DISCOUNTED CASH FLOW
TECHNIQUES
Advanced Investment Appraisal
BY SABI AKTHER
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What to focus on?
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Net Present Value (NPV)
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Internal Rate of Return (IRR)
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Discounted Payback Period (DPP)
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Macaulay duration
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Multi-period capital rationing
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Monte Carlo Simulation & sensitivity analysis
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Value at Risk (VAR)
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Net Present Value (NPV)
•
Capital investment projects are best evaluated using the NPV technique.
•
Discounting the relevant cash flows for each year of the project at an appropriate cost of
capital
•
Some basic NPV concepts (relevant cash flows, discounting, the impact of inflation, the
impact of taxation
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Net Present Value (NPV)
Relevant cash flows
Relevant costs and revenues
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future
•
incremental.
Some basic NPV concepts are revised as follows: You should therefore ignore:
•
sunk costs
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committed costs
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non-cash items
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apportioned overheads.
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Net Present Value (NPV)
The impact of inflation
Real and nominal rates are linked by the Fisher formula:
(1 + i) = (1 + r)(1 + h)
Or
(1 + r) = (1 + i)/(1 + h)
in which:
r = real rate
i = money/nominal interest rate
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h = general inflation rate.
Net Present Value (NPV)
The impact of inflation
Calculating the free cash flows of a project under inflation
•
In project appraisal the impact of inflation must be taken into account when calculating
the free cash flows to be discounted.
•
The impact of inflation can be dealt with in two different ways – both methods give the
same NPV.
Note:
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The real method can only be used if all cash flows are inflating at the general rate of
inflation.
•
In questions involving specific inflation rates, taxation or working capital, the
money/nominal method is usually more reliable.
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Net Present Value (NPV)
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Net Present Value (NPV)
The impact of taxation
There are two main impacts of taxation in an investment appraisal:
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tax is charged on operating cash flows, and
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tax allowable depreciation (sometimes referred to as capital allowances or writing down
allowances) can be claimed, thus generating tax relief.
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Net Present Value (NPV)
Revision of Tax Allowable Depreciation
In the year of sale or scrap a balancing allowance or charge arises.
$
Original cost of asset
X
Cumulative TAD claimed
(X)
–––
Written down value of the asset
X
Disposal value of the asset
(X)
–––
Balancing allowance or charge
X
–––
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Internal Rate of Return (IRR)
The IRR of a project has the following features:
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It represents the discount rate at which the NPV of an investment is zero.
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It can be found by linear interpolation.
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Standard projects (outflow followed by inflows) should be accepted if the IRR is greater
than the firm’s cost of capital.
The steps in linear interpolation are:
1 Calculate two NPVs for the project at two different costs of capital
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Internal Rate of Return (IRR)
The steps in linear interpolation are:
2 Use the following formula to find the IRR:
where:
L = Lower rate of interest.
H = Higher rate of interest.
NL = NPV at lower rate of interest.
NH = NPV at higher rate of interest.
3 Compare the IRR with the company’s cost of borrowing. If the IRR is higher than the cost of
capital, the project should be accepted
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The Modified IRR (MIRR)
Problems with using IRR
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The assumptions. IRR is often mistakenly assumed to be a measure of the return from
a project, which it is not. The IRR only represents the return from the project if funds
can be reinvested at the IRR for the duration of the project
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The decision rule is not always clear cut. For example, if a project has 2 IRRs (or more),
it is difficult to interpret the rule which says "accept the project if the IRR is higher than
the cost of capital".
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Choosing between projects. Since projects can have multiple IRRs (or none at all) it is
difficult to usefully compare projects using IRR.
It is therefore usually considered more reliable to calculate the NPV of projects for
investment appraisal purposes.
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The Modified IRR (MIRR)
A more useful measure is the modified internal rate of return or MIRR.
This measure has been developed to counter the above problems since it:
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is unique
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gives a measure of the return from a project
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is a simple percentage.
It is therefore more popular with non-financially minded managers, as a simple rule can be
applied:
MIRR = Project's return
If Project return > company cost of finance - Accept project
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The Modified IRR (MIRR)
The interpretation of MIRR
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MIRR measures the economic yield of the investment under the assumption that any
cash surpluses are reinvested at the firm’s current cost of capital.
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Although MIRR, like IRR, cannot replace net present value as the principle evaluation
technique it does give a measure of the maximum cost of finance that the firm could
sustain and allow the project to remain worthwhile.
•
For this reason it gives a useful insight into the margin of error, or room for negotiation,
when considering the financing of particular investment projects.
