Example 7.3
We are to make a decision as to the best choice of two investment options, Investment Option A and Investment Option B —— based on their Expected Returns. Investment Option A offers a 12% p.a. Risk Free Return on our investment. The frequencies of the yearly Returns on investments in Investment Option B, a Managed Investment Fund, are as given in the following Distribution Table.
Investment Option B
Frequency Distribution of Return of x% p.a. over the past 20 years




Return
(x% p.a.)

Frequency
(years)


Probability of achieving this Return of x% p.a. in any one year

4%

2


0.10

8%

2


0.10

11%

6


0.30

13%

5


0.25

15%

2


0.10

17%

1


0.05

19%

2


0.10

We know our Expected Return from Investment Option A is 12 % p.a. and that investment in Investment Option A is Risk Free (i.e. Standard Deviation _{A} = 0). There is Certainty as to the outcome of our investment in Investment Option A.
We compute the Expected Return from an investment in Investment Option B using Formula (F9).
E (x) 
= 
n
i = 1

PRi xi 
Expected Return from Investment Option B
n
i = 1

= (0.10 x 4%) + (0.10 x 8%) + (0.30 x 11%) + (0.25 x 13%)
+ (0.10 x 15%) + (0.05 x 17%) + (0.10 x 19%)
= 12% p.a.

We compute the Variance (2) of the Returns from Investment Option B using Formula (F10).
Variance (_{B}2)
= 
0.10 ( 4 – 12 )2 + 0.10 ( 8 – 12 )2 + 0.30 ( 11 – 12 )2
+ 0.25 ( 13 – 12 )2 + 0.10 ( 15 – 12 )2
+ 0.05 ( 17 – 12 )2 + 0.10 ( 19 – 12 )2 
= 
15.6 (% p.a.)2 
The Standard Deviation (_{B})
=15.6=3.95 %p.a.
So our Investment Choice presents itself to us as follows:
(a) 
Invest in Investment Option A with a Certainty Return of 12 % p.a. (i.e. with an Expected Return of 12 % p.a. and No Associated Risk; Standard Deviation (_{A})= 0)

OR
(b) 
Invest in Investment Option B with an Expected Return of 12 % p.a. and an Associated Risk, as measured by its Standard Deviation (_{B}), of 3.95 % p.a.

Obviously it makes no sense whatsoever to choose Investment Option B as it offers us no Risk Premium, in excess of the 12 % Certainty Return available from Investment Option A, to compensate us for the Risk Associated with its Standard Deviation of 3.95% p.a.
There is NO REWARD for taking the Risk.
So, what Risk Premium, in excess of the 12% Certainty Return available from investment in Investment Option A, would be required to COMPENSATE us for the Risk (as measured by the Standard Deviation of 3.95% p.a.) inherent in investment in Investment Option B ?
In other words, what REWARD would justify accepting the Risk taken?
The Risk Premium required (to JUSTIFY a decision to accept the RISK) would obviously be directly proportional to the Risk itself, i.e. the Risk Premium required would be directly proportional to the associated Standard Deviation ().

As part of the Statistical Analysis Data relevant to a particular Investment Fund, Financial Investment Institutions provide a standard objective measurement (of the Risk Premium) by which the performance of the fund can be evaluated. This standard performance evaluation measurement is based on the historical returns data (where available) for the particular Fund.
For example, Financial Investment Institutions commonly express the Risk Premium in terms of the ratio between the Risk Premium (in excess of the Certainty Return) and the Standard Deviation of the return distribution. This ratio is called the Sharpe Ratio.
Sharpe Ratio 
= 
Risk Premium Standard Deviation 
The Sharpe Ratio therefore provides an objective measurement by which the performance of a particular Investment Fund can be evaluated; the Sharpe Ratio is a Performance Evaluation Measurement. A relatively good performance Sharpe Ratio measurement would be of the order of between 0.60 and 0.75, i.e. a Risk Premium return equivalent to between 60% and 75% of the Risk taken.

