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P2 Advanced Management Accounting
Module: 13
Measuring Risk
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Description Step
1. Measuring risk using standard deviations
The most commonly used single measure of risk is that of standard
deviation. A standard deviation is calculated by taking the possible outcomes of a decision and then calculating how wide ranging these outcomes are
in comparison to the average. A wide range of outcomes is considered
riskier than outcomes that are closely grouped.
For example, if the possible outcomes of a luxury project were £0, £100,000
profit or £100,000 loss, it would be considered riskier than a budget project
with possible outcomes of £100 profit, £1,000 profit or £1,000 loss. In both
scenarios the average is identical; a profit of £100. However, the first project
has a more extreme range of possible outcomes, making it a riskier project.
When comparing standard deviations, direct comparisons are only possible
when the EV is the same for each option. If the EVs are different, we must
take the standard deviation as a percentage of the EV and use that for
comparison purposes instead. This percentage figure is known as the
coefficient.
Calculating a standard deviation:
Calculate the EV
Take the difference between the EV and the NPV for each possibility
Square these answers
Times by probability
Total these answers
Square root to find the overall probability
We'll consider an example for Toy Town a toy manufacturer for the example.
Example
Toy Town have two new luxury playhouses which they want to start building
called the ‘Classic’ and the ‘Vintage’. They would like to start producing both,
but capacity is limited to one. The NPV and probabilities of each product are
outlined below, but which one should Toy Town start producing?
The Classic model:
NPV
(£) Probability Weighted amount (£)
10,000 0.3 3,000
11,000 0.1 1,100
12,000 0.2 2,400
13,000 0.2 2,600
14,000 0.2 2,800
11,900 EV
The Vintage:
NPV (£)
Probability Weighted amount (£)
10,000 0.1 1,000
12,000 0.4 4,800
14,000 0.3 4,200
15,000 0.1 1,500
18,000 0.1 1,800
13,300 EV
This analysis tells us that Toy Town should start producing the Vintage, as it
has the highest EV. However to assess the risk of each project we will also calculate their standard deviations, starting with The Classic.
The Classic:
Step 2 - We already have the NPV's for The Classic, along with the EV, so we
can start at step 2 and calculate the difference between the NPV and EV:
Deviation
NPV (£)
EV (£) from EV (£)
10,000 11,900 -1,900
11,000 11,900 -900
12,000 11,900 100
13,000 11,900 1,100
14,000 11,900 2,100
Step 3 – Square these answers
Deviation Deviation²
NPV (£)
EV (£) from EV (£) (£)
10,000 11,900 -1,900 3,610,000
11,000 11,900 -900 810,000
12,000 11,900 100 10,000
13,000 11,900 1,100 1,210,000
14,000 11,900 2,100 4,410,000
Step 4 – Times by the probability (this gives us the weighted amount)
Deviation Deviation² Weighted
NPV (£)
EV (£) from EV (£) (£) Probability Amount (£)
10,000 11,900 -1,900 3,610,000 0.3 1,083,000
11,000 11,900 -900 810,000 0.1 81,000
12,000 11,900 100 10,000 0.2 2,000
13,000 11,900 1,100 1,210,000 0.2 242,000
14,000 11,900 2,100 4,410,000 0.2 882,000
Step 5 – Total these answers
Deviation Deviation² Weighted
NPV (£)
EV (£) from EV (£) (£) Probability Amount (£)
10,000 11,900 -1,900 3,610,000 0.3 1,083,000
11,000 11,900 -900 810,000 0.1 81,000
12,000 11,900 100 10,000 0.2 2,000
13,000 11,900 1,100 1,210,000 0.2 242,000
14,000 11,900 2,100 4,410,000 0.2 882,000
2,290,000
Step 6 – Square root this answer!
