Discounted measures

DISCOUNTED MEASURES OF PROJECT WORTH

Net Present Value (NPV)

The cashflows estimated for the project are in the future; they are not yet realised

The future is not here yet, but decisions would have to be taken in the present time


The question then is, what is the value of these future estimated cashflows in the present or current period, or better still today?

future estimated cashflows would have to be ‘brought’ to the current or present period


Tt t

tt=1

B -CNPV =

(1+r)


T Tt t

t tt=1 t=1

B CNPV = -

(1+r) (1+r)


:

cost

t

Where

B is periodic benefit

C is periodic t

is the summation sign


Decision Rule: NPV > 0; project is viable, accept. NPV < 0; project is not viable, reject. NPV = 0; project is neither viable nor not

viable


The value of NPV suggests how much a project is adding in value terms to an existing entity or how much value the project is creating.

A positive NPV means that the project is expected to add value to the firm and will therefore increase the wealth of the owners.


Since the goal of projects is to add value or increase owner’s wealth, NPV is a direct measure of how well this project will meet the goal.

NPV has units of currency such as Rs or US dollars (US$).


Year 0 1 2 3 4 5 6 7

A -100 30 30 40 20 10 0 0

B -100 30 30 30 30 30 10 10


Cashflow Analysis for Project A and B

Cashflow Discount Factor Discounted Cashflow

Year A B (1+0.30)^-t A B

0 -100 -100 1.0000 -100.00 -100.00

1 30 30 0.7692 23.08 23.08

2 30 30 0.5917 17.75 17.75

3 40 30 0.4552 18.21 13.65

4 20 30 0.3501 7.00 10.50

5 10 30 0.2693 2.69 8.08

6 0 10 0.2072 0.00 2.07

7 0 10 0.1594 0.00 1.59

Net Present Value -31.2691 -23.2675


Advantages Takes opportunity cost of money into

account. A single measure, which takes the

amount and timing of cashflows into account.

With NPV one can consider different scenarios.


Results are expressed in value terms units of currency. So one is able to know the impact the value that the project would create.

It is based on cashflows, which are less subjective than profits.

Net Present Value (NPV) Disadvantages Complex to calculate and communicate. Meaning of the result is often misunderstood. Only comparable between projects if the initial

investment is the same.


It can be difficult to identify an appropriate discount rate.

Cashflows are usually assumed to occur at the end of a year, but in practice this is over simplistic.

Net Benefit Investment Ratio

Investments are required for project benefits to be realised.

These investments in the project cashflow can be identified as negatives.


The procedure: discount all the positive cashflows

separately discount all the negative cashflows

separately. Sum each of them The sum of positive discounted cashflows

is divided by sum of negative discounted cashflows.


1

1

(1 )

(1 )

T

tt

t

T

tt

t

B

iNBIR

K

i

where K is sum of negative net benefit or investment


The decision rule: NBIR > 1 accept; NBIR < 1 reject.

Net Benefit Investment RatioCashflow Analysis for Project A and B

Cashflow Discount Factor Discounted Cashflow

Year A B (1+0.30)^-t A B

0 -100 -100 1.0000 -100.00 -100.00

1 30 30 0.7692 23.08 23.08

2 30 30 0.5917 17.75 17.75

3 40 30 0.4552 18.21 13.65

4 20 30 0.3501 7.00 10.50

5 10 30 0.2693 2.69 8.08

6 0 10 0.2072 0.00 2.07

7 0 10 0.1594 0.00 1.59

Sum of +ves 68.7309 76.7325

Sum of -ves 100.00 100.00

NBIR 0.687309 0.767325


NBIR is also referred to as Profitability Index by the accounting profession.

It is often used for ranking projects especially if rationing is in place.

Benefit – Cost Ratio (BCR)

A variant of the formula for NPV uses the subtraction of discounted cash outflow from discounted cash inflow.

In the case of BCR, the discounted cash inflow is expressed in terms of the discounted cash outflow.


(1 )

(1 )

Tt

ttT

tt

t

B

rBCR

C

r


This can be viewed as: how many times the discounted cash

inflow covers the discounted cash outflow over the project horizon.


Decision criteria For a single project, a B/C ratio which is

greater than 1 indicates acceptability For multiple (competing) projects, the

project(s) with the highest B/C ratios (greater than 1) should receive highest priority


NPV measures totals, indicates the amount by which benefits exceed (or do not exceed) costs.

B/C measures the ratio (or rate) by which benefits do or do not exceed costs.

They are clearly similar, but not identical. With multiple projects, some may do better

under NPV analysis, others under B/C.

Internal Rate of Return (IRR)

IRR is the rate of return or discount rate that makes the NPV = 0.