In this instance (i.e. in the circumstances of Example 7.3), we would therefore reasonably require a Risk Premium differential of the order of 3% p.a. or more to compensate us for the associated Standard Deviation risk measurement of 3.95 % p.a.; i.e., we would reasonably require a Risk Premium in excess of 75% of the Standard Deviation as a Reward for Risk taken.
In other words, we would require that the Expected Return from investment in Investment Option B would be of the order of 15% p.a. or more, where the associated Standard Deviation (_{B}) is 3.95% p.a.
We would therefore require that the frequency of the yearly returns on investment in Investment Option B would yield similar investment and risk parameters to those from the following Distribution Table.
Investment Option B
Frequency Distribution of Return of x% p.a. over the past 20 years



Return
(x% p.a.)

Frequency
(years)


Probability of achieving this Return of x% p.a. in any one year

7%

2


0.10

11%

2


0.10

14%

6


0.30

16%

5


0.25

18%

2


0.10

20%

1


0.05

22%

2


0.10

i.e. We would require that :
the Expected Return from investment in Investment Option B
= (0.10 x 7 % ) + (0.10 x 11 % ) + (0.30 x 14 % ) + (0.25 x 16 % )
+ (0.10 x 18 % ) + (0.05 x 20 % ) + (0.10 x 22 % ) = 15 % p.a., or more
and
the Variance (_{B}2)
= 
0.10 ( 7 – 15 )2 + 0.10 ( 11 – 15 )2 + 0.30 ( 14 – 15 )2
+ 0.25 ( 16 – 15 )2+ 0.10 ( 18 – 15 )2
+ 0.10 ( 18 – 15 )2 + 0.10 ( 22 – 15 )2 
= 
15.6 (% p.a.)2 
yielding the Standard Deviation (_{B})=3.95% p.a.
IMPORTANT NOTE !
A continued serious misinterpretation of the individual’s Attitude to Risk is evidenced in a June 2000 ruling by the U.K. Personal Investment Authority Ombudsman on the matter of the sale of Endowment Mortgages. (Refer to The Sunday Times — Money Section, 18/6/2000 Article on Endowment Mortgages by Robert Winnett.)
The PIA Ombudsman adjudged that, because a particular individual was a riskaverse investor, who tended to save money in deposit accounts, a stockmarket endowment was completely unsuitable.
This ruling promoted acceptance of the erroneous interpretation of a riskaverse investor as being one who places most of his money in Bank or Building Society savings accounts, rather than speculating in stockmarket investments. —— SUCH IS NOT THE CASE.
Almost all investors dislike risk; they have an aversion to risk; they are riskaverse.
This is no subjective maxim; it is an empirical fact, backed by years of financial data, and backed by the studies of ALL who are knowledgeable in such matters.
It is THE objective maxim genetic to all authoritative work on the subject of Investments and Financial Decisions.
This does not mean that persons who are riskaverse are averse to speculating in the stockmarket or in investment products linked to the stockmarket.
What it does mean is that they are averse to speculating in such an investment product UNLESS such investment affords them the objective expectation of a Risk Premium commensurate to the Risk associated with that investment product,
THAT IS TO SAY —— the riskaverse person will require that it can be shown, objectively, that there is an expectation of a REWARD in excess of the alternative Certainty Return available.
For a riskaverse investor, Acceptance of the Risk must, therefore, be JUSTIFIED on the basis of objective facts.
You will note that there is a marked interpretative parallelism between what is the reasoned investment conduct of a riskaverse person and the common law Negligence understanding of ‘reasonableness of conduct’. (See Section 2.3.3: Negligent Misrepresentation.)

While, with investment in stockmarket products or in funds linked to stockmarket products, the alternative Certainty Return is the riskfree cash investment rate available from the investment marketplace, SUCH IS NOT THE CASE with investment in an Endowment Mortgage.
Relative to investment in an Endowment Mortgage, the alternative Certainty Return is the return available from investment in a Repayment Mortgage.
This in an ABSOLUTE FACT.
Justification as to choice of Mortgage type can, therefore, only be properly assessed on the basis of a truthful and objective Investment Analysis comparison between an Endowment Mortgage and a Repayment Mortgage, i.e. on the basis of an objective comparison of the fundamental Financial Analysis parameters of Internal Rate of Return, associated Risk, and Reward for Risk taken.
To limit the interpretation of a riskaverse investor to the narrow focus applied by the U.K. Personal Investment Authority Ombudsman (as in the June 2000 instance cited above) is to shirk from a proper confrontation of the issues involved.
Example 7.4
We are to make a decision as to the best choice of two investment options, Investment Option C and Investment Option D —— based on their Expected Returns. Investment Option C offers a 12% p.a. Risk Free Return on our investment. The frequencies of the yearly Returns on investments in Investment Option D, a Managed Investment Fund, are as given in the following Distribution Table.
Investment Option D
Frequency Distribution of Return of y% p.a. over the past 20 years



Return
(x% p.a.)

