Measurement of Risk and Calculation of Portfolio Risk

1. Measurement of Risk and
Calculation of Portfolio Risk
Made By:
Dharti Shah 46
Dhrumil Shah 47
Kavisha Shah 48
Param Shah 49
Shairavi Shah 50

2. 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

3. 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.

4.  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%

5.  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

6.  The variance and standard deviation for stock A is
calculated as follows:
s2
A = .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%.

7.  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.

8. Portfolio Risk
 Portfolio 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.

9.  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:
N
E[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

10.  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%

11.  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.

12.  The Covariance between the returns on two stocks can
be calculated as follows:
N
Cov(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

13.  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

14.  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

15.  Using either the correlation coefficient or the covariance,
the Variance on a Two-Asset Portfolio can be calculated
as follows:
sd2
p = (wA)2s2
A + (wB)2s2
B + 2wAwBrA,B sAsB
OR
sd2
p = (wA)2s2
A + (wB)2s2
B + 2wAwB sA,B
 The Standard Deviation of the Portfolio equals the
positive square root of the variance.

16.  Let’s calculate the variance and standard deviation of a
portfolio comprised of 75% stock A and 25% stock B:
s2
p=(.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.