The document discusses key concepts related to risk and return including: - Normal distribution and how it relates to standard deviation and z-scores. - How to calculate the minimum and maximum expected returns over 1, 2, and 3 standard deviations from the mean for a stock with an 8% mean return and 10% standard deviation. - Formulas for calculating the probability of returns above or below certain levels for a stock with a given mean and standard deviation. - Definitions of covariance and correlation as ways to measure co-movement between assets. - Examples of calculating covariance using historical and probabilistic stock price data. - An introduction to the concept of beta as a measure of systematic risk.

1. Risk and Return

2. Normal distribution and standardization

3. Obs (CP) Mean Obs - Mean (Obs – Mean)/SD
355
515.5
-160.5 -0.65748
385
515.5
-130.5 -0.53458
410
515.5
-105.5 -0.43217
510
515.5
-5.5 -0.02253
84
515.5
-431.5 -1.7676
304
515.5
-211.5 -0.86639
853
515.5
337.5 1.382541
770
515.5
254.5 1.042539
864
515.5
348.5 1.427602
620
515.5
104.5 0.428076
Mean value = = 515.5 Mean value=0
SD = = 244.11 SD= 1

4. If X is a random variable and has a normal distribution with mean µ and standard deviation σ,
then the Empirical Rule says the following:
About 68% of the x values lie between –1σ and +1σ of the mean µ (within one standard
deviation of the mean).
About 95% of the x values lie between –2σ and +2σ of the mean µ (within two standard
deviations of the mean).
About 99.7% of the x values lie between –3σ and +3σ of the mean µ(within three standard
deviations of the mean).
The z-scores for +1σ and –1σ are +1 and –1, respectively.
The z-scores for +2σ and –2σ are +2 and –2, respectively.
The z-scores for +3σ and –3σ are +3 and –3 respectively.

5. If a stock has a mean return of 8% and a Std. Dev of 10%. Find its max and min
value upto 3 std dev assuming that the data points are distributed normally
Mean = μ = 0.08
Std dev = σ = 0.1
Formula for min and max value μ ± zσ
Min and max values for 1 std dev
μ ± zσ = μ ± 1σ = 0.08 ± 1*0.1 = -0.02 to 0.18 (About 68% of the x values lie Between -0.02 and 0.18)
Min and max values for 2 std dev
μ ± zσ = μ ± 2σ = 0.08 ± 2*0.1 = -0.12 to 0.28 (About 95% of the x values lie Between -0.02 and 0.18)
Min and max values for 3 std dev
μ ± zσ = μ ± 3σ = 0.08 ± 3*0.1 = -0.22 to 0.38 (About 99.7% of the x values lie Between -0.02 and 0.18)

6. What is the probability that a stock with mean return of 8% and standard
deviation of 10% would yield a return of 6% or less.
The probability that return is less than 6% [P(x<6)=0.4207]

7. What is the probability that a stock with mean return of 8% and standard
deviation of 10% would yield a return of 15% or less.
The probability that return is less than 15% [P(x<15)=0.7580]

8. What is the probability that a stock with mean return of 8% and standard deviation of 10% would
yield a return between 12% and 18%
[P(x<12)=0.6554]
[P(x<18)=0.8413]
[P(12<x<18) = 0.8413 – 0.6554 = 0.1859]

9. Covariance and correlation
• Covariance: Co-movement between two
assets
• Correlation: Standardized version of
covariance

10. Covariance (Historical data)
Given below are the closing stock prices of two companies. Find
out the covariance between them.
Year Raincoat comp Sunglass comp
2017 100 300
2018 90 330
2019 80 350
2020 75 360

11. Covariance (Probabilistic data)
You bought a share of Raincoat company for Rs 120 and one
share of Sunglass company for Rs 280. Under the current
scenario, three probabilities of share price movements for the
companies are likely
Prob RC SC
0.3 100 300
0.2 90 330
0.5 80 350

12. Beta

13. Beta (Sensitivity) Systematic risk
Historical data

14. Beta (Sensitivity) Systematic risk
Probabilistic data