Deviation Deviation² Weighted
NPV (£)
EV (£) from EV (£) (£) Probability Amount (£)
10,000 11,900 -1,900 3,610,000 0.3 1,083,000
11,000 11,900 -900 810,000 0.1 81,000
12,000 11,900 100 10,000 0.2 2,000
13,000 11,900 1,100 1,210,000 0.2 242,000
14,000 11,900 2,100 4,410,000 0.2 882,000
2,290,000
Standard deviation 1,513.27
(square root of weighted squared amount)
If this was repeated for The Vintage, we would get these figures:
The Vintage:
Deviation Deviation² Weighted
NPV (£)
EV (£) from EV (£) (£) Probability Amount (£)
10,000 13,300 -3,300 10,890,000 0.1 1,089,000
12,000 13,300 -1,300 1,690,000 0.4 676,000
14,000 13,300 700 490,000 0.3 147,000
15,000 13,300 1,700 2,890,000 0.1 289,000
18,000 13,300 4,700 22,090,000 0.1 2,209,000
4,410,000
Standard deviation 2,100.00
(square root of weighted squared amount)
Because the EV of each project is different, we cannot compare them
directly. This is because you can't compare two things which don't match!
Let's say you have a very fast man and a very fast car, you can't compare the
two to see which is faster; instead we compare them against the rest of their
own kind.
Going back to our example, we cannot compare EV's so instead we find the
coefficient of each one, which is calculated by dividing the standard
deviation by the project’s EV.
The Classic The Vintage
Standard deviation £1,513.27 £2,100
EV £11,900 £13,300
Coefficient 12.72% 15.79%
In this example, you can see that the coefficient of the Vintage model
15.79%, higher than that of the Classic model’s 12.72%. This tells us that the
Vintage model has a wider range of possible outcomes and is therefore
considered a riskier project.
Management is now better placed to make an educated decision. While the
EV of the Vintage model is higher, it also carries a higher level of risk.
Management will choose the best option available to them depending on their
goals and their risk appetite.
2. Sensitivity analysis
Risk can be included in the investment appraisal process by using
sensitivity analysis. Sensitivity analysis allows us to consider the impact
that a change in one variable, for example revenue, would need to be for
the NPV of a project to fall to 0.
When we are calculating future revenues or costs in an NPV calculation, no
matter how good our assumptions are, this is only a best guess, as
unforeseeable events could easily make this forecast irrelevant.
Sensitivity analysis considers is how much flex we have in a particular
factor before the project is no longer viable. For instance, would it require
a 20% drop to our revenue forecast or a 2% drop?
It is a good way of testing how risky a project is. If you were someone
making an investment, you might be quite happy that your predictions are right
to a 20% level of variability, but if you saw the sensitivity was just 2% you
might decide that there's simply too much risk and decide not to proceed.
The sensitivity can be calculated using the following formula:
Example - Sensitivity analysis
Big Fortune Ltd is reviewing a project with an NPV of £166.70 with an asset it
can sell in year 5 for £525. The review is to include sensitivities of the
project to key variables.
Requirement
How sensitive is the NPV to the change in the disposal value? i.e. how much
lower would the disposal value have to be before the project is not viable?
Assume the cost of capital is 12%.
Answer The first step is to work out the PV of the asset disposal figure [525 x 0.567
(the year 5 discount factor from the tables) = £297.68], and then apply the
formula:
Sensitivity =
NPV of project
PV of disposal
value
Sensitivity = NPV of project
PV of one variable (e.g. sales)
Thus,
= 166.70
297.68
= 0.56 (i.e. 56%) The answer of 56% means that the disposal value can fall by 56% before the
project is no longer viable. 56% of £525 is a £294 drop in value.
We can then look at the market for the asset and any likely changes in that
market to decide whether this is likely or not.
If there is a liquid market and the prices are typically stable then there would
appear to be a small likelihood the value would fall this far and we could
proceed with the project. If prices are highly volatile and we have great
uncertainty in final prices we may decide the project is too risky.
Multiple variables
In a larger project you can calculate the sensitivity to a variety of
different variables and then assess the likelihood that each will vary to a
certain degree.
Do note though, that a limit of this analysis is that it only considers one
variable at a time. If two or three variables changed adversely at the same
time the effect would be severe on the NPV, but this would not necessarily
be seen in any single figure. For example, if prices fell by 10%, costs rose by
15% and the final resale value fell by 20% - the combined effect of these
changes would be significant, and not clearly demonstrated in any single
sensitivity calculation!
