Unit 5

Project Appraisal

The time value of money

lends £100

Ozzy Sharon

The time value of money

lends £100

Ozzy Sharon

Ozzy Sharon

repays £100 either tomorrow or,

£100 in one year’s time

Which will Ozzy choose?

The time value of money

lends £100

Ozzy Sharon

Ozzy Sharon

repays £100 either tomorrow or,

£100 in one year’s time

Which will Ozzy choose?

Clearly Ozzy will choose to have his money tomorrow. Why is this such an easy decision?

Time value of money

Ozzy wants to be compensated for waiting.

Three reasons:

Risk – Sharon is more likely to be able to repay him tomorrow than one years time

Inflation - £100 will be worth less in one year’s time than tomorrow

Reinvestment - Ozzy can use the money from tomorrow to make a return

The time value of money

lends £100

Ozzy Sharon

Ozzy Sharon

repays £100 either tomorrow or,

£1000 in one year’s time

Which will Ozzy choose?

The time value of money

lends £100

Ozzy Sharon

Ozzy Sharon

repays £100 either tomorrow or,

£1000 in one year’s time

Which will Ozzy choose?

Now Ozzy will probably choose to wait a year – 900% is a very good return!

The time value of money

So at £100 in one year’s time it is a “no-brainer” – Ozzy will turn down the deal

At £1,000 in one year’s time it is also a no-brainer – Ozzy will accept.

The time value of money

So at £100 in one year’s time it is a “no-brainer” – Ozzy will turn down the deal

At £1,000 in one year’s time it is also a no-brainer – Ozzy will accept.

This means that there is a tipping point somewhere between 0% and 900% return. At this point Ozzy is indifferent – a little less and he will not lend, a little more and he will.

The time value of money

So at £100 in one year’s time it is a “no-brainer” – Ozzy will turn down the deal

At £1,000 in one year’s time it is also a no-brainer – Ozzy will accept.

This means that there is a tipping point somewhere between 0% and 900% return. At this point Ozzy is indifferent – a little less and he will not lend, a little more and he will.

This point reflects Ozzy’s “time value of money” – how much he needs to be compensated for waiting for his repayment.

The time value of money

Let us assume that Ozzy’s time value of money is 10%.

If he lends Sharon £100 and she offers to repay him £110 he is indifferent – he has been exactly compensated for waiting – it makes absolutely no difference to him if he lends (or invests) or not

The time value of money

Let us assume that Ozzy’s time value of money is 10%.

If he lends Sharon £100 and she offers to repay him £110 he is indifferent – he has been exactly compensated for waiting – it makes absolutely no difference to him if he lends (or invests) or not

If Sharon offers him £109 in one year’s time then he will not invest – he has not been adequately rewarded.

If she offers £111 in one year’s time, he will invest.

The time value of money

Let us assume that Ozzy’s time value of money is 10%.

If he lends Sharon £100 and she offers to repay him £110 he is indifferent – he has been exactly compensated for waiting – it makes absolutely no difference to him if he lends (or invests) or not

If Sharon offers him £109 in one year’s time then he will not invest – he has not been adequately rewarded.

If she offers £111 in one year’s time, he will invest.

But at £110 he does not care either way. So if your time value of money is 10% then:

£100 today has the same value to you as £110 in one year’s time

The time value of money

Instead of one year, let us look at a two year deal.

If the time value of money is now 9%. Then Sharon would need to offer more than £100 x 1.09 x 1.09 = £118.81 in two years time. The tipping point is now £118.81.

The time value of money

Instead of one year, let us look at a two year deal.

If the time value of money is now 9%. Then Sharon would need to offer more than £100 x 1.09 x 1.09 = £118.81 in two years time. The tipping point is now £118.81.

So (for 9%) £100 today has the same value as £118.81 in two years time.

This is a mathematical statement of common sense: £100 today is worth more than £100 in the future. This technique allows us to quantify how much more valuable it is.

Time value of money

Let us look at this a different way.

Assume that the time value of money is 10%. We can state the future value of £100.

Today (0) 1 2 34

100 110 121 133.1146.4

Time value of money

Let us look at this a different way.

Assume that the time value of money is 10%. We can state the future value of £100.

Today (0) 1 2 34

100 110 121 133.1146.4

[100 x 1.1 x1.1 x1.1 x1.1]

Remember these sums are all identical = £100 today is worth the same as £146.4 in 4 years time (at 10%).