Decision Rule: Accept the project if the IRR is greater than

the required return


This is the most important alternative to NPV. It is often used in practice and is intuitively

appealing. It is based entirely on the estimated cashflows and

is independent of interest rates found elsewhere. Without a financial calculator, this becomes a trial

and error process.


A critical thing to note is that there should be at least one change of sign in order to realise IRR.

there should be a negative net cashflow among positive net cashflows or a positive cashflow among negative cashflows.

The change in sign is crucial.


Using a spreadsheet; Start with the cashflows. You first enter your range of cashflows,

beginning with the initial cash outlay (negative).

Internal Rate of Return (IRR) Call the IRR function

Choose insert on the menu bar Select function Choose IRR from among the list Select the range of cashflows Enter a guess rate, but it is not necessary; Excel

will start at 10% as a default The default format is a whole percent – you will

normally want to increase the decimal places to at least two to get the most accurate output.


NPV and IRR will generally give us the same decision.

There are however some exceptions. Non-conventional cashflows

cashflow signs change more than once Mutually exclusive projects

Initial investments are substantially different Timing of cashflows is substantially different


When the cashflows change sign more than once, there is more than one IRR.

When we solve for IRR it would be noticed that we are solving for the root of an equation and when we cross the x-axis more than once, there will be more than one return that solves the equation.

Therefore, IRR may be unreliable if we have any negative cashflows after our original investment.


Suppose an investment will cost ¢90,000 initially and will generate the following cashflows: Year 1: 132,000 Year 2: 100,000 Year 3: -150,000

The required return is 15%. Should we accept or reject the project?


Year 0 -90,000

Year 1 132,000

Year 2 100,000

Year 3 -150,000

IRR 10.11% reject

NPV fx 15% 91,770

Less inv. -90,000

NPV at 15% 1,770 accept

IRR says to reject, but NPV says to accept. Go with NPV.


Mutually exclusive projects If you choose one, you can’t choose the

other Example: You can choose to attend

graduate school next year at either Legon or Central, but not both


Intuitively you would use the following decision rules: NPV – choose the project with the higher

NPV IRR – choose the project with the higher

IRR


Period Project A Project B

0 -500 -400

1 325 325

2 325 200

IRR 19.43% 22.17%

NPV 64.05 60.74


The required return for both projects is 10%.

Which project should you accept and why?

(Accept Project A because of NPV)

Internal Rate of Return (IRR) Conflicts between NPV and IRR

NPV directly measures the increase in value to the firm.

Whenever there is a conflict between NPV and another decision rule, you should always use NPV.

IRR is unreliable in the following situations Non-conventional cashflows Mutually exclusive projects

Internal Rate of Return (IRR) Advantages of IRR It takes into account the time value of money,

which is a good basis for decision-making. Results are expressed as a simple percentage,

and are more easily understood than some other methods.

It indicates how sensitive decisions are to a change in interest rates.


Advantages of IRR It is a simple way to communicate

the value of a project to someone who doesn’t know all the estimation details.

If the IRR is high enough, you may not need to estimate a required return, which is often a difficult task.

Internal Rate of Return (IRR) Disadvantages For mutually exclusive projects: timing and scale

differences. This may lead to incorrect decisions in comparisons of mutually exclusive investments.

Assumes funds are re-invested at a rate equivalent to the IRR itself, which may be unrealistically high.


IRR will produce more than one mathematically correct rate for each year in which inflows are followed by outflows and vice versa. This is common with projects with unconventional cashflows. This can create some confusion to the user.

Choice of Discount Rate Cost of capital - weighted average and marginal

(financing rate) ‘Opportunity cost’ of capital - what could they

earn if that money was elsewhere Current capital position and expected capital

position over next few years The rates of return for alternative investments. Market sentiments.

Sources of discount rate

Banks Long term government papers Ministry of Finance Sponsors

Suggestions

For industrial projects use market rate or cost of borrowing funds.

For public sector projects use social time preference rate.

For public projects to be funded from international loans use the cost of borrowing.

Suggestions Generally, in financial analysis, the market rate is

used, whilst the social time preference rate is used for public sector projects.

When funding comes from various sources or from the same source but at different rates, then, compute and use the weighted average.

Choosing Year 0 or Year 1

World Bank World Bank believes that since investment

is made and some returns may accrue from the first year, then discounting should start from 0 to first year.

In this case, the initial year is Year 1.

Choosing Year 0 or Year 1

Others Other international originations use Year 0. Their argument is that investment must

take place before benefits accrue. Thus, discounting should start from the

second year. Choose any convention but be

consistent.

Deciding on a Project

We should consider several investment criteria when making decisions.

NPV and IRR are the most commonly used primary investment criteria.

Payback is a commonly used secondary investment criteria, but only because of its ease of use.

Deciding on a Project

For a single project, a positive NPV indicates acceptability.

For multiple (competing) projects, the project(s) with the highest NPVs should receive highest priority.

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Discounted measures