discounted cash flow

Discounted cash flows

When working with capital budgeting techniques, future cash inflows and outflow are discounted to its present value. This is undertaken for number of reasons. Firstly, according to the time value of money, the dollar to be received in the future has less value than the dollar to be received in the present.

Moreover, adjustment for risk and inflation must also be made. Therefore, compensation is necessary to equalize future and present values of cash flows. Such compensation is known as the required rate of return. The higher is the required return (which is used as a discount rate) the lower will be the present value of the future cash flows from the proposed project.

To ensure that such compensation is in place, the future cash inflows are discounted to its present value with the use of required return. To do so, the following calculation can be made with the help of the financial calculator:

FV: (future cash flow)

N: number of periods between future cash flow and present

I: required return

PV: calculate

Test yourself:

ABC expects to have cash inflows of $5,000 over the next 5 years from the project A. The required return of ABC is 8%. What is the present value of the cash inflows?

Solution:

In this problem we have an annuity, which refers to a terminating an equal periodic cash flows over specific period of time, such as 10 years. Due to the fact that we are dealing with an annuity, we can calculate the present value of all cash inflows much faster and easier than if we were dealing with the mixed stream of cash flows. If we were dealing with the mixed stream, which refers to unequal cash flows during specific period with no precise pattern, than we would have to calculate present value of each cash flow separately and than add them together. We will do such calculation in the next problem. An annuity allows us to calculate present value of all cash flows in one step.

The calculation for this problem is as follows:

PMT: 5,000 (PMT here refers to equivalent cash inflows over period of time)

N: 5

I: 8

PV: calculate = 19,963.55

Test yourself:

ABC also have option to invest in project B. Cash flows from that project will be $3,000 at the end of year 1, $5,000 at the end of year 2, $4,000 at the end of years 3 and 4 and $8,000 at the end of year 5. Calculate the present value of above cash flows if required return of ABC is 8%.

Solution:

As promised, now we are dealing with the mixed stream of cash inflows. We will need to calculate present value of each cash flow separately and than add all present values. With the help of financial calculator, calculation will be as follows:

Cash flow in year 1:

FV: 3,000 (cash inflow at the end of year one)

N: 1

I: 8

PV: calculate 2,777.78

Cash flow in year 2:

FV: 5,000 (cash inflow at the end of year two)

N: 2

I: 8

PV: calculate 4,286.69

Cash flow in year 3:

FV: 4,000 (cash inflow at the end of year three)

N: 3

I: 8

PV: calculate 3,175.33

Cash flow in year 4:

FV: 4,000 (cash inflow at the end of year four)

N: 4

I: 8

PV: calculate 2,940.12

Cash flow in year 5:

FV: 8,000 (cash inflow at the end of year five)

N: 5

I: 8

PV: calculate 5,444.67

Now we will add up all present values to obtain the final answer:

=2,777.78 + 4,286.69 + 3,175.33 + 2,940.12 + 5,444.67

=18,624.59

Therefore, the present value of a mixed stream of cash flows from project B is $18,624.59.

If we look at this a previous problem we can conclude that, assuming that projects A and B require equal investment, project A is a better choice for capital expenditure because it offers a higher present value of future cash inflows ($19,963.55) from investment than does project B ($18,624.59).

Discounted Cash Flows

Risk Adjusted Discount Rate: Dealing with Risk in Capital Budgeting

Breakeven cash inflow analyses, risk adjusted discount rate (RADR) and scenario analyses are tools that facilitate better insight into managing risk in capital budgeting.

Risk in capital budgeting especially refers to variability of the returns (variability of cash inflows), because the initial investment is more or less known with some level of confidence. Therefore, we need to ensure that present value (PV) of cash inflows will be large enough to ensure that project is acceptable.

To adjust the present value of future cash inflows for risk embodied in particular project, we can either adjust cash inflow directly or we can adjust the discount rate. Because adjusting cash inflow is highly subjective, we will rather adjust discount rate. This is when risk adjusted discount rate technique comes into play.

