note

So far we have based the analysis of the “desirability” of an investment on the “ worth ” it would produce, called it NPW in Chapter 5 and EUAW in Chapter 6. These two methods do not really measure the “attractiveness” of an investment. For example, suppose you are considering investing $1,000 in a venture which would produce a Net Present Worth of $900, the NPW is positive, but is the investment “attractive”?

We now introduce another parameter, is the relation between “ worth ” produced and the amount of “ invested capital ” necessary to produce it.

7-1 RATE OF RETURN (ROR) vs. RETURN ON INVESTMENT (ROI)

The performance of an investment is measured by a parameter called Return On Investment:

RETURN ON INVESTMENT (ROI) refers to the percentage of increase (or decrease) of the Net Present Worth of the Investment over the Present Worth of the Invested Capital, during the life of the investment

ROI = [(NPW Total Investment)/(PW Invested Capital)] -1

For example, suppose an investment of $1,000 generates a cash flow whose NPW is $1,100, then the Return On Investment would be

ROI = (1,100 / 1000) – 1 = 1.1 – 1 = 0.1 = 10%

Recalling earlier discussions, the NPW value is a function of three parameters:

· The investment’s cash flow

· The timing of such cash flow

· The cost of capital for the investing Entity, i.e. the applicable interest rate i

The first two parameters are characteristics of the specific investment, while the third, the interest rate, depends on the money market, i.e. the market price of the availability of money for the Investing Entity.

Example 7.1 - Let us consider a simple example of how the NPW varies as a function of i: Consider an investment of $150,000 over a period of 7 years, producing a yearly string of benefits of $50,000. The variation of NPW with i is easily calculated by using Excel’s NPV Financial Function:

The Excel generated graph shows how NPW value varies as a function of i.

Considering now the ROI:

The ROI value becomes negative between the interest rates of 3% and 4% as clearly indicated by the fact that at such interest rates the value of NPW becomes smaller than the invested capital P.

The graphs above also clearly indicate that interest rates and ROI (as well as NPW) are inversely proportional to each other: as one increases, the other decreases

To calculate the correct ROI, the applicable interest rate i must be known .

Note in Example 7.1 that NPW becomes negative when the interest rate is greater than 27% but the investment becomes economically unattractive when the interest rate is greater than approx. 3.65%

7-2 INTERNAL RATE OF RETURN

As previously mentioned, the “return” of an investment is determined by three parameters:

1. The cash flow generated by the investment - The face values of the cash flows generated by the investment are a specific characteristic of the type of investment activity, it is “internal” to the investment activity

2. The timing of such cash flow - The timing of these cash flows, i.e. the times each cash sum flows in or out, also is a specific characteristic of the type of investment activity, it is “internal” to the investment activity

3. The cost of capital for the investing Entity, i.e. the applicable interest rate i - The cost of capital for the investing entity, on the other end, is determined by the money market, it is “external” to the investment activity.

We now define INTERNAL RATE OF RETURN (IRR) the specific interest rate which makes the NPW value equal to zero; in other words, when i = IRR the present value of Benefits equals the present value of Cost:

When i=IRR

NPW= PVbenefits – PVcosts = 0

EUAW = EUAB – EUAC = 0

The adjective “internal” indicates that the determination of i=IRR depends exclusively from the “internal” characteristics of the investment activity, with no relation to the money market.

In Example 7.1 the IRR of the investment is IRR = 27.12%, as can be found using the Excel Financial Function IRR:

The importance of IRR appears evident: the value of interest rate i=IRR is the boundary between gain and loss (i.e. NPW ≥0 or NPW ≤0) for an investment because at that value the NPW becomes zero, however it is not an indication of the “desirability” of proceeding with the investment, “gain” does not means “acceptable profit”, an investor most likely will want a certain level of profit rather than just “getting even”. This consideration requires the introduction of another concept:

7-3 MINIMUM ACCEPTABLE RATE OF RETURN (indicated as MARR in the text)

As discussed above, the NPW and the ROI of an investment are functions of the cost of capital, which is the interest rate applicable to the specific investor for the specific investment activity. It follows that once an investor identifies the minimum ROI he/she would accept , then the corresponding maximum cost of capital he/she could accept is also identified ( remember NPW and ROI are inversely proportional to i ). That maximum acceptable cost of capital is defined in our text as MARR since it corresponds to the Minimum Acceptable Rate of Return

Example 7.2 – Your boss is considering an investment to improve the productivity of the Company for the next 15 years. He/she is considering acquiring a new machine which will cost $ 50,000, and have a useful life of 10 years, at which point he/she plan to sell the old machine for a salvage price of $ 10,000, and buy a new identical one for same price, and sell it after 5 years, at the end of the project life, for a salvage price of $ 20,000. The new machine would produce a steady stream of $ 35,000 of yearly benefits and cost $ 500 per year for Operation and Maintenance. Your boss tells you that he/she want to make at a minimum a Return Of Investment of 70% , and he/she asks you to find out what interest rate must be obtained from the Lending Institution to reach such ROI level.

With the use of Excel the solution is easily found:

As shown in the Figure above, the MARR for this investment is between 13% and 14%, while the IRR is between 68% and 69%.

The example also clearly shows that

IRR IS DETERMINED BY THE STRUCTURE OF THE INVESTMENT, WHILE MARR IS DETERMINED BY THE INVESTOR

Example 7.3 – You have put $150,000 into a saving account and want to withdraw $20,000 per year for the next 10 years. What is the interest rate that you must obtain from the Saving Institute?