Example
Big Fortune Ltd is reviewing a project before deciding whether or not to go
ahead with it. The project is for the purchase of a new machine that will allow
them to produce their product (gold chains) at a faster rate.
The project is expected to last for 5 years and the initial investment required
is $200,000. The residual value of the machine at the end of the 5 years will
be 10% of its purchase price.
The gold chains will sell for $100 a piece with variable unit costs of $30. In
addition to this there will be fixed costs of $20,000 a year. Annual sales are
expected to be 1,000 units in years 1 and 2. 1,200 in years 3 and 4, before
falling to 800 in year 5.
The cost of capital at Big Fortune Ltd is 8%.
As a shrewd organisation, Big Fortune Ltd are not willing to proceed until
conducting a sensitivity analysis on the following factors:
• The initial investment
• Sales volume
• Fixed cost
• Discount rate
• The project’s life
• The sales price
They have asked us, as their management accountant, to conduct the
aforementioned analysis.
NPV
The first step in conducting sensitivity analysis is to calculate the NPV as per
the following formula.
Occasionally this (the NPV) may be given to you, but if it is not you need to
work it out.
We know that the initial investment is for $200,000 with a cost of capital
of 8%. We also know that the project will last for 5 years with the machine
having a residual value of 10% of the purchase price:
$200,000 x 10% = $20,000
Using this information and the present value tables (see formulas) we can
begin building our NPV:
NPV
The next step is to add our contribution for each year. The expected sales are:
Years 1-2 = 1,000
Years 3-4 = 1,200
Year 5 = 800
The sales price is $100 with variable costs of $30 giving a contribution (sales
price – variable cost per unit) of $70. Annual fixed costs are $20,000. With
this in mind we can calculate our total contribution for each year by
multiplying our expected sales by the contribution per unit:
Year
1 $70 x 1,000 = $70,000
2 $70 x 1,000 = $70,000
3 $70 x 1,200 = $84,000
4 $70 x 1,200 = $84,000
5 $70 x 800 = $56,000
These figures can now be added to our NPV. Fixed costs can be entered as a
cumulative figure because they remain consistent throughout the 5 years
and this helps later on in our calculation as we'll need to know the present
value of the fixed costs on their own to work out the sensitivity to fixed costs:
Year
0
Investment
Cashflow $
(200,000)
DF – 8%
1.000
Present value $
(200,000)
1 Contribution 70,000 0.926 64,820
2 Contribution 70,000 0.857 59,990
3 Contribution 84,000 0.794 66,696
4 Contribution 84,000 0.735 61,740
5 Contribution 56,000 0.681 38,136
1-5 Total fixed costs (20,000) 3.993 (79,860)
5 Residual value 20,000 0.681 13,620
NPV 25,142
We have an NPV of $25,142. If the company goes ahead with the project
they will, in theory, be $25,142 better off. But, things do not always go so
smoothly. Hence the sensitivity analysis! Big Fortune need to know by how
much these various factors can change before the project stops being
worthwhile.
Now that we have the NPV we can begin calculating the sensitivity of the
variables using the formula. Here it is once again:
The initial investment
The machine costs $200,000. So to calculate the sensitivity we put it into the
formula:
Sensitivity of the initial investment = 12.6%. What this means is that a
12.6% increase in the price of the machine would leave the NPV at 0 and
thus no longer a profitable investment.
Sales volume
As sales change, revenues go up but so do costs. The net change therefore is
the to the contribution (sales – variable costs). To calculate the sensitivity of the sales volume, we must first calculate our total contribution across the
5 years:
Year Total contribution $
1 64,820
2 59,990
3 66,696
4 61,740
5 38,136
291,382
The total contribution is $291,382. So to calculate the sensitivity we put it into
the formula:
Sensitivity to sales volume =
$25,142
$291,382
Sensitivity to sales volume changes = 8.6%.