RADR is a discount rate that must be earned to compensate an investor for the risk undertaken. Under RADR the value of the firm must be at least maintained or must increase. Risk adjusted discount rate is the most popular risk adjustment technique that utilize NPV.

The higher is the risk of specific project, the higher RADR will be.

The deployment of RADR is best illustrated by the use of an example:

EXAMPLE

Amanda can invest in two shares, A and B. Both shares presently cost $50 and Amanda wants to hold shares for 4 years. Annual dividends from share A expected to be $7. Annual dividends from share B are expected to be $12. However, shares B are more risky. In 4 years time Amanda expects to be able to sale shares A for $55 each and shares B for $70 each. Amanda’s required return is 8%. However, for shares B she adjusts her return so that her risk adjusted discount rate becomes 12%. Calculate risk adjusted net present values (NPVs) of shares A and B and recommend which shares should Amanda purchase.

Solution:

We will use financial calculator to find risk adjusted net present values (NPVs) of shares A and B.

Risk adjusted NPV of shares A:

Clear calculator: second function, C ALL

CFo: -50

CF1: 7

CF2: 7

CF3: 7

CF4: 62 (7+55)

I: 8

Second function, NPV: $15.38

Risk adjusted NPV of shares B:

Clear calculator: second function, C ALL

CFo: -50

CF1: 12

CF2: 12

CF3: 12

CF4: 82 (12+70)

I: 12

Second function, NPV: $30.94

Since investment in shares B offers higher risk adjusted NPV, Amanda should choose to invest in shares B.

The main difficulty in using risk adjusted discount rate (RADR) technique is in determining level of risk and approximating an appropriate risk adjusted discount rate (RADR). There is currently no systematic way to adjust required return to risk adjusted discount rate (RADR). Management usually determines risk adjusted discount rate (RADR) subjectively.

Sometimes risk index is determined which reflects risk adjusted discount rate (RADR) for every subsequent level of risk. For example, risk can be categorized into below average, average, above average and very high. Past experience and CAPM can be used to subjectively determine the risk adjusted discount rate (RADR) appropriate for each subsequent level (category) of risk.

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Risk Scenario Analysis: Dealing with Risk in Capital Budgeting

Another way to evaluate risk of the project is to undertake scenario analysis.

Scenario analysis focus on developing few alternative scenarios and evaluating variability between returns, which can be measured by net present value (NPV).

For example, we can generate 3 scenarios (optimistic, most likely and pessimistic) and than find NPVs for each of the scenarios. When we know net present values for each scenario, we can find the range.

The range here is found by taking NPV of optimistic outcome less NPV of pessimistic outcome, as shown below:

range (1) = NPV of optimistic outcome – NPV of pessimistic outcome

Alternatively, the range is found by taking annual cash inflow from optimistic outcome and subtracting annual cash inflow from pessimistic outcome, as shown below:

range (2) = annual cash inflow from optimistic outcome – annual cash inflow from pessimistic outcome

Range shows us variability between returns.

Risk in Capital Budgeting

When it comes to capital budgeting, risk refers to probability that project will proof to be unacceptable with net present value (NPV) less than zero or internal rate of return (IRR) less than cost of capital. Particularly, it refers to variability of the returns.

To find the minimum cash inflow level acceptable, we need to calculate breakeven cash inflow.

Breakeven cash inflow refers to the minimum cash inflow that it required for the project to be acceptable. It is calculated as follows:

PV – Initial investment

N – number of periods over which cash inflow is received

I – required cost of capital

Find PMT – breakeven cash inflow

Test yourself:

ABC Corporation have an option to invest in project A which requires investment of $120,000. The duration of the project is 5 years and ABC’s cost of capital is 9%. What is the breakeven cash inflow?

Solution:

We can find breakeven cash inflow of project A with the help of financial calculator.

PV: -120,000

N: 5

I: 10

Find PMT: $31,448.94

The above calculation helps us to determine that the minimum annual cash inflow that will be acceptable for project A is $31,448.94.

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Using a financial calculator

Using a financial calculator is a skill, similar to typing. You just need to know which steps to take and then you need to practice to the point when you feel comfortable with using a calculator.

