Examining the Principles of the Net Present Value

In this essay I would be elaborating on the principles of the Net Present Value and its advantages. Also writing about Internal Rate of Returns and comparing the two methods and analyzing it.

When making and investment with regards to capital fixed assets such as plant and equipment the decisions are important given the significance of the capital outlay involved. In this essay we will be talking about two of the main methods which involves discounted cash flow methods

Net present value method ( NPV)

Internal rate of return (IRR)

However when it comes to these two methods the cash flow and the time value of money place a very important role

Cash flow

"Cash flow is essentially the movement of money into and out of your business; it's the cycle of cash inflows and cash outflows that determine your business' solvency" [1] when it comes to decision making you take into consideration the incremental cash flow ( cash flows that occurs directly as a consequence of taking on the project). This will depend on the prospect of the investment. Example - Opportunity cost, Taxes, depreciation

Time value of money

It's the amount of interest receive for a given amount of money over the amount of time. Invest look for the best return that they could get in the fastest time as possible, therefore when making an investment this concept is very important,

Net Present Value

"NPV is the method of discounting future streams of income using an expected rate of return to evaluate the current value of expected earnings. It calculates future value in today's money. NPV may be used to determine the current value of a business being offered for sale or capitalized". [2] 

NPV in project finance could be defined as the sum of present values (PVs) of cash flows expected from the project minus initial investment made. The formula for calculating NPV could be written as:

frac{R_t}{(1+i)^{t}}

Rt -net cash flow occurs at the end of each period 1

i - Discount rate used to Discount the cash flow (DCF)

t- Time period of the project

There are two kinds of NPV

Project NPV - The cash flow used to calculate the NPV would be the future operational cash flows of the project less initial project capital costs

Equity NPV - The cash flow used to calculate the NPV would be the equity distributions minus initial equity investment

However finance models are often presented in more detailed during construction period as opposed to during operations. For example we often find many project finance models have monthly calculations during construction and perhaps semi-annual / annual during operations. We might want to use XNPV instead of NPV, as returns the net present value for a schedule of cash flows that is not necessarily periodic.

Discounted Factor in NPV

"A discounted cash flow (DCF) is the most fundamentally correct way of valuing an investment." [3] 

There are some basic information that should be needed for this valuating approach;

Life of the Asset

The Cash flow during the time period of the Asset

The discount rate ( ex : the rate of interest) to apply to these cash flows to get the present value

The rate used to discount future cash flows to their present values is an important variable when it comes to finance and calculating NPV, some of the important once are ;

Weighted average cost of capital (WACC)

 Reinvestment rate

 Target rate of return

WACC

"A calculation of a firm's cost of capital in which each category of capital is proportionately Weighted" [4] , Common stock, bonds, preferred stock and other long term liabilities are included.

When finding WACC as an increase in WACC notes a decrease in valuation and a higher risk. However some may say it's appropriate use higher discount rates to adjust for risk of "Riskier" projects.

"The WACC equationA is the cost of each capital componentA multiplied by its proportional weight and then summing" [5] 

Weighted Average Cost Of Capital (WACC)

Where:

Re = cost of equity

Rd = cost of debt

E = market value of the firm's equity

D =A market value of the firm's debt

V = E + D

E/V = percentage of financing that is equity

D/V = percentage of financing that is debt

Tc =A corporate tax rate

a company's assets are financed by either debt or equity. WACC is the average of the costs of these financing sources, By taking a weighted average we can identify how much interest the company has to pay for every amount it finances. This is the overall required return on the firm as a whole and determines economic feasibility of an expansion opportunities and mergers. It is a good discount rate to use for cash flows with risk that is similar to that of the firm.

Reinvestment rate

This can be defined as the rate of investment from a firm on average, when analyzing projects in a capital constrained environment its more appropriate to use Reinvestment rate than WACC as the discounting rate. This is often calculated by considering the return on an alternative investment that can be made if the current project is not taken.

Target Rate of Return

"A method of pricing that estimates the desired return on investment to be achieved from the

fixed and working capital investment and includes that return in the price of a product/service" [6] this is used my monopolists or market leader

The general outcome of the NPV is without paying taxes the maximum amount that a firm could pay for the opportunity of keeping the investment without making a deficit.

NPV = Present Value Benefits - Present Value Cost

This could be easily modelled in a one shot occurrence project, however when it comes to mutually exclusive projects with unequal lives the theory may have to be altered.