Time value of money

So what is the present value of future amounts? In other words, what is £100 in the future worth in today’s money?

Today (0) 1 2 34

? 100

? 100

? 100

?100

Time value of money

So what is the present value of future amounts? In other words, what is £100 in the future worth in today’s money? Before, we multiplied by the time value (“compounding”). To go backwards, we divide by the time value (“discounting”)

Today (0) 1 2 34

? 100

? 100

? 100

?100

Time value of money

So what is the present value of future amounts? In other words, what is £100 in the future worth in today’s money? Before we multiplied by the time value (“compounding”). To go backwards, we divide by the time value (“discounting”)

Today (0) 1 2 34

90.9 100

? /1.1 100

? 100

?100

Time value of money

So what is the present value of future amounts? In other words, what is £100 in the future worth in today’s money? Before we multiplied by the time value (“compounding”). To go backwards, we divide by the time value (“discounting”)

Today (0) 1 2 34

90.9 100 [ie 90.9 x 1.1 = 100]

? /1.1 100

? 100

?100

Time value of money

So what is the present value of future amounts? In other words, what is £100 in the future worth in today’s money? Before we multiplied by the time value (“compounding”). To go backwards, we divide by the time value (“discounting”)

Today (0) 1 2 34

90.9 100

82.6 /1.1 100

? /1.1 100

?100

Time value of money

So what is the present value of future amounts? In other words, what is £100 in the future worth in today’s money? Before we multiplied by the time value (“compounding”). To go backwards, we divide by the time value (“discounting”)

Today (0) 1 2 34

90.9 100

82.6 /1.1 100

75.1 /1.1 100

68.3 /1.1100

/1.1

Discounted Cash Flow (DCF)

So at 10%, £100 in 2 years time is worth £82.60 today - £100 in 4 years time is worth £68.3 today.

This discounting can be done either by dividing (1 + discount rate/100) or by using discount table (see Vital Statistics)

Discounted Cash Flow (DCF)

So at 10%, £100 in 2 years time is worth £82.60 today - £100 in 4 years time is worth £68.3 today.

This discounting can be done either by dividing (1 + discount rate/100) or by using discount table (see Vital Statistics)

If I want to assess an investment (or “project”) then I do not want to compare cash flows in year 4 with those in year 2. We now accept that these amounts have different values. So I will restate all of the cash flows in the same year (normally today – the present – year 0) and compare.

Showing the cash flow like this is called a discounted cash flow (dcf)

DCF - example

Let us assume that an investment has the following cash flows – discount rate 9%:

0 1 2 34

CF (1000) 100 200 400600

DCF - example

Let us assume that an investment has the following cash flows – discount rate 9%:

0 1 2 34

CF (1000) 100 250 400600

Notice the layout – to do a DCF you need to total each year’s cash flow. Then discount it

[0.9174 0.8417 0.7722 0.7084]

DCF (1000) 91.7 210.4 308.9425.1

DCF - example

Let us assume that an investment has the following cash flows – discount rate 9%:

0 1 2 34

CF (1000) 100 250 400600

Notice the layout – to do a DCF you need to total each year’s cash flow. Then discount it

[0.9174 0.8417 0.7722 0.7084]

DCF (1000) 91.7 210.4 308.9425.1

Eg. 0.8417 = 1/(1.09 x 1.09) = 1/ (1.09)2

So 250 x 0.8417 = 250/(1.09 x 1.09) = 250/(1.09)2

DCF - example

Let us assume that an investment has the following cash flows – discount rate 9%:

0 1 2 34

CF (1000) 100 250 400600

Notice the layout – to do a DCF you need to total each year’s cash flow. Then discount it

[0.9174 0.8417 0.7722 0.7084]

DCF (1000) + 91.7 + 210.4 + 308.9 +425.1 + = 36.1

The total of this is called the net present value (NPV) = 36.1.

This is positive – the present value of the future cash flows (+ve) is greater than the present value investment– so the investor has been more than adequately rewarded. Ozzy has received more than his tipping point! If funds are available then the investor should invest.

Relevant Cash Flows

Note that it is a discounted “cash flow”. We do not use profit forecasts for this analysis. This is because only cash has a present value – it is subject to inflation, may be reinvested etc.

Also ensure only “relevant” cash flows are included. These are any cash flows (and only those cash flows ) that arise because of the decision to invest.

Net present value

In the earlier example – what would have happened to the NPV if the discount rate had been 12% instead of 9%.