What this means is that an 8.6% decrease in the sales volume each and
every year from budgeted levels leave the NPV at 0 and thus no longer a
profitable project. It's worth noting that this assumes the 8.6% fall applies in
each year compared to the budgeted sales in that year.
This is not that big a fall in sales volumes and so it is worth double checking
the sales estimates to ensure we are confident they are accurate.
Fixed costs
The fixed costs for the length of the project amount to $79,860. So to calculate
the sensitivity we, again, put it into the formula:
Sensitivity of fixed costs =
$25,142
$79,860
Sensitivity of fixed costs = 31.5%. What this means fixed costs would need
to rise by 31.5% to reduce the NPV to 0 and render the project unprofitable. This means that it is NOT that sensitive to fixed costs and fixed costs are
unlikely to be the high risk factor.
The discount factor
This is where it gets a little more complicated. We need to work out what the
fall is in discount rate that will mean the NPV is zero. You may remember
from previous studies that the discount rate when the NPV is zero is what we
call the Internal Rate of Return (IRR).
The IRR is calculated by working out the discount factor that leaves the NPV
at 0 and the first step is to conduct another NPV that leaves us with a
negative figure.
We know that a cost of capital of 8% gives an NPV of $25,142 so we need to
do one with a higher discount factor than this. We’ll use 13%:
Year Cashflow $ DF –
13% Present value $
0 Investment (200,000) 1 (200,000)
1 Contribution 70,000 0.885 61,950
2 Contribution 70,000 0.783 54,810
3 Contribution 84,000 0.693 58,212
4 Contribution 84,000 0.613 51,492
5 Contribution 56,000 0.543 30,408
1-5 Total fixed costs (20,000) 3.517 (70,340)
5 Residual value 20,000 0.543 10,860
NPV
(2,608)
The two NPVs then need to be put in the IRR formula which is as follows:
IRR = A +
NPVa
NPVa -
NPVb
x (B - A)
Where:
A = The discount rate used to calculate NPVa
B = The discount rate used to calculate NPVb
NPVa = The NPV at the first discount rate
NPVb = The NPV at the second discount rate
In Big Fortune's case:
A = 8%
B = 13%
NPVa = 25,142
NPVb = (2,608)
These figures can now be used to calculate the IRR:
IRR = 0.08 +
25,142
27,750
X 0.05
IRR = 0.08 + (0.91 x 0.05)
IRR = 0.08 + 0.455
IRR = 0.1255 = 12.6%
Let's work out the sensitivity now:
Percentage change in discount rate =
(12.6%-8%)
8%
= 60%
The discount rate would have to rise to 12.6% which is an increase of 60% on
the current rate of 8% for the project to be unviable. The project is relatively
insensitive to a rise in discount rate therefore.
The project’s life
The current project is 5 years in length, but what if it were just 4 years or
even 3? Would the project still be viable? Here we need to calculate the time
when the project will pay itself back. If it will pay itself back in 2 years Big
Fortune have plenty of leeway, if it is going to take 4.9 years then any
additional cost could push it over the 5 year mark. Making the 5 year project
unprofitable.
To do this we use the discounted payback period method. This involves
returning to our NPV table. This time we need to split fixed costs out by
year. This is because the payback table needs to know the specific annual
costs:
Year 0
$
Year 1
$
Year 2
$
Year 3
$
Year 4
$
Year 5
$
Investment
Contribution
(200,000) 70,000
70,000
84,000
84,000
56,000
Fixed costs (20,000) (20,000) (20,000) (20,000) (20,000)
Residual value
20,000
Net Cash Flow
(200,000) 50,000 50,000 64,000 64,000 56,000
Discount
Factor
(8%)
1.000 0.926 0.857 0.794 0.735 0.681
Present Value (200,000) 46,300 42,850 50,861 47,040 38,136
Net Present
Value
25,142
Now we need to work out the outstanding balance at the end of each of
the 5 years. We do this by deducting the present value of each year from
the initial investment year on year:
Investment
Contribution
Fixed costs
Residual
value
Net Cash
Flow
Discount
Factor
(8%)
Present
Valu
e
Balance
outstanding
At the end of year 4 Big Fortune have an outstanding balance of $12,944 and are due to receive $38,136 in year 5. Meaning the payback period will be
reached sometime during the 5th year.
