1. Check out Vanguard Index Chart … 1984 to 2014 ….

2. Portfolio Theory
Returns on Share Investments are made up of…..

4. (3.55 – 3.20) +.25 / 3.20 = …….

9. Batsman A has four innings and scores 25, 25, 25, 25
Batsman B scores 0, 0, 0, 100
What are their averages ?
What are their Standard Deviations?

10. Batsman A has four innings and scores 25, 25, 25, 25
Batsman B scores 0, 0, 0, 100
What are their averages ?
What are their Standard Deviations?
Both have an average of 25 but Batsman A has a standard deviation of 0 and Batsman B has a Standard Deviation of 43.3.
Using the calculator Stat Mode 1,1 then 25, xy, 0, ENT, 25,xy, 0, ENT, 25, xy, 0, ENT, 25, xy, 100, ENT

11. What is the difference between the Population and a Sample?
All questions in Portfolio theory revolve around data for the population so we will use RCL 6 or RCL the standard deviation for the population.

12. Back to our batsmesn ….
Batsman A has four innings and scores 25, 25, 25, 25
Batsman B scores 0, 0, 0, 100
What are their Standard Deviations? If we took a sample of their batting scores – perhaps there were 20 innings and we sampled 4 innings – or the population that is they had only batted 4 times – these were the complete scores
Batsman A has a standard deviation of 0 whether it is a sample or not (RCL 5, RCL 6) and Batsman B has a Standard Deviation of 50 if it was a sample (RCL 8) and 43.3 if it was the population (total data) (RCL 9)
Long Hand calculation : -

13. Long Hand calculation : Sample for A (0^2 + 0^2 + 0^2 + 0^2) / 3 = 0
Population for A (0^2 + 0^2 + 0^2 + 0^2) / 4 = 0
Dev
Scores B
From mean
Squared 1
-25
625 2 3 4
100 75
5625
Total
7500
Sum of deviations divided by 3
2500
Now find the square root 50
Sum of deviations divided by 4
1875

14. The Mean is the expected return and the Standard Deviation is a measure of Risk. How volatile are the returns?
How close can we expect them to be to the mean ( the expected returns for the portfolio?
Mode, 1, 0, 2nd F, Alpha , 0,0,
.02, xy, 10 ent …. RCL 4 for the mean RCL 6 for the Std Dev

15. Doing it long hand…..
Standard Deviation

17. Strong Positive Correlation

18. Strong Negative Correlation

19. No Correlation

21. Mode, 1, 1 (this time since we have two sets of data Company A and Company B – think of it as an X axis and a Y axis – thus your calculator says Line on STAT Mode 1, 2nd F, Alpha , 0,0,
.3, xy, .10, xy, .30, ENT,
(Note convert the probability / frequency to a whole number – move the decimal place 2 places to the right)
.15, xy, .12, xy, .40, ENT,
.09, +/-, xy, ..09, xy, .30, ENT,
RCL 4 for the mean of X = .1230, RCL 6 for the Std Dev of X = .1526
RCL 7 for the mean of y = .1050, RCL 9 for the Std Dev of y = .0128
RCL, r or “(“ = Correlation Coefficient = .4361, Note Perfect Positive Correlation is +1
Note Perfect Negative Correlation is -1, No Correlation = 0
.4361 suggests expected returns are slightly going the same way – that is they are both dropping

22. Ideally you are looking for a negative correlation – when one share is going up the other one is going down or vice versa
For example you invest in a firm that is an exporter and another that is an importer and you flatten out the impact of currency rate changes
Or you invest in a firm that is financed predominantly by debt (High Gearing) and another company that has very little debt (low gearing) – it is financed by shares (Equity) and you flatten out the impact of interest rate changes

23. Long Hand Calculation - Company A pretty big deviation – 15% - which is about right – variation from 30%

25. Portfolio theory is trying to get us to see that we can reduce risk by diversifying – particularly if we have a negative correlation
Risk /
Std Deviation
Of Portfolio
Returns
No. of Securitities (different companies you have invested in

26. Systematic Risk – affects the entire economy – recession, war
Unsystematic risk – is avoidable risk – affects a small sector of the market – one firm vs another – for example technology, drought, flood, management style, strike

33. Now for the portfolio measurement – pro rata the returns

34. Investing in two companies with a perfectly negative correlation
State
Probability
Share A
Share B
Boom %
0.3
0.1
Normal
0.2
Recession
Calculate the mean and std deviations for both companies
Calculate the Correlation Coefficient
Expected Returns for A is - .2 or 20% for B is .2 or 20%
Std Dev or Risk for A is or 8.16% for B is or 8.16%
With a 50/ 50 mix the weighted average risk is - .5 x x = 8.16%
The correlation coefficient is -1.0 – perfectly negative so we should see some risk reduction from 8.16%

35. Investing in two companies with a perfectly negative correlation State
Probability
Share A
Share B
Boom %
0.3
0.1
Normal
0.2
Recession
Calculate the mean and std deviations for the portfolio and quantify the risk reduction
50 /50 mix Portfolio Returns
Average Returns
State
Probability
Share A
Share B
for portfolio
Boom %
0.3 x .5
.1 x .5
0.2
Normal
0.2 x .5
Recession
Stat mode 1,0,now - .2, xy, , ENT, .2, xy, , ENT, .2, xy, , ENT, – RCL 4 = .2 – expected returns for the portfolio is 20%
Expected Risk for the portfolio is RCL 6 = zero – that is no deviations from the mean – we’re expecting 20% whether it is a recession or a boom
Reduction in Risk – 8.16 – 0 = 8.16