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The Modified IRR (MIRR)
Calculation of MIRR
where
PVR = the present value of the 'return phase' of the project
PVI = the present value of the 'investment phase' of the project
re = the firm's cost of capital.
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Discounted Payback Period
Discounted payback period
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Discounted payback period measures the length of time before the discounted cash
returns from a project cover the initial investment.
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The shorter the discounted payback period, the more attractive the project is. A long
discounted payback period indicates that the project is a high risk project.
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Macaulay Duration
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Duration measures the average time to recover the present value of the project (if cash
flows are discounted at the cost of capital).
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Duration captures both the time value of money and the whole of the cash flows of a
project. It is also a measure which can be used across projects to indicate when the
bulk of the project value will be captured.
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Projects with higher durations carry more risk than projects with lower durations.
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Discounted payback may lead a project to be discarded that has highly favourable cash
flows after the payback date
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Macaulay Duration
Comparison of Macaulay Duration and Modified Duration
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Macaulay Duration is the name given to the weighted average time until cash flows are
received, and is measured in years.
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Modified Duration is the name given to the price sensitivity and is the percentage
change in price for a unit change in yield.
•
Macaulay Duration and Modified Duration differ slightly, and there is a simple
relationship between the two (assuming that cash flows are discounted annually),
namely:
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Modified Duration = Macaulay Duration/(1+ cost of capital)
Therefore, in the above illustration, where the project cash flows were discounted at 8% 18
and the Macaulay Duration was 1.94 years, the Modified Duration is (1.94/1.08 =) 1.80
Multi-Period Capital Rationing
Capital rationing is where there are insufficient funds to do so.
This shortage of funds may be for:
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a single period only – dealt with as in limiting factor analysis by calculating profitability
indexes (PIs)
PI = NPV/PV of capital invested
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more than one period – extending over a number of years or even indefinitely.
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Multi-Period Capital Rationing
Revision of linear programming (LP)
The linear programme is formulated in three stages:
1 Define the unknowns.
2 Formulate the objective function.
3 Express the constraints in terms of inequalities including the non-negativities
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Monte Carlo Simulation
Overview of methods
The input variables in an investment appraisal are all estimates of likely future outcomes.
There are several methods of incorporating risk into an investment appraisal, for example:
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expected values (probability analysis)
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use of the CAPM model to derive a discount rate
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sensitivity analysis, and simulation.
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Monte Carlo Simulation
Probability analysis
For example, if sales are expected to be either $1,000,000 or $1,500,000 with probabilities
of 35% and 65% respectively, the expected sales can be calculated as:
($1,000,000 × 0.35) + ($1,500,000 × 0.65) = $1,325,000
The main problem with the expected value calculation is that the value might not
correspond to any of the possible outcomes, so although the calculation gives a useful
long-run average figure, it is not useful for one-off calculations
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Monte Carlo Simulation
Conditional probabilities
The article 'Conditional probability' in the Technical Articles section of the ACCA website
provides further details on this topic
Sensitivity analysis
Sensitivity analysis measures the change in a particular variable which can be tolerated
before the NPV of a project reduces to zero.
It can be calculated as
(NPV of project)/(PV of cash flows affected by the estimate) × 100%
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Monte Carlo Simulation
Simulation
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The main problem with sensitivity analysis is that it only allows us to assess the impact
of one variable changing at a time. Simulation addresses this problem by considering
how the NPV will be impacted by a number of variables changing at once.
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Simulation employs random numbers to select specimen values for each variable in
order to estimate a ‘trial value’ for the project NPV. This is repeated a large number of
times until a distribution of net present values emerge.
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By analysing this distribution, the firm can decide whether to proceed with the project.
For example, if 95% of the generated NPVs are positive, this might reassure the firm
that the chances of suffering a negative NPV are small.
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Value at Risk (VAR)
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Value at risk (VaR) is a measure of how the market value of an asset or of a portfolio of
assets is likely to decrease over a certain time, the holding period (usually one to ten
days), under ‘normal’ market conditions.
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VaR is measured by using normal distribution theory
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VaR = amount at risk to be lost from an investment under usual conditions over a given
holding period, at a particular 'confidence level'.
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Confidence levels are usually set at 95% or 99%,
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e.g. for a 95% confidence level, the VaR will give the amount that has a 5% chance of
being lost.
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Summary
1) Choose project which gives positive NPV
2) IRR is not the most appropriate method to appraise investment compared to MIRR and
NPV
3) Macaulay duration & modified duration takes into account time value of money
4) Only discuss multi-period capital rationing
5) Monte Carlo simulation overcomes sensitivity analysis problem which deals with one
variable only
6) VAR is the measure of risk at certain confidence level usually 99% and 95%.
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