In all explanations with a financial calculator we will be using an HP 10bll. Other financial calculators are similar, yet we find it easier to work with the HP. Most text books use HP calculators when providing guidance on using a financial calculator, so if you have a different calculator you may need to spend more time learning slightly different calculation steps. Before investing further time, it may be wise to get a universally used calculator.

Before using a financial calculator to make specific calculations such as calculating NPV or IRR, it is important to make sure that you:

1 – Clear the calculator – by pressing second function followed by “C All”

2 – Set calculator for the “END” by pressing second function followed by “BEG/END” and ensuring that the word “BEGIN” is not displayed. Exceptions to this rule occur when it is specifically stated in the problem that cash flows occur at the beginning of the period (for example, at the beginning of the year).

Again, if no sign appears on the display then you do not need to reset it as it is set for “END” by default. If it says “BEGIN” on the display, you need to press second function followed by “beg/end.”

When you set the calculator for the “END” of the period you do that because in the problem you are working with, cash inflow or outflow occurs at the end of the period. If the problem does not state when cash flows occur, you need to assume that it occurs at the “END” of the period.

The majority of calculations will require the “END” setting. If it is by mistake set for “BEGIN” but cash flows occur at the end of the period, then incorrect answers will be generated.

Therefore, it is advisable to keep it set for the “END” at all times as a default and only reset it for “BEGIN” when a calculation requires that to be done. Right after a calculation is completed that requires the “BEGIN” setting, it is important to develop a habit to reset it to the “END”.

In the explanations using a financial calculator, for convenience and clarity purposes, we will generally display explanations of calculations as presented in the example below:

PV: -900 I: 7 N: 5 FV: 1,262.3

When using a HP 10bll financial calculator, or using any financial calculator, you need to first insert the number (number, e.g. -900) and then insert the purpose of the number (e.g. PV).

For example, as per above, you need to press:

900 followed by the minus sign followed by PV

7 followed by I

5 followed by N

Than press FV, and the calculator will display the correct answer

Financial calculators sometimes give false answers. It is advisable to check each calculation 3-4 times to make sure that the same answer is given consistently.

Throughout the site, if you ever struggle with a calculation, always come back to this page for some simple tips on using a financial calculator.

Test yourself


ABC Corporation plans to invest in project C which has an initial investmentof $500,000. ABC’s cost of capital is 8%. The operating cash flows to be generated from the project will be as follows:

End of 1st year: $100,000 End of 2nd year: $300,000 End of 3rd year $250,000

1 – What is the Profitability Index (PI) for project C?

2 – What is the NPV for project C?

3 – Taking the NPV found in the previous step into account, is the project acceptable according to the NPV technique?

4 – Based on the Profitability Index (PI), is project C acceptable?

SOLUTION:

1 – First we need to find present values of the mixed stream of operating cash inflows. Using a financial calculator, we need to take the following steps:

End of 1st year:

FV: $100,000

N: 1

I: 10

Calculate PV: $90,909.09

End of 2nd year:

FV: $300,000 N: 2

I: 10

Calculate PV: $247,933.88

End of 3rd year:

FV: $250,000

N: 3

I: 10

Calculate PV: $187,828.7

Next we need to add up all present values from operating cash inflows to obtain the total PV of operating cash inflows:

= $90,909.09 + $247,933.88 + $187,828.7

Total PV of operating cash inflows = $526,671.67

Next we will follow the equation for Profitability Index (PI):

PI = Total present value of cash inflows/Initial investment

PI=$526,671.67/$500,000

PI=1.05

Therefore, the profitability index (PI) for project C is 1.05.

2 – To find NPV, we follow the formula for NPV:

NPV=Present value of cash inflows – Initial investment

Therefore, NPV for project C = $526,671.67 – $500,000

NPV for project C = $26,671.67

3 – Since NPV is more than zero ($26,671.67), project C is acceptable according to NPV technique.

4 – Since Profitability Index (PI) is greater than 1 (1.05), the project may be considered to be acceptable.