The difference is calculated using the following formula:
Years passed +
Balance at the start of the final year
Present value of the final year
= Payback period
Applying this to Big Fortune:
12,944 4 +
38,136
= Payback period
4 + 0.34
The payback period is 4.34 years (roughly 4 years and 4 months)
meaning it will break-even then. This means Big Fortune must keep on top of costs as they only have 8 months of leeway over the 5 years.
Generally speaking, it is much better to have a short payback period, since
the investor's initial outlay is at risk for a shorter period of time. As such, Big
Fortune should try and keep their payback period as short as possible in
order to minimise risk!
Year 0
$
Year 1
$
Year 2
$
Year 3
$
Year 4
$
Year 5
$
(200,000) 70,000
70,000
84,000
84,000
56,000
(20,000) (20,000) (20,000) (20,000) (20,000)
20,000
(200,000) 50,000 50,000 64,000 64,000 56,000
1.000 0.926 0.857 0.794 0.735 0.681
(200,000) 46,300 42,850 50,861 47,040 38,136
(200,000) (153,700) (110,850) (60,034) (12,944)
The sales price
If our selling price changes then this will impact our revenue and so this is
the relevant present value to use to calculate its sensitivity.
However, we do not yet know the revenue and so we must work it out. We
know the selling price per unit is $100 and expected sales are 1,000 in the
first two years. 1,200 in the next two and 800 in the final year. We must also
keep in mind the 8% discount factor:
Expected sales x sales price x discount factor = present value of revenue
Year Expected sales Sales price $ Discount
factor PV Revenue $
1 1,000 100 0.926 92,600
2 1,000 100 0.857 85,700
3 1,200 100 0.794 95,280
4 1,200 100 0.735 88,200
5 800 100 0.681 54,480
Total revenue 416,260
The revenue for the length of the project amounts to $416,260. So to calculate
the sensitivity we put it into the formula:
Sensitivity of sales price =
$25,142
$416,260
Sensitivity of the sales price = 6.04%.
What this means is if the sales price drops by any more than 6.04% it will
reduce the NPV to 0 and it would no longer be a profitable project. Of all the factors measured, we can see that this project is MOST sensitive to the
sales price.
Advantages and disadvantages
Sensitivity analysis has a number of advantages and drawbacks:
Advantages
Simple to understand and calculate
Identifies the critical variables i.e. those that have low
percentage changes before the project is not viable.
Disadvantages
Does not consider multiple variables changing at the
same time. It only considers a change to one variable at
a time.
Does not indicate the probability of a change in the
future value of the key or critical variable. Further research
would be needed to investigate this.
Does not identify the overall decision. Unlike NPV
which tells you one way or the other this just gives a
figure on which further analysis is done.
3. Risk-based decisions
“In general, I am happy to take professional and entrepreneurial risks, but I’m
quite risk-averse when it comes to putting my body in danger”.
Robin Chase, Co-founder of Zipcar
Despite this quote, Robin Chase illegally travelled into Kenya from Tanzania at
the age of 23, risking prison and worse! What's more, Chase then launched
Zipcar – a car sharing service - with just $78 in her bank account. Things
seem to have worked out, however, when by 2016 Zipcar had over 350,000
members!
This is a great example of something we've already discussed: that business
decisions are largely dependent on the risk profile of the decision maker.
The maximin, maximax and minimax regret model is a behavioural model
which is used to illustrate an individual’s decisions relative to their risk
appetite.
Example:
Let’s imagine we sell our home made juice at the market each Sunday.
There are 3 Sunday markets in town – Downtown, Uptown and Beachside.
Our sales are greatly dependent on the weather for the day. At Beachside,
good weather always results in fantastic sales. However if the weather is poor,
we barely break even. On the other hand, sales at the Downtown market are
pretty consistent regardless of the weather conditions. As we need to
reserve and pay for our market stall a week in advance, we are unable to
wait and see what the weather is like on the day before choosing our location.
