WORKED EXAMPLE OF TYPICAL FINAL EXAM QUESTION
NOTE: Examinable questions on this topic relate to a two (2) asset portfolio. In the real
world a two (2) asset portfolio would not give an investor sufficient spread to diversify
unsystematic risk ( ). Calculations for any (n) asset portfolio would have to be done on a
spreadsheet rather than on a financial calculator. If you’re groping for topics to discuss
between yourself and your friends, here are the relevant formulǽ for an (n) asset
portfolio, which you could implement for a bit of fun, into an appropriate Excel financial
model:
=
p=(
r = Cov(
+2
Pij
)½
)/
MEANING OF SYMBOLS USED:
(symbols have been chosen to closely match those used in most of the common
financial calculators)
r := correlation co-efficient of returns from investments in portfolio,
:= expected return from security j,
:= expected portfolio return,
s := state of economy,
P := probability of s happening,
j := standard deviation of returns from security j,
p := unsystematic portfolio risk (standard deviation),
Wj := weight (proportion) of security j in the portfolio.
Example:
s :=
Good
P :=
0.3
j:
A
-20.00%
B
50.00%
Average
0.5
Bad
0.2
18.00%
18.00%
50.00%
-20.00%
Required (for a portfolio comprising 50% A and 50% B):
1. Calculate
for each investment (A and B)
2. Calculate j for each investment (A and B)
3. Calculate
and p
4. Calculate the straight (weighted) average of j of investments A
and B,
5. Calculate the correlation co-efficient r of returns from
investments A and B in the portfolio
6. Calculate the risk avoided (diversified away) by investing in this
portfolio (the portfolio effect).
CALCULATIONS USING THE SHARP EL-735S (NEW model Sharp)
Let A := x and B := y
DATA ENTRY STEPS:
nd
Alpha
0
0
1. Clear the calculator 2 F
2. Get into 2 Variable STATISTICS 1 mode
MODE
1
1
3. DATA ENTRY
Generalised: Rx (x,y)
Ry (x,y)
Ps
ENT
State:
Good: 20 +/-
(x,y)
50
(x,y)
30
ENT
Average:
18 (x,y)
18
(x,y)
50
ENT
Bad:
50 (x,y)
20 +/-
(x,y)
20 ENT
That’s it! Voila! All the hard work has been done and your calculator
has been fed. Now let it spit out the answers:
Q1
Alpha
4
:=
= 13.00% pa
Double check: 30% * -20% + 50% * 18% + 20% * 50% = 13.00%
Alpha
7
:=
= 20.00% pa
Double check: 30% * 50% + 50% * 18% + 20% * -20% = 20.00%
Q2
Alpha
6
:=
= 24.76% Alpha
9
:=
= 24.33%
Q5
Alpha
r = -0.9927
(
r can be checked by inspection. You will notice that company A returns
are a reflection of company B and vice versa. Indeed, A is a perfect
reflection of B so one would expect r = -1. However because Ps is not
normally distributed, so in this question r > -1 (slightly)
Q3 Will slow you down in the exam because you have to transform the
data in the question as follows:
Ps
30%
50% * RA -10%
50% * RB 25%
TOTAL
15%
50%
9%
9%
18%
20%
25%
-10%
15%
2nd F
1. Clear the calculator
Alpha
0
0
2. Get into single Variable STATISTICS 0 mode
MODE
1
0
Now feed it as follows:
15 (x,y)
30 ENT
18 (x,y)
50 ENT
15 (x,y)
20
ENT
To spit out the answer:
Alpha
4
:=
= 16.50% pa
Double check: From Q1, 50% * 13% + 50% * 20% = 16.50%
Alpha
6
:=
= 1.50%
Double check by inspection: as r => -1 then
=> 0
Q4 From Q2, 50% * 24.76% + 50% * 24.33%  24.55%
Q6
24.55% (Q4 above) – 1.50% (Q3 above) = 23.05% (risk avoided)