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Profitability index (PI)

Profitability Index (PI) is another sophisticated technique used in capital budgeting decisions. PI is related to NPV as it indicates the net present value (NPV) per each dollar invested. PI is calculated as follows:

PI =

Total present value (PV) of cash inflows divided by Initial investment.

PI is an especially advantageous technique if the company operates under capital rationing. If the PI is greater than one, the project is acceptable.

 

Equivalent Annual Annuity Approach (EAA)

Equivalent Annual Annuity approach (EAA) is another sophisticated technique used in capital budgeting decisions. EAA determines the annual cost of the project over its economic life. It is determined by dividing net present value (NPV) of the project by the present value interest factor for annuity (PVIFAr,n) for a specific period and at a specific discount rate. PVIFAr,n can be found in financial tables.

EAA is helpful when a project has to be selected from mutually exclusive projects with unequal lives. The project with the highest Equivalent Annual Annuity (EAA) is more attractive. If two mutually exclusive projects have equal EAA than the project with the shorter economic life is more acceptable.

 

The Difference Between IRR and NPV

This article answers two questions:

1 – What is important difference between IRR and NPV?

2 – Based on these differences and other considerations, which method is more popular and which method is theoretically superior?

What is important difference between IRR and NPV?

Net Present Value method assumes that cash inflows are reinvested at cost of capital, which is more realistic than assumption made in Internal Rate of Return method (IRR) that cash inflows are reinvested at IRR.

Based on these differences other considerations, which method is more popular and which method is theoretically superior?

Theoretically, it is advisable to use the Net Present Value method because it assumes that cash inflows are reinvested at cost of capital. However, in real life, the Internal Rate of Return method is more common because it considers the rate of return instead of dollar amount considered in the Net Present Value method and the former seems to be more intuitive to users of techniques. There are, however, ways to deal with shortcomings of Internal Rate of Return method and therefore IRR is still considered a sophisticated and reliable technique.

Internal Rate of Return method (IRR)

Sophisticated capital budgeting techniques include Net present value method (NPV), Internal Rate of Return method (IRR), Profitability index (PI) and Equivalent Annual Annuity (EAA). Internal Rate of Return method (IRR) is discussed below.

Internal Rate of Return (IRR) is a widely used technique.

It is also very easy to utilize Internal Rate of Return with the help of a financial calculator. It is much more challenging to calculate it by hand. Again, as in utilizing the NPV method, it is important to first understand the logic behind the calculation.

In simple terms, the IRR is the rate of return that would equate NPV with zero. If IRR higher than cost of capital than project should be accepted and vice versa. If IRR at least equals cost of capital than we know that business will earn at least rate equal to its cost of capital on this particular project.

Below is shown how to calculate IRR using the financial calculator.

IRR for annuity is calculated as follows:

Initial investment, minus sign – CFi

Annual cash inflow – CFi1

Number of periods – second function Ni

Second function IRR

IRR for a mixed stream is calculated as follows:

Initial investment, minus sign – CFi

Put in each cash inflow separately following with CFi1, CFi2 etc

Second function IRR

Both NPV and IRR will show whether the project is acceptable. However, the ranking of specific acceptable projects may differ between two techniques.

Test yourself

ABC have an option to invest in project B. The initial investment for project B is $35,000. Operating cash inflows from project B expected to be $5,000 per year for 8 years. The cost of capital of ABC is 5%.

What is the Internal Rate of Return (IRR) for project B?

Find out if project B is acceptable based on IRR calculation.

Solution:

With the help of financial calculator, we can determine IRR of project B as follows:

CFio: -35,000

CFi1: 5,000 (annual operating cash inflow)

Second function Nj: 8 (8 years)

Second function IRR: calculate – 3.07

The IRR of project B is 3.07%. The cost of capital of ABC is 5%. Since IRR (3.07%) is below cost of capital (5%), the project is not acceptable.

Test yourself

ABC have an option to invest in project D. The initial investment is $300,000. The operation cash inflows are expected to be $100,000 at the end of year 1, $110,000 at the end of year 2 and $130,000 at the end of year 3. The cost of capital of ABC is 10%.