Net present value

In the earlier example – what would have happened to the NPV if the discount rate had been 12% instead of 9%.

The future (positive) cash flows would have been worth less today, the present day investment would be the same, so the npv would have been less.

Net present value

In the earlier example – what would have happened to the NPV if the discount rate had been 12% instead of 9%.

The future (positive) cash flows would have been worth less today, the present day investment would be the same, so the npv would have been less.

In fact the npv would become -45.4 (so you would not invest) [this is Ozzy receiving less than the tipping point]

Between 9 and 12% there is a rate at which the investment yields exactly zero (the present investment = present value of future cash flows) or NPV = 0.

This rate is called the “internal rate of return” – it is the annual rate of return for the project allowing for the timings of the cash flows (in this example it is 10.3%)

Competing projects

You have £100m to invest – which project/s should you invest in? (assume that none of the investments are scaleable)

Project Investment/£m NPV IRR%

A 30 69

B 50 1010

C 85 1511

D 15 310

Competing projects

You have £100m to invest – which project/s should you invest in? (assume that none of the investments are scaleable)

Project Investment/£m NPV IRR%

A 30 69

B 50 1010

C 85 1511

D 15 310

You would choose the combination which gives the highest NPV: A+B+D = 18

Geared beta

Last time we discussed the beta of a share. This is a key input for determining the cost of equity and therefore the weighted average cost of capital.

A company’s beta, [the “equity beta” as measured on a stock exchange] reflects two things:

Operational risk - what is does

Financial risk – how much it has borrowed

Geared beta

Last time we discussed the beta of a share. This is a key input for determining the cost of equity and therefore the weighted average cost of capital.

A company’s beta, [the “equity beta” as measured on a stock exchange] reflects two things:

Operational risk - what is does

Financial risk – how much it has borrowed

So the measured (ie on the stock market) beta is called the equity beta. However if the company is going to take on a lot of debt to fund an investment then its equity beta will change to reflect the increased debt (it has become riskier and so its beta should increase)

Geared beta

So to arrive at its new beta :

1 Calculate its beta with no debt ( = asset beta)

β asset = β equity x E/(D+E) (D=old debt, E = old equity)

Geared beta

So to arrive at its new beta :

1 Calculate its beta with no debt ( = asset beta)

β asset = β equity x E/(D+E) (D=old debt, E = old equity)

2 Calculate the new equity beta using the new D/E ratio

β equity = β asset x (D+E)/E (D= new debt, E= new equity)

Tax in DCF calculations

The first thing to realise is that the method given in the OU material is a simplification of the UK corporation tax system – you will probably need to be more rigorous in practice

Tax in DCF calculations

The first thing to realise is that the method given in the OU material is a simplification of the UK corporation tax system – you will probably need to be more rigorous in practice

Tax is payable on the profit of “revenue” items (sales – operating costs)

Tax relief is sometimes possible for “capital” items (investment) – this is called “capital allowance” (CA)

Tax in DCF calculations

Let us consider an example. The pre-tax cash flow is:

0 1 2 3

Investment -360

Net operating 100 300 500

Tax in DCF calculations

Let us consider an example. The pre-tax cash flow is:

0 1 2 3

Investment -360

Net operating 100 300 500

Calculate tax in a separate box – assume 25% rate

0 1 2 34

Operating -25 -75-125

CA

Note that the tax is “paid” one year after the year in which it “arises”

Tax in DCF calculations

Let us consider an example. The pre-tax cash flow is:

0 1 2 3

Investment -360

Net operating 100 300 500

Calculate tax in a separate box – assume 25% rate. Let us assume CA spread over 3 years

0 1 2 34

Operating -25 -75-125

CA 30 30 30

[360 x 25% = 90. = 90/3 each year]

Tax in DCF calculations

Let us consider an example. The pre-tax cash flow is:

0 1 2 3

Investment -360

Net operating 100 300 500

Calculate tax in a separate box – assume 25% rate. Let us assume CA spread over 3 years

0 1 2 34

Operating -25 -75-125

CA 30 30 30

0 0 5 -45-95

Tax in DCF calculations

Let us consider an example. The pre-tax cash flow is:

0 1 2 34

Investment -360

Net operating 100 300 500

Tax 5 -45-95

Cash flow -360 100 305 455-95

0 1 2 34

Operating -25 -75-125

CA 30 30 30

0 0 5 -45-95

Next Time

We shall look at company valuations from unit 6