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PV =

(1 + r )n

Note that the formula works irrespective of whether we are dealing with cash

inflows or cash outflows. A cash inflow is given a positive sign while an out-

flow is negative. The objective is always to choose the option with the highest

present value. A positive number is higher than a negative one and a small

negative number is higher than a large negative number. So if we have a com-

mitment to pay $100 now but are given the choice of paying $120 in three

yearsâ€™ time we can choose which option is better once we know our time

value of money. If our rate was 7% the calculation would look like this:

Immediate payment

PV = âˆ’$100

Delayed payment

âˆ’$120

PV =

(1.07 )3

âˆ’$120

=

1.225

= âˆ’$97.96

In this case we can see that the highest value is â€“$97.96 and so we can con-

clude that delayed payment is the better option.

Up to now we have only worked with whole numbers of years. It is easy to

conceptualise the idea of, say, (1.07)3. What should we do if the payment was,

say, two years and ten months into the future? Well, the formula works for

11 Building block 1: Economic value

non-integer values of n as well. Ten months is 0.833 of a year so the equation

would be (1.07)2.833. The arithmetic might appear a little complicated to those

who are not familiar with mathematics but todayâ€™s spreadsheets and even

many calculating machines can make the calculation easy. The spreadsheet

formula 1.07^2.833 tells us the answer is 1.211. So the present value would be

â€“$120 Ã· 1.211 or â€“$99.07. So we could conclude that it would still pay to elect

for the late payment option, but this time the present value benefit of doing

so has reduced from $2.04 to $0.93.

Introducing the idea of discount factors

We can make a useful simplification if we introduce the idea of a discount

factor. A discount factor is the amount by which you must multiply a future

value in order to compute its present value. The equations are:

PV = FV Ã— DF

where DF is the discount factor calculated as:

1

DF =

(1 + r )n

In the days before spreadsheets and electronic calculating machines, financial

analysis depended on so-called discount factor tables. These listed the discount

factor for different values of r and n. They can still be useful for doing quick

calculations without having to fire up a spreadsheet. An example follows:

Discount factors â€“ year-end cash flows

Number of years into the future â€“ n

r 1 2 3 4 5 6 7 8 9 10

1% 0.990 0.980 0.971 0.961 0.951 0.942 0.933 0.923 0.914 0.905

2% 0.980 0.961 0.942 0.924 0.906 0.888 0.871 0.853 0.837 0.820

3% 0.971 0.943 0.915 0.888 0.863 0.837 0.813 0.789 0.766 0.744

4% 0.962 0.925 0.889 0.855 0.822 0.790 0.760 0.731 0.703 0.676

5% 0.952 0.907 0.864 0.823 0.784 0.746 0.711 0.677 0.645 0.614

6% 0.943 0.890 0.840 0.792 0.747 0.705 0.665 0.627 0.592 0.558

7% 0.935 0.873 0.816 0.763 0.713 0.666 0.623 0.582 0.544 0.508

8% 0.926 0.857 0.794 0.735 0.681 0.630 0.583 0.540 0.500 0.463

9% 0.917 0.842 0.772 0.708 0.650 0.596 0.547 0.502 0.460 0.422

10% 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 0.386

12 The five financial building blocks

From the table we can see that the discount factor corresponding to 7% over

three years is 0.816. So in our delayed payment example above the present

value of delayed payment is:

âˆ’$120 Ã— 0.816 = âˆ’$97.92

The small difference of 4 cents between $97.96 and $97.92 is due to a round-

ing effect because the example table only shows discount factors to three sig-

nificant figures.7

One particular feature of discount factors which should be immediately

obvious is the way that the discount rate becomes increasingly important as

one considers sums of money that are further into the future. The discount

factor does not change much as you look down the one year column but there

are big changes in the year 10 column. Every year of delay reduces the present

value factor by a further amount of (1 + r).

Another point to note is the reference in the title of the table to year-end

cash flows. The discount factors have been calculated on the assumption

that the first cash flow occurs a full year into the future. There are many

occasions when this is exactly what is required. There are also, however,

situations when it is not. The most common of these is when dealing with

annual plans.

At the time the plan was formulated the present would be the beginning

of the current financial year. The cash flow shown in the first year of a plan

will usually be earned across the full year ahead. So would it be right to treat

it as though it happened all at the end of the year? Surely not. It is usually a

more accurate representation if the cash flows in a plan were all considered to

happen in the middle of each year. The discount factors for the various years

in a plan then become 1 Ã· (1 + r)0.5, 1 Ã· (1 + r)1.5 and so on. In pre-computer

days this was a bit of a chore but with a spreadsheet it is quite simple to do

these sums. We will explore this timing effect further in the second and third

examples in part 3 below.

Introducing annuity factors

An annuity is a fixed sum payable at specified intervals, typically annually,

over a period such as the recipientâ€™s life. How much might one have to pay in

The â€˜correctâ€™ number, to the nearest cent, is $97.96. One has to get used to small differences if tables,

7

as opposed to mathematical formulae, are used. It always pays to understand the limits of accuracy

of any answer. The usual situation is that one cannot be very sure what the exact time value of money

should be and so worrying about a few cents caused by rounding errors is usually a waste of time!

13 Building block 1: Economic value

order to buy one of these? If we apply the present value formula you would

expect to have to pay the present value of the future receipts.

We can see from the discount factor table above that a single payment

of $100 in a yearâ€™s time is worth $95.2 if the time value of money is 5%. If

we receive $100 a year for two years the present value will be $95.2 + $90.7

(i.e. $185.9). So if we produce a second table where the discount factors are

added across we will have a table that tells us the worth of an annuity for

any given number of years and discount rate. This is done in the following

table.

Annuity factors â€“ year-end cash flows

Present value of 1 per annum for n years

r 1 2 3 4 5 6 7 8 9 10

1% 0.990 1.970 2.941 3.902 4.853 5.795 6.728 7.652 8.566 9.471

2% 0.980 1.942 2.884 3.808 4.713 5.601 6.472 7.325 8.162 8.983

3% 0.971 1.913 2.829 3.717 4.580 5.417 6.230 7.020 7.786 8.530

4% 0.962 1.886 2.775 3.630 4.452 5.242 6.002 6.733 7.435 8.111

5% 0.952 1.859 2.723 3.546 4.329 5.076 5.786 6.463 7.108 7.722

6% 0.943 1.833 2.673 3.465 4.212 4.917 5.582 6.210 6.802 7.360

7% 0.935 1.808 2.624 3.387 4.100 4.767 5.389 5.971 6.515 7.024

8% 0.926 1.783 2.577 3.312 3.993 4.623 5.206 5.747 6.247 6.710

9% 0.917 1.759 2.531 3.240 3.890 4.486 5.033 5.535 5.995 6.418

10% 0.909 1.736 2.487 3.170 3.791 4.355 4.868 5.335 5.759 6.145

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