# Firmsconsulting

## Risk Adjusted Discount Rate: Dealing with Risk in Capital Budgeting

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:28 pm

Breakeven cash inflow analyses risk adjusted discount rate (RADR) andscenario 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 the shares for 4 years. Annual dividends from share A are expected to be \$7. Annual dividends from shares B are expected to be \$12. However, share B is more risky.

In 4 years time Amanda expects to be able to sale share A for \$55 each and share B for \$70 each. Amanda’s required return is 8%. However, for share B she adjusts her return so that her risk adjusted discount rate becomes 12%.

We need to calculate the risk adjusted net present value (NPV) of shares A and B (with the help of deployment of risk adjusted discount rate) and recommend which shares Amanda should purchase. SOLUTION:

We will be using a financial calculator to find a risk adjusted net present value (NPV) 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.

Sometimes a risk index is determined which reflects the 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 thecapital asset pricing model (CAPM)can be used to subjectively determine the RADR appropriate for each subsequent level (category) of risk.

## Risk Scenario Analysis: Dealing with Risk in Capital Budgeting

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:27 pm

Breakeven cash inflow analyses ,risk scenario analysis and risk adjusted discount rates are tools that facilitate better insight into managing risk in capital budgeting. Here we focus on risk scenario analyses.

This tool can be used to evaluate the risk of a project. It focuses on developing few alternative scenarios and evaluating the variability between returns, which can be measured by net present value (NPV).

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

The range is found by taking NPV of optimistic outcome and subtracting the NPV of pessimistic outcome, as shown below:

Range (Option 1) = NPV of optimistic outcome – NPV of pessimistic outcome.

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

Range (Option 2) = annual cash inflow from optimistic outcome – annual cash inflow from pessimistic outcome.

Range shows us variability between returns. When conducting risk analysis, remember to always use the MOST positive and negative numbers. Choosing anything between them will not produce the correct result.

## Risk in Capital Budgeting

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:26 pm

Risk in capital budgeting refers to the probability that a project will prove to be unacceptable with a net present value (NPV)less than zero or aninternal rate of return (IRR)less than the cost of capital. Particularly, it refers to variability of the returns (variability of cash inflows) because theinitial investment is more or less known with some level of confidence. Therefore, we need to ensure that cash inflows will be large enough to ensure that project is acceptable. Breakeven cash inflow, scenario analyses and risk adjusted discount rates are tools that facilitate better insight into managing these risks.

## Using a financial calculator

In Capital Budgeting, Finance on October 27, 2010 at 6:25 pm

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 a 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.

## Profitability index (PI)

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:24 pm

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)

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:23 pm

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

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:01 pm

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)

In Capital Budgeting, Finance, MBA on October 27, 2010 at 6:00 pm

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 one of the sophisticated capital budgeting techniques. It is a widely used technique.

It is also very easy to utilize Internal Rate of Return with the help of afinancial 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.

Lets use a simple example to explain this. IRR is WACC, but only when the WACC results in the NPV equal to zero. That means IRR is another form of cost of capital. IRR is a theoretical number, while WACC is a real number. In simple terms, the IRR is the rate of return that would equate the NPV with zero. If IRR is higher than the cost of capital, then a project should be accepted, and vice versa. If IRR at least equals the cost of capital than we know that the business will at least earn a rate equal to its cost of capital on this particular project.

### Example

Let’s see how to utilize Internal Rate of Return method with the help of a financial calculator.

IRR 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

Find IRR: second function, IRR

IRR 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

Find IRR: second function, IRR

Both Net present value method (NPV) and Internal Rate of Return method (IRR) will show whether the project is acceptable. However, the ranking of specific acceptable projects may differ between the two techniques.

### Test yourself

ABC Corporation has an option to invest in project B. The initial investment for project B is \$35,000. Operating cash inflows from project B are 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 the Internal Rate of Return method.

SOLUTION:

With the help of a financial calculator, we can determine the 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 the IRR (3.07%) is below the cost of capital (5%), the project is not acceptable.

TEST YOURSELF:

ABC Corporation has 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 the Internal Rate of Return method, project D is acceptable.

SOLUTION:

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

STEP ONE

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%

STEP TWO

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

### Comparing NPV and IRR

You will notice these next two paragraphs appears verbatim on 3 oher pages. There is a reason for this. Understanding the difference between NPV and IRR is critical. Make sure you understand the differences, and are able to apply both techniques.

Theoretically, it is advisable to use Net Present Value method because NPV 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.

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 can be considered a sophisticated and reliable technique.

## Net Present Value Method

In Capital Budgeting, Finance, MBA on October 27, 2010 at 5:58 pm

Sophisticated capital budgeting techniques include Net present value method (NPV), internal rate of return (IRR), Profitability index (PI) and Equivalent annual annuity (EAA). NPV and IRR are discussed below.

### NPV

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).

### Comparing NPV and IRR

Theoretically, it is advisable to use Net Present Value method because NPV 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 reinvested at IRR.

However, in real life, the IRR 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 IRR and therefore IRR is still can be considered a sophisticated and reliable technique.

## Unsophisticated capital budgeting techniques

In Capital Budgeting, Finance, MBA on October 27, 2010 at 5:57 pm

Simple (unsophisticated) capital budgeting techniques include average rate of return (ARR) and the payback method (also called PB or payback period).

### Average Rate of Return

Average Rate of Return (ARR) is an unsophisticated budgeting technique and generally considered to be ineffective. Average Rate of Return (ARR) evaluates relative profitability of the investment. In other words, it evaluates how project affects accounting profits. Average Rate of Return (ARR) is calculated as follows:

ARR = Average income / Average investment

Average income refers to annual average net profits after tax (refer toincome statement to see how net profits after tax are determined). Annual average net profits after tax is found by taking total net profits after tax over the useful life of the project and dividing it by number of years over useful life of the project. Average investment refers to average investment over the economic life of the project. The ARR capital budgeting technique does not consider the time value of money. It also considers net profits rather than cash inflows. Consequently, the technique overlooks the possibility of reinvestment of returns.

The positive side of this technique, as compared to payback period discussed below, is that it considers returns on investment over entire useful life of the project. However, this technique is generally not recommended.

### Payback method

Payback period (PB), also called a payback method, is another unsophisticated budgeting technique. It determines how long it takes to recover the initial investment by taking into account cash inflows from the investment. If we deal with an annuity (an equal periodic cash flow over a specific period) than all we need to do is to divide initial investment by an annuity.

However, if we deal with a mixed stream of cash inflows (unequal cash flows during specific period with no precise pattern) than we need to add up cash flows until the initial investment is recovered.

Management needs to subjectively determine the maximum payback period and then projects are evaluated according to this. If the project’s payback period is below maximum than the project is acceptable and vice versa.

The payback period budgeting technique measures business’s risk exposure because the project’s risk level depends on how long it takes to recover the initial investment. However, it does not explicitly consider the time value of money.

Moreover, this budgeting technique is weak because it is subjective in nature since the minimum payback period is subjectively determined. Furthermore, it does not take into account the cash flows that occur after the payback period.

A variation of payback period capital budgeting technique allows to account for time value of money and risk (due to usage of discount rate which incorporates risk). Such variation is called discounted payback period technique. This technique determines how long it takes for discounted cash flows to recover the investment. However, this variation still does not consider cash flows after the payback period.

### Test Yourself

ABC Corp has a proposed project A, which has expected cash inflows of \$4,000 over 10 years period. The initial investment is \$30,000. Find the payback period.

SOLUTION:

Payback period = 30,000/4,000 = 7.5 years

This means that it will take 7.5 years for ABC to recover its investment in project A.