As a result a little guesswork is required. Over the past few months we’ve
managed to summarise our expected results as follows:
Expected profits by weather and location
Downtown Uptown Beachside
Rainy £1,000 £700 £100
Overcast £1,200 £1,300 £800
Sunny £1,500 £1,800 £2,500
With this information, we can determine what decision an individual will make
based on their risk appetite.
Maximin approach
This approach involves maximising our minimum profit. In other words,
we are looking at the choice that has the best worst case scenario.
In this case, the worst case scenario for each market is when it is raining, as
this is when the lowest profits are made. Therefore we will choose the
scenario that has the highest profit in rainy conditions. Using this logic, the
maximin choice will be the Downtown market with a minimum profit of £1,000.
This approach is usually taken by highly risk averse individuals. They have
a pessimistic view of most situations and are willing to forego potential
profits if it means avoiding heavy failure.
Maximax approach
In this scenario we are looking to maximise profits by all means
necessary, in other words, maximising our maximum profit.
In all 3 markets the best results are seen when the weather is sunny.
However the highest of the 3 is the Beachside market, which potentially
returns £2,500 profit if the weather turns out well. This market would
therefore be the choice under a maximax approach.
Such decisions are entertained by high risk takers, who take the most
optimistic view of each situation. A high possibility of profit at the Beachside
market is the key draw, despite the fact that a dose of bad luck will result in a
profit of only £100.
Minimax regret approach
This approach looks to minimise the amount of regret one might feel if a
poor decision is made. In other words, we are looking to find the option
with the smallest opportunity cost.
If we decide to choose the Beachside market and the weather turns out to be
sunny, we will have no regret at all as we made the highest possible profit. If
it ends up raining however, we will regret having chosen the Beachside
market as we could have had a higher profit at the other markets. In this
case we can actually quantify our regret; we know that we could have made
£1000 at the Downtown market but instead made £100 at the Beachside,
therefore our regret amounts to £900 (£1000-£100).
The minimax regret approach looks to find the option that will minimise this
quantity of regret.
The first step in this approach is to quantify the amount of regret for each
available option:
Regret experienced by weather and location
Downtown Uptown Beachside
Rainy - £300 £900
Overcast £100 - £500
Sunny £1,000 £700 -
From this analysis we can see that no regret is experienced if we choose the
Downtown market during rainy weather, as this is the best possible choice.
Likewise for the Uptown market during an overcast day and the Beachside
market during a sunny day.
The highest regret is experienced on a sunny day at the Downtown market, as
our profit is £1,000 lower than the best alternative, which is Beachside. The
next highest level of regret is at Beachside on a rainy day, which gives a profit
£900 lower than what was available to us at Downtown. According to our
table, the option with the lowest maximum regret is the Uptown market,
which will never have a level of regret that exceeds £700. Because this option
minimises the amount of regret we might feel, it would be the first choice
under a minimax regret approach.
4. Simulation
In business, however, simulating future scenarios can be a valuable tool in
assessing risk because it recognises that a large amount of variables are
uncertain and simulation can be used to find the combination that is most likely
to play out. Simulation is often carried out using computer programs which
can model a range of various outcomes and their probabilities.
Example:
Let’s imagine we are selling tickets to a show. We do this via agents and
want to know the chances of us making a profit. The breakeven point is 50
tickets.
We know that ticket sales are affected by a variety of unknown factors, such
as the number of tourists in town, the number of other shows on, the
success of social media advertising, even the weather on the day affects last
minute sales. Because of the various uncertainties it is helpful to enter these
probabilities into a computer program and run a simulation, which would
produce something like this:
Ticket sales Frequency
0-10 2
10-30 8
30-50 21
50-70 59
70-100 6
100+ 4
100
This is helpful because it has simulated a collection of variables with
different probabilities and given us a set of likely outcomes. With this
information we have a better indication of the level of sales we might expect.
For example, we know that there is only a 2% chance that we will sell less
than 10 tickets, and only a 4% chance that we’ll sell more than 100.