Since the PV of the string of $20,000 payments must be equal to the deposited $150,000 at the present, this is an IRR calculation because the NPW is zero

20,000 (P/A, i, 10) = 150,000

This equation could be solved by trial and error changing the values of i until we get the correct answer, or more simply by using the IRR Function in Excel to find the interest rate which will set the NPW of the cash flow (initial sum and all the subsequent withdraws) equal to zero

This value can be doublechecked easily :

20,000 (P/A, 5.6, 10) = 20,000 x 7.502 = 150,040

(with usual negligible discrepancy due to rounding up)

Example 7.4 (similar to example 7-4 in text) – a student borrows $ 10,000 per year throughout his/her college, i.e. for four years at the beginning of each year. No repayment is due while he/she is in college. Yearly end-of-year payments must begin after graduation at an interest rate of 6.5%, and the loan must be paid off in 5 years. What is the real interest rate charged?

The Cash Flow Diagram is

At Graduation the student owes $10,000 x 4 = $ 40,000 which must be repaid in 5 yearly installments at an interest of 6.5%, so the yearly payments are

Payment x (P/A, 6.5, 5) = 40,000

Payment = 40,000 / (P/A, 6.5, 5) = 40,000/4.156 = $9624.64/yr

Using the IRR Function in Excel:

The real interest rate charged to the student is 3.43%

7-4 INCREMENTAL ANALYSIS

When an investor must decide between two (or more) competing potential investment, the decision is based on the difference between the two alternatives, in other words, according to his/her DECISION CRITERION, the investor considers the difference in investment between investments as an investment in itself and calculates the ∆IRR of the ∆Cash Flow, or the MARR set as the goal of the investment

Note that this is not different from calculating the IRR for one single potential investment, because in such case the IRR actually identifies the boundary between proceeding with the potential investment and the other potential alternative which is “do nothing”.

The difference between the two alternatives is computed as “the more costly alternative minus less costly alternative” and the investor must decide if it is worth investing in the difference.

Let us consider the following simple case:

Example 7.5 – An investor wants to buy a new piece of machinery for his/her shop, and is considering two potential candidates

Machine A Machine B

Price 70,000 55,000

O&M/year 5,000 3,000

Benefit/year 30,000 25,000

Salvage Value 15,000 8,000

Life (years) 10 10

Using an Excel spreadsheet:

As we can see from the graph, if the cost of capital for the investor is less than 17.7% alternative A is the better one since its NPW is greater, in other words the extra investment to buy the more expensive machine is worth from the standpoint of maximizing NPW. From 17.7% up to 34.218% alternative B is better. If the cost of capital is greater than 34.218% none of the two alternatives is financially feasible since both would generate a negative NPW.

However, if the selection criterion is to maximize the ROI of the investment, the IRRs and ∆IRR do not necessarily indicate the best decision. Using the example above, assume the investor wants a minimum ROI of 50%, the corresponding MARR is approximately 7% for Machine A and approximately 9% for Machine B; the graph clearly indicates that within such range the ROI for Machine B is better than the one for Machine A so, if maximizing ROI is the criterion, Machine B should be the selected one although in that range the NPW of Machine A is greater than the one for Machine B: Machine B produces a better ROI because although its NPW is lower, its invested capital is smaller.

The decision between Machine A and Machine B is dictated by the selection criteria adopted by the investor:

· Maximize NPW of the investment: Choose Machine A for i ≤ 17.7%, Machine B for 17.7% ≤ i ≤ 38.7%. Do not invest when i ≥ 38.7%

· Maximize ROI : Choose Machine B for i ≤ 13.7%; regardless of the choice, the ROI will be negative for i ≥ 13.7%.

· Minimum Acceptable ROI : assume the investor accepts a minimum ROI of 50%, the corresponding MARR is about 7% for Machine A and 9% for Machine B; in such range Machine B outperforms Machine A for ROI, but Machine A produces a better NPW

· To be considered : the ROI of the incremental investment A-B, i.e. deciding of making the extra investment of purchasing the more expensive Machine, becomes negative for i ≥ 3.5%, therefore form the ROI standpoint buying Machine A is never a good decision

7-5 ANALYSIS PERIOD

It is worth repeating a concept already explained earlier:

When comparing two or more alternatives, the investment horizon considered must be the same for all alternatives.

In case one of the alternatives has a shorter life, but a multiple of the longer life of the other alternatives, the simplifying assumption of “equal replacements” must be applied.

In case one of the alternatives has a shorter life, but not a multiple of the longer life of the other alternatives, the simplifying assumption of “equal replacements” is still applied, and the shorter last period of the alternative is accounted for with an appropriately larger salvage value, or similar considerations.

Example 7.6 – from the ROI standpoint, which of the following alternatives should be chosen?

Alternative A Alternative B

Investment 60,000 200,000

Benefits/year 30,000 30,000

O&M/year 10,000 8,000

Life (years) 4 15

End-of- life Salvage 10,000 25,000

Mid-life Salvage 20,000 N/A

Using an Excel Spreadsheet:

The ROI analysis of the two alternatives, as well as of the differential investment, clearly indicates that none of the Alternatives will ever produce an acceptable level of return.

As far as NPW are concerned, Alternative B is better for i ≤ 3.56% while Alternative A is better when 3.56% ≤ i ≤ 16.94%; for i ≥ 16.94% the NPW becomes negative. Even without calculating the ROIs, a quick comparison of the NPW values versus the Cost of Investment indicates that both investments are at a loss.