi Shah 50

Risk

When there is a expectation of loss from a given loss, it is termed as Risk. In other words, the probability or threat of quantifiable damage, injury, liability, loss, or any other negative occurrence that is caused by external or internal vulnerabilities and that may be avoided through preemptive action.

Higher investment Higher risk Higher return

Measurement of Risk

Risk reflects the chance that the actual return on an investment may be different than the expected return.

One way to measure risk is to calculate the variance and standard deviation of the distribution of returns.

We will use a probability distribution in our calculations.

Probability Distribution:

E[R]A = 12.5%E[R]B = 20%

State Probability Return on Stock A

Return on Stock B

1 20% 5% 50%

2 30% 10% 30%

3 30% 15% 10%

4 20% 20% -10%

Given an asset's expected return, its variance can be calculated using the following equation:

N

Var(R) = s2 = S pi(Ri – E[R])2

i=1

Where:◦ N = the number of states ◦ pi = the probability of state i ◦ Ri = the return on the stock in state i◦ E[R] = the expected return on the stock

The standard deviation is calculated as the positive square root of the variance:

SD(R) = s = s2 = (s2)1/2 = (s2)0.5

The variance and standard deviation for stock A is calculated as follows:

s2A = .2(.05 -.125)2 + .3(.1 -.125)2 + .3(.15 -.125)2 + .2(.2 -.125)2 = .002625

sA = (.002625)0.5 = .0512 = 5.12%

By applying the same method for stock b, we will get .042 and 20.49%.

Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's.

This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio.

Portfolio RiskPortfolio risk refers to the possibility that a

portfolio will not earn the expected or desired rate of return.

In simple words Portfolio can be explained as a list of financial assets of an individual or a bank or other financial institution.

The practical example for this can be Investments, stock exchange etc. People invest in TATA’s & Adani’s rather than investing in other companies due to the goodwill.

The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio.

The weights reflect the proportion of the portfolio invested in the stocks.

This can be expressed as follows: NE[Rp] = S wiE[Ri] i=1

Where:◦ E[Rp] = the expected return on the portfolio◦ N = the number of stocks in the portfolio◦ wi = the proportion of the portfolio invested in

stock i ◦ E[Ri] = the expected return on stock i

For a portfolio consisting of two assets, the above equation can be expressed as:

E[Rp] = w1E[R1] + w2E[R2]

If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then the expected return of the portfolio is:

E[Rp] = .50(.125) + .50(.20) = 16.25%

The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together.

Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient.

The Covariance between the returns on two stocks can be calculated as follows:

NCov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB]) i=1

Where:◦ sA,B = the covariance between the returns on

stocks A and B ◦ N = the number of states ◦ pi = the probability of state i ◦ RAi = the return on stock A in state i ◦ E[RA] = the expected return on stock A ◦ RBi = the return on stock B in state i◦ E[RB] = the expected return on stock B

The Correlation Coefficient between the returns on two stocks can be calculated as follows:

sdA,B Cov(RA,RB)Corr(RA,RB) = pA,B = sdAsdB = SD(RA)SD(RB)Where:

◦ pA,B=the correlation coefficient between the returns on stocks A and B

◦ sdA,B=the covariance between the returns on stocks A and B,

◦ sdA=the standard deviation on stock A, and ◦ sdB=the standard deviation on stock B

The covariance between stock A and stock B is as follows:

sdA,B = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) + .3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) =

-.0105

The correlation coefficient between stock A and stock B is as follows:

-0.0105 pA,B = (.0512)(.2049) = -1.00

Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:

sd2p = (wA)2s2

A + (wB)2s2B + 2wAwBrA,B sAsB

ORsd2

p = (wA)2s2A + (wB)2s2

B + 2wAwB sA,B

The Standard Deviation of the Portfolio equals the positive square root of the variance.

Let’s calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B:

s2p=(.75)2(.0512)2+(.25)2(.2049)2+

2(.75)(.25)(1)(.0512)(.2049)= .00016

sp =√.00016=.0128=1.28%

Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A.

This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual stocks can be eliminated.