Mutually exclusive projects with unequal lives

When comparing two mutually exclusive alternatives with significant different lives, The Net Present Value does not give the correct answer, therefore adjustments to the model is important. First thing when dealing with projects of this nature we need to find out if the project can be repeated and if so we must take this into accounts when estimating the projects profitability.

The example that I have taken is from Greg MacKinnon, Sobey School of Business, Saint Mary's University. In his example he has two brands of the same machine where Brand A costs more but lasts longer than Brand B.

Example: Machine A costs $10,000 and increases profits by $5000 per year. The machine lasts for six years (at which time it breaks down and is no longer usable). Machine B costs $5500 and increases profits by $5000 per year. This machine is expected to last for 3 years. Assume that the discount rate is 10%.

According to the above calculations Machine A has the best Net Present Value and therefore ideally the best option would be to choose machine A. If this project had no repetition and was a one shot deal Machine A would be the correct option, however machines have to be replaced due to wear and tear and depreciation. To overcome this matter and to make the best deal there are two methods to compare these alternatives

Matching Cycles or Replacement Chain ( Common life) Approach

This is method of comparing projects of unequal lives that assumes that each project can be repeated as many times as necessary to reach a common life span; the NPVs over this life span are then compared, and the project with the higher common life NPV is chosen.

find a common multiple of the two life lengths

use this as the total project length for both alternatives, where each alternative is repeated the necessary number of times

calculate NPV for both over this common time frame and choose the best alternative" [8] 

if we take the previous example of Machine A and B six years is a common multiple of its lives.

For Machine A, NPVA=11776.31 over six years.

For Machine B, the cash flows over six years are:

Year:

0

1

2

3

4

5

6

-5500

5000

5000

5000

-5500

5000

5000

5000

these cash flows are equivalent to:

Year:

0

1

2

3

4

5

6

6934.26

6934.26

With the comparisons made with the matching cycles method Machine B is better than Machine A.

2) Equivalent Annual Cost (EAC) or Uniform Annuity Series (UAS)

A method which calculates the annual payments a project would provide if it were an annuity. [10] When comparing a project of unequal lives the one with the higher annuity rate should be chosen

"EAC and UAS are the same. One term (EAC) is used when referring to problems of minimizing costs and the other term (UAS) is used when referring to problems of maximizing returns." [11] 

We know that Net Present Value of Machine A is 11776.31 and it lasts for six years. Therefore we are setting this model for six years annuity.

What this means is that the amount that you will receive per annum through the investment of Machine A will be £2703.93.

Similarly a three year annuity for Machine B

Both methods would give the same decision as to which Machine is the better option to choose. These methods are valid only if it is reasonable to assume that the machines will be replaced as they wear out and tear.

Internal Rate of Return (IRR)

"IRR is the rate at which the project NPV equals 0. It also provides the expected return rate of the project, assuming certain conditions are met." [14] Higher the rate of return more the investment looks promising. If all other factors are equal when comparing a project since Internal Rate of Return is a rate of quantity it indicates the efficiency, quality or the yield of an investment this is contrast to the NPV where it looks at the value or magnitude of the company. There are three ways of determining a project using IRR

Graphically

Using table of annuity factors

Trial & Error ( the most common one)

When calculating IRR the most important thing is that all details should be equivalent to one particular period

Calculating IRR

Using Tables of Annuity

Example - A company is investing in a new machine costing £104,320. The current annual cash flow of the company is £20,000 per year in operating cost over the machine life of 10years. What is the IRR of the project?

To find this we need to get the factor of IRR, afterwards we check it using the annuity table to find out the percentage

Factor of IRR =

5.216 =

The factor of 5.216 corresponds to the rate of return of 14%

Trial and Error Method

Take a discount rate and check the NPV. If it's a positive answer recalculate at a higher rate till you get a negative NPV and then interpolate the two rates.

Comparisons of NPV and IRR

When it comes to mutually exclusive projects using IRR would not be the best method because it could rank it incorrectly.

IRR

NPV

Project A

24%

£1540

Project B

18%

£1730

According to IRR method you should choose Project A but Project B has a higher NPV. Calculating IRR is hard since it may involve solving of a polynomial. More often than not the Trial & Error method is use for calculation.

IRR believes internal Cash flows are reinvested at the same rate, therefore it should not be used to compare projects of different durations, NPV assumes more realistic that they are reinvested according do debt and equity.

Projects with mixtures of positive and negative cash flows can result in multiple IRRs, therefore it's hard to find out which one is right or wrong

Despite the positivity of NPV, executives prefer IRR because of its easier to compare in percentages than in currencies.

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