  1. Calculate IRR
  2. Recommend if based on IRR technique project D is acceptable.

Solution:

1. With the help of financial calculator, the calculation is as follows:

Clear calculator: second function followed by C ALL

CFo: -300,000

CF1: 100,000

CF2: 110,000

CF3: 130,000

Second function IRR: calculate – 6.24%

2. Since cost of capital of ABC is 10% and IRR is only 6.24% (less than cost of capital), based on IRR technique, project D is not acceptable.

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Net Present Value Method

Sophisticated capital budgeting techniques include Net present value method (NPV), Internal Rate of Return method (IRR), Profitability index (PI) and Equivalent Annual Annuity (EAA). NPV is discussed below.

NPV is a sophisticated capital budgeting technique. Theoretically, Net Present Value (NPV) is the best technique out of sophisticated capital budgeting techniques but it is difficult to use it in practice. Sometimes Net Present Value method is referred to as the “gold standard” for investment decisions.

It is very easy to use Net Present Value with the help of a financial calculator if all necessary data is available. However, it is important to firstly understand the logic behind this calculation. NPV is determined by finding present value of cash inflows and then subtracting an initial investment.

NPV=Present value of cash inflows – initial investment

Now, after we understand the logic behind usage of Net Present Value method, we can calculate NPV using a financial calculator. We will always use a HP 10bll financial calculator throughout the website. Other calculators are similar but may have some small differences.

Before you make any calculations, make sure that you:

1 – Clear the calculator – by pressing the second function followed by “C All”

2 – Ensure that it is set for end if cash flows occur at the end of the period and that it is set for beginning if cash flows occur at the beginning of the period.

To set for end/beginning – press second function followed by beg/end. If it is set for the beginning than word “begin” will be displayed. If it is set for the end than no word will be displayed.

Majority of calculations will be with the “end” setting (used when cash flows occur at the end of the period). Therefore, it is important to acquire a habit of re-setting your calculator to the “end” setting after every calculation with the “begin” setting. Otherwise, you are running a risk of forgetting to re-set the calculator and obtaining an incorrect result from future calculations.

NPV FOR ANNUITY IS CALCULATED AS FOLLOWS:

Initial investment: amount, minus sign, CFi

Annual cash inflow: amount, CFi1

Number of periods: number of periods, second function, Ni

Cost of capital: number, i

Find NPV: second function, NPV

NPV FOR A MIXED STREAM IS CALCULATED AS FOLLOWS:

Initial investment: amount, minus sign, CFi

Put in amount for each cash inflow separately following with CFi1, CFi2 etc

Cost of capital: number, i

Find NPV: second function, NPV

If NPV is higher than zero than we know that this project will earn returns higher than the business’s cost of capital. Further, the owner’s wealth will increase by the amount equal to NPV.

Test yourself


ABC Corporation has an option to invest in projects A. Project A has aninitial investment of $15,000, and operating cash inflows of $3,000 over the economic life of the project, which is 8 years. The cost of capital (also called discount rate or rate of return) is 8%.

Find the net present value (NPV) of project A?

SOLUTION:

With the use of a financial calculator, we can find the net present value (NPV) as follows:

Clear the calculator by pressing second function followed by “C ALL”.

Make sure calculator is set to the “end”. This setting is used because in this problem cash flows occur at the end of each period. It is commonly accepted that if problem does not state when cash flows occur, you need to assume that cash flows occur at the end of the period, not at the beginning of the period.

WE KNOW THAT NPV FOR AN ANNUITY IS CALCULATED AS FOLLOWS:

Initial investment: amount, minus sign, CFi

Annual cash inflow: amount, CFi1

Number of periods: number of periods, second function, Ni

Cost of capital: number, i

Find NPV: second function, NPV

NOW YOU NEED TO PLUG IN THE NUMBERS:

NPV for annuity:

Initial investment: 15000, minus sign, CFi

Annual cash inflow: 3000, CFi1

Number of periods: 8, second function, Ni

Cost of capital: 8, i

Find NPV: second function, NPV

= $2,239.92

The above calculation makes it clear that project A is an acceptable project for ABC because the NPV is higher than zero ($2,239.92).

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