More importantly, we can see that there is a 69% chance of us reaching our
breakeven point of 50 tickets (the sum of all probabilities over 50 tickets, that
is 59% + 6% + 4%).
We can also see that we should realistically expect sales at the upper end of
the 30-70 tickets, as this is the range where the highest probabilities lie.
Simulations work best in situations where there is a range of various
uncertainties. Other examples might include queuing times (uncertain
factors include number of customers, service time required) and stock
management (uncertain factors include sales levels, returns, damaged
goods etc.)
5. Payoff tables
A payoff table is a tool for analysing a scenario where there are several
outcomes based on various choices. The table shows the profit or loss
that will occur if the combination of factors happens. Let's illustrate this
with an example.
Susan’s' bakery make cupcakes on a daily basis. Cupcakes are sold for $1
and cost $0.50 to produce. Cakes are made in batches of 10 and unsold
cakes are thrown away at the end of the day.
Demand at the bakery is dependant on weather. The probability of the weather
outcomes is as follows:
• Heavy rain: probability of 0.15 and expected sales of 20 cakes
• Light rain: probability 0.25 and expected sales of 30 cakes
• Overcast: probability of 0.27 and expected sales of 40 cakes
• Sunny: probability of 0.33 and expected sales of 50 cakes
From this information, the following table can be constructed:
Daily Supply
Expected Profits Based on Cake
Production Numbers
Daily
Demand
Weather
Probability Expected
Sales 20
cakes
30
cakes
40
cakes
50
cakes
Heavy
Rain 0.15 20 $10 $5 $0 ($5)
Light
Rain 0.25 30 $10 $15 $10 $5
Overcast 0.27 40 $10 $15 $20 $15
Sunny 0.33 50 $10 $15 $20 $25
As you should notice, the maximum profit Susan can hope to make is $25.
This will occur if she decides to produce 50 cakes AND it is a sunny day
where demand is usually for 50 cakes [(50 x $1)- (50 x $0.5) = $25].
However, should Susan make 50 cakes, but the weather turns out to be
heavy rain, the bakery will make a $5 loss as she only sells 20 cakes [(20 x
$1) – (50 x $0.50) = -$5].
All the other figures in the table can be calculated in the same way. Different management styles will make different choices:
Risk seekers only concern themselves with the best possible outcome,
no matter how small the probability of it occurring, therefore they will choose
the option with the highest possible profit. This is the maximax approach. In
this example, this would be to bake 50 cakes.
Risk neutral decision makers are not swayed by best and worst case
scenarios and are only concerned with the best long term average (or the
Expected Value).
In this example, the option with the highest EV is 40 cakes. This is
calculated by multiplying the profit by the probability of getting the profit:
(0.15 x0)+(0.25 x10)+(0.27 x 20)+(0.33 x 20) = £14.50
Of course you would have to work this out for each of the production runs and
see which was highest to make a choice.
The risk neutral manager will always go for the outcome with the highest
expected value.
Risk averse decision makers prefer a low variation of outcomes, and
always consider the possibility of the worst-case scenario no matter how
unlikely it may be. This is the maximin approach and a maximin manager
would choose the option with the highest minimum outcome. In the above
example this is to bake 20 cakes for a guaranteed profit of $10.
As seen previously, a manager choosing a minimax regret approach will
seek to choose the option with the least regret which is calculated by
taking the outcome of a choice and finding the difference between it and the
best possible outcome.
For example, if Susan made a $5 loss because she baked 50 cakes but there
was heavy rain, the regret would be $15 because she should have chosen to
bake 20 cakes, which would have led to a profit of $10. The difference
between a $5 loss and $10 profit is $15. In summary:
50 cakes = $15 (Worst regret: actually make-$5 when best choice gave $10))
For the other options the maximum regret is as follows:
40 cakes = $10 (Worst regret: actually make £0 when best choice gave $10)
30 cakes = $10 (Worst regret: actually make £15 when best choice gave $25)
20 cakes = $15 (Worst regret: actually make £10 when best choice gave $25)
The option with the minimum regret on the example could be to bake 30 or 40
cakes.
