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]]>Unfortunately, this decision rule is correct only for certain cash flow profiles; for other profiles it would be incorrect and could lead to exactly the wrong decision.
The rule makes sense only if the initial cash flow(s) are negative and all other cash flows are positive, with only the one change of sign i.e. from negative to positive (zero cash flows are not considered a change of sign). For example, if you can invest $100 this year and earn $200 next year (cash profile -100, 200), the IRR is 100%, and this would in principle be a sensible and profitable investment. On the other hand, if you could earn $100 this year but have to pay back $200 next year (cash profile 100, -200), then the IRR is still 100%, but the decision to proceed would in principle not make sense or be unprofitable. Note that the IRR is the same even if the cash flow profile is reversed in sign (since the discount rate to create a zero sum for the discounted cash flows is the same for the original and for the reversed-sign cash flow profile). Thus, for the profile in which income is earned first, the decision rule would have to be reversed “accept the project if the IRR is below the cost of capital”.
The rule can also be misleading or inapplicable if the cash flow profiles change sign and if one relies on the default settings for the Excel IRR function. For example, if one could invest $100 this year to earn $120 next year (cash profile -100, 120), then the IRR is 20%. Of course, if one could delay some of the initial investment and incur the cost in year 3, then the IRR would increase. So, a cash profile of -70, 120, -30 would have an IRR of 41%. If a delay of even more investments is made so that the cash flow profile is -30, 120, -70, then the Excel IRR function will be default show a value of -29%. In other words, it appears that the ability to delay the investment (but not the return) has reduced IRR, and potentially made the project unprofitable. In fact, due to the two change of sign, there are two values for the IRR (in each case). For the latter case, the IRR function has “jumped” to a second solution (-29%) instead of showing the first solution (229%). This is because the function uses an iterative process with a default guess of 10% to initiate the iteration. However, even if one is aware of the two solutions, one no longer has a decision rule that is easy to apply. In fact, in this case, by doing an NPV analysis of the cash flows at different discount rates, one can derive the rule to apply for this particular set of cash flows: “do the project if the cost of capital is above -29% and the return requirements are below 229%”. But such rules are not clear, not least as the reverse would be true if cash flow signs were reversed.
An analysis of the properties of the IRR and its appropriate use in decision support is covered within Module A of our Certificate in Financial Modelling and Data Analysis. Participants in this course also have free access to our in-depth library of proprietary materials (such as our 80-page guide to fully understanding IRR), which can be referred to on an optional basis for those wishing to really understand the topic in depth.
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Belief in the Sufficiency of an Existing Approach
The biggest challenge to making good decisions is the belief that one already is or will make a good decision.
Presence of Multiple Objectives, Criteria and Stakeholders
There can be a challenge to formulate problems appropriately for analysis. For example, to distinguish what items are objectives (and therefore a priorities) from those that are constraints (and therefore may not be met in a final proposed solution). Whereas it is known that in linear continuous optimisation problems, all constraints are met at the optimal solution point, this is not the case for most practical problem in business, which have some non-linear or discrete characteristic.
Oversimplification
Biases in Decision-Making, analysis or Preferences
Lack of Feedback (Objective Results Analysis)
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A simple example of using optimisation to find model parameters is linear regression, where the slope and intercept of the best-fitting linear regression line can be found by minimising the sum of the least-squares difference between the observations and the predictions for any hypothesised line. One could attempt to apply similar concepts to unknown aspects relating to a model of COVID-19 infections.
With respect to cases in France, the graph above shows that the case-number growth rate is around 10% per day currently. This means that new case numbers per week are about ten times as high as they were prior to the lock-down on 17th March (56 941 new cases in the week to 7th April, compared to 5946 cases in the week to 17th March).
In simple terms, given an incubation period of up to two weeks, one would – at first consideration at least – expect the number of new cases to fall significantly, if not close to zero. Of course, there are reasons why this is not so: There would still be transmission within households, and of course in reality the lock-down is not total (“essential” shops are still open, which in France includes the local boulangeries ?, seriously!). Also, improved awareness and testing may mean that there is a higher detection rate of infections.
However, the main driver of a significant increase could also be that there were many more non-detected infections prior to the lock-down, which have resulted in new cases since.
This led me to build a small and simple model. It assumes that there are two unknown parameters: The growth rate in daily new cases prior to and after the lock-down. Other required inputs (such as incubation period, and % detection of infections), are assumed to be known and constant.
I then used Solver in Excel to find these two growth rates that minimise was the squared difference on a weekly basis in the number of new cases predicted and those which are observed. Once this is done, the absolute numbers of cases can be calculated:
Cumulated Cases | ||
Week Ending | Observed Cases | Modelled |
03/03/2020 | 212 | 360 |
10/03/2020 | 1784 | 6834 |
17/03/2020 | 7730 | 129693 |
24/03/2020 | 22304 | 292433 |
31/03/2020 | 52128 | 659377 |
07/04/2020 | 109069 | 1486763 |
As an example, the model predicts that there had been a total 130 000 infections when the lock-down started, compared to the total number of people known to have been infected of 7 730.
These modelled figures may be pessimistic: if awareness and testing has improved significantly since the lock-down (which I have no information about), then the number of cases (and growth rate) required to optimise the model parameters against observed cases would be lower, since more of the modelled cases would be detected. But a simple reading suggests also that about 1.5 m people have been infected on a cumulative basis. In fact, this tallies approximately with an alternative method to estimate the figures: as of 7^{th} April there have been 10328 deaths, so with a mortality rate in the region of 1%, case numbers would be around 1000000.