Frequency
(years)


Probability of achieving this Return of x% p.a. in any one year

0%

4


0.20

4%

2


0.10

6%

4


0.20

12%

4


0.20

24%

3


0.15

28%

2


0.10

32%

1


0.05

We know our Expected Return from Investment Option C is 12 % p.a. and that investment in Investment Option C is Risk Free (i.e. Standard Deviation _{C} = 0). There is Certainty as to the outcome of our investment in Investment Option C.
We compute the Expected Return from an investment in Investment Option D using Formula (F9).
Expected Return from Investment Option D
= (0.20 x 0 %) + (0.10 x 4 %) + (0.20 x 6 %) + (0.20 x 12 %)
+ (0.15 x 24 %) + (0.10 x 28 %) + (0.05 x 32 %)
= 12% p.a.
We compute the Variance (_{D}2) of the Returns from Investment Option D using Formula (F10).
Variance (_{D}2)
= 
0.20 ( 0 – 12 )2 + 0.10 ( 4 – 12 )2 + 0.20 ( 6 – 12 )2
+ 0.20 ( 12 – 12 )2+ 0.15 ( 24 – 12 )2
+ 0.10 ( 28 – 12 )2 + 0.05 ( 32 – 12 )2 
= 
109.6 (% p.a.)2 
The Standard Deviation (_{D})
=109.6=10.47% p.a.
So our Investment Choice presents itself to us as follows:
(a) 
Invest in Investment Option C with a Certainty Return of 12% p.a. (i.e. with an Expected Return of 12 % p.a. and No Associated Risk; Standard Deviation (_{C})= 0).

OR
(b) 
Invest in Investment Option D with an Expected Return of 12 % p.a. and an Associated Risk, as measured by its Standard Deviation, of 10.47% p.a.

Again, it is obvious that it makes no sense whatsoever to choose Investment Option D as it offers us no Risk Premium, in excess of the 12 % Certainty Return available from Investment Option C, to compensate us for the considerable Risk associated with its Standard Deviation of 10.47% p.a.
There is NO REWARD for Risk taken.
What Risk Premium, in excess of the 12% Certainty Return available from investment in Investment Option C would be required to COMPENSATE us for the Risk (as measured by the Standard Deviation of 10.47% p.a.) inherent in investment in Investment Option D ?
You will recall, from Example 7.3 above, that for a Standard Deviation of 3.95 % p.a. we would reasonably require a Risk Premium differential of the order of 3 % p.a. or more above the Certainty Return Value of 12 % p.a.
We now have a Standard Deviation of 10.47 % p.a., so we will obviously require a much higher Risk Premium above the Certainty Return Value in order to balance the much increased Risk associated with this higher Standard Deviation.
On the basis of requiring 60% of the Standard Deviation as a reasonable Reward for Risk taken, we would require a Risk Premium differential of the order of 6.3 % p.a. or more to compensate us for the associated Standard Deviation Risk measurement of 10.47 % p.a.
In other words, we would require that the Expected Return from investment in Investment Option D would be of the order of 18.3% p.a. or more, where the associated Standard Deviation (_{D}) is 10.47% p.a.
From the above Examples it is clear that :
(a) 
Information, regarding the Expected Return and the measure of the Expected Deviation from this Expected Return, is essential to Risk Assessment.

(b) 
A Risk Premium differential, in excess of the Certainty Return available, is required to compensate the investor for the Risk associated with any Risk Type Investment.

(c) 
The measure / quantification of the Risk Premium differential required in excess of the Certainty Return available is directly proportional to the measure / quantification of the associated Risk. In other words —— as the Measured Risk increases, the Risk Premium differential required to compensate the investor for this Risk should also increase proportionately.