The COVID-19 outbreak provides many opportunities to remind ourselves of the richness of the analytic toolkit available to model it, which do not need to be restricted to “risk analysis”!
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]]>The function dialogue describes the Rate parameter as “the interest rate per period of the loan. For example, use 6%./4 for quarterly payment at 6% APR”.
This reference to APR is confusing and generally wrong. The function is implicitly assuming that the periodic interest is compounded at the end of every period. But a loan with a “stated rate” of 6% p.a. that is compounded quarterly would have an APR of around 6.14% (i.e. 1.015^4 -1). Thus, if one wishes to use an interest rate as an input to this function, but where this interest rate is defined with reference to a period length (e.g. annual) that is not the same as the length of the periods for loan repayments (e.g. monthly), then two things need to be done. First, one must assume that the interest is compounded at the end of every loan period (in general this might not be valid e.g. if one were to use the function to try to calculate daily payment requirements for a loan that is compounded only monthly). Second, one would apply the relevant time conversion formula to find the equivalent loan-period rate from the stated annual rate. For example, if the stated rate is 12% p.a. and the loan period is monthly, then on the assumption that interest is compounded monthly, the periodic rate is 1.12^(1/12)-1 or approximately 9.5%.
The complexity of this function with respect to “Rate” is therefore due to the incorrect example within the Help menu, coupled with the subtleties of interest compounding.
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Excel does not miscalculate, so any spreadsheet will show correct figures. For these figures to be a model, they need to be a representation of a real-life situation. Where the worksheet items are clearly labelled, and there is a clear and transparent logic flow, then it should be possible to identify the corresponding situation(s) in real-life to which the calculations correspond (e.g. a multiplication of two numbers could correspond to the calculation of sales revenue from the unit price and the volume sold). Alternatively, if the labelling and flow is not particularly clear, but sufficient detailed supporting documentation is provided which explains the calculations as they correspond to a real-life situation, then one may consider that the calculations also form a model.
There are unfortunately far too many cases where people who are in principle highly skilled in Excel create “models” which are neither documented nor understandable by others (even others within their organisation who are working on the same project!). It should be incumbent on anyone who wishes to be considered as a competent modeller (rather than having competence in Excel) to create transparency in their work, making it as easy as possible to understand to others. All too often, it is the fate of the recipient to try to make head or tail of something that does not deserve the label of “model”, but who is left in the situation of feeling inadequate or lacking knowledge for not being able to interpret the model. In reality, due to the number of combinations of possible formula and calculations that are possible to create in Excel, it is relatively easy to create a set of calculations that correspond to a real-life situation but which no-one, however smart, would be able to correctly identify or be able to judge to be correct or not.
As a modelling community, I believe that we need to create an environment of push-back, where these poor “models” are not considered as models, but only as a “set of calculations (of something)”. We need to help to create a cultural environment within companies that allows such work to be sent back for rework!!
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]]>Once complete, we will have several urls including www.cinfm.co.uk (and .com), and www.certificateinfinancialmodelling.co.uk, www.cinfm.co.uk
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Real options are most frequently described within situations in which there is uncertainty. In such situations, value may be created by the ability to make the final decision only when some of the uncertainty has been resolved. However, there is no requirement for uncertainty to be present, only that there is an additional decision possibility that has been created or is available for some reason.
It is worth recalling that best practices in decision-making involve ensuring that a full range of decision options are generated and considered. Therefore, the creation of real options is at least implicit in any such best practice decision process. The choice of which decision possibility to follow does not therefore always necessitate an explicit calculation of the real options value, only the assessment of which decision alternative is the best one.
The explicit calculation of real options value can be challenging for several reasons, including:
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The PMT function has the default parameters PMT(rate,nper,pv,[fv],[type]), and returns the constant-per-period payment required to repay a loan of amount “pv” over “nper” periods, where the periodic interest rate is “rate”. The optional arguments are the future value of the loan “fv”, and the timing type (payments made at the end or beginning of the periods). It is worth noting that that if the interest rate is set to 0%, then the PMT function effectively shows a straight-line amortisation of the principal value. This can be a useful cross-check method. The PPMT function calculates only the principal part of the repayment within each period and is different in each period (when interest rates are not zero).
In terms of mistakes that are easy to make when using these functions:
First, the functions by default return negative values. For example, a loan of $1200 to be completely repaid over 12 periods with an interest rate of 0% would have a PMT of -$100, rather than (the perhaps expected value) +$100. This can be overlooked when the function is embedded within other calculations.
Second, a more subtly, the optional fv may need to have opposite sign to the mandatory pv, even though one may have not expressed this in the Excel entry cells or model calculations. For example, if a loan of $1400 is to be paid down to $200 over 12 periods when the interest rate is zero, the periodic amount to pay down is $100. One might expect the function PMT(0%, 12, 1400, 200) to return -$100. However, the return value is -$133.33. The correct value is returned by using PMT(0%, 12, 1400, -200). This is overlooked in many texts, which provide examples of the function, as they very often do not use the fv argument (so that it is implicitly zero, and therefore its sign is not important). However, the issue becomes clear if a non-zero value is used and the formula checked with simple values – something that is often hard in practice as the inputs are often themselves items that cannot be manually varied, and the function may be embedded in more complex calculations that mask incorrect formulae (e.g. within Min, MAX or IF statements).
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