Hi All,

In my textbook problem, I have the following:

.............
If during each year i from the beginning of the system lifetime, the system economical losses are V_i, the total losses during the entire lifecycle (m years) in the present values can be obtained as:
L = sum(i=1 to m) (V_i / (1+r)^i )

The net present value of the profit from the investments into the system reliability improvement is:
C = L - C_(investment)
............

Usually in the texts I see that V_i is (cash inflow - outflow). But in this particular textbook problem, V_i is only economcal losses.

My Questions:

(1) Is it valid to have V_i as only losses in the NPV formula?


(2) Does it mean that finally L is a negative amount ? Like
In 2001 (since 2000), V_1 = -100
In 2002 (since 2000), V_2 = -40
In 2003 (since 2000), V_3 = -120


(3) What does "NPV of the profit from the investments" C mean in this case?
Does C mean net present value of the total loss in say 10 years? If so, why the word "Profit" in the above sentence? Is C negative?

(4) When comparing multiple alternatives (multiple values of C, assuming C is negative), how can the best alternative be chosen? Least absolute value of C?


Thanks,
Tims

There is a system which presently incurs losses ("economical losses"). The problem is whether you want to make investments to improve the reliability of the system SUCH THAT THE SYSTEM DOES NOT INCUR THESE LOSSES. In effect, the more the present losses, the more you would save by investing in such a system to improve its reliability. It's a matter of interpretation.

For instance, the system is losing Rs 100 annually because of lack of reliability (e.g. it does not execute trades reliably, or does not turn off a valve fast enough). To improve its reliability you need to invest Rs 500 today. Would you invest? Every rupee of loss saved is a rupee earned. So, NPV of Rs 100 lost annually is actually Rs 100 saved (V_i in the text) by improving the system's reliability.

The terms in the equation

C = L - C_investment are defined thus,

L = NPV of the present losses or stated alternately, NPV of the potential savings by investing to improve reliability (e.g. NPV of Rs 100 saved annually).

C_investment = Cash outflow due to investment (e.g. Rs 500).

C = Net Savings.

I don't have much idea about the things that you have asked about.

Still let me guess the answers:

--------------------------

Let's assume rate of interest in an economy is 5%.

If a company makes a loss of $100, it needs $100 to make up for it.

What if the company is going to make a loss of $100 after a year?

In that case, the company needs only $95.24 now to make up for $100 loss that will come after a year.

The company's $95.24 will turn into $100 after a year as interest rate in the economy is at 5%.

If the company makes a provision of $95.24 (sets asides $95.24) and invests it in a interest generating assets, this $95.24 will turn into $100 in a year and compensate the $100 loss.

Thus, the NPV of $100 loss, which will happen after a year, is $95.24.

In this simplified example,

Here $95.24 is L and $100 is V_i.

(What is more and what is less will depend on whether you use the minus sign for loss or not). --------------------------

There are many businesses and government undertakings that are eternally loss-making. These are run for reasons which are not economic in nature.

e.g. Instead of outsourcing and reducing costs, some companies prefer to continue their loss-making divisions because they don't want their technology to be known to others.

These companies may use NPV analysis to estimate the amount of cash that they need to set aside now to compensate of future losses.

If you use a minus sign for a loss,

V_i in such cases will always be negative.

Obviously, L too will be negative, but less negative the sum of all V_i.

--------------------------

This is the part that I am least sure about.

I don't know what C_(investment) is.

I think the above formula is for the cost benefit that one gets for making an investment to improve system reliability and thus prevent the losses V_i.

and

I am assuming C_(investment) is the investment that can prevent these losses (L).

Lets again assume an economy where the interest rate is 5%.

Should a company invest $98 to save $100 after a year?

No,

because in a year $98 will turn into $102.9 after a year, which can more than compensate for a $100 loss.

Any amount greater than $95.24 can turn into an amount greater than $100 in a year.

An investment to prevent this $100 loss will make sense only if its cost is less than $95.24.

In other words, the investment makes sense only if its cost is less than the NPV of the loss.

--------------------------

That's what the equation is about.

These losses are represented V_i.

(i=1 to m)

NPV of all losses is represented by

L = sum(i=1 to m) (V_i / (1+r)^i )

C_(investment) represents this investment for improving system reliability and prevent these losses.

C is the difference between the losses that will occur and investment needed to prevent these losses.

The investment is worth making, only if value of C_(investment) is less than the NPV of all losses.

In other words, the investment is worth making only if C is be positive.

Greater is the value of C, better the investment.

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Thank you Rainman and the Alchemist.

I appreciate both of your detailed explanations.

Simulated some numbers comparing NPV for investing on varying number of extra components N with a fixed annual loss of V for each component.

L = V * sum(i=1 to m) (1 / (1+r)^i )
C = L - C_inv



C is decreasing by increasing the number of spare components, V seems to stay constant beyond 6 spares.

(1) However C still seems to be doubtful,

With 0 spares, C is large and positive. The whole idea is to say that investing in a few number of spares is advantageous over 0 spares. So? I am still confused.

(2) if C is +ve only for 1 spare, does it mean having more than one spare is a loss?

Thank you
Tims

Some additional info.

V above comes from a mathematical model that accounts for number of spares, cost of a repair, time to repair & the cost of energy not supplied (ENS).

m=40.

The first thing that came to my mind when I read your message last evening was where did you get the values of V from. Anyway, you answered the question in your next message.

If you take a look at the table you would come to the conclusion that making no improvements would save you more money. That obviously does not make sense.

I think the values of V in the table are ACTUAL LOSSES (and not savings)
Loss @ Spares = 0 is 125.42 (This is the current state of the system)
Loss @ Spares = 1 is 3.38
Loss @ Spares = 2 is 0.077
and so on.

Thus the Potential Savings V can be calculated as,
Savings @ Spares = 0, V = 125.42 - 125.42 = 0
Savings @ Spares = 1, V = 125.42 - 3.38 = 122.04
and so on.

By definition, you need to use values of Potential Savings to calculate L. After you make the changes, you should notice that the value of C is maximum @ Spares = 2.

P.S. I will be able to answer your questions in detail only as long as the stock markets continue to drift.

True; V are the actual annual average losses.

As per your suggestion, I calculated the potential savings and then determined the NPV of savings.

Here are the results, as expected 2 spares seems optimal.
Plot of savings:

http://cid-21b3123f14081604.office.l...lic/spares.jpg

Modified Table:

http://cid-21b3123f14081604.office.l...LMinusCInv.jpg

What if I look at optimal loss L+C_inv instead of gains? Does the following table make sense?

http://cid-21b3123f14081604.office.live.com/self.aspx/Public/Spares^_LPlusCInv.jpg

Thanks,

Tims.

That works too. One less column to deal with.

Although, on the other hand, when you present the plot with losses on Y-axis, someone's bound to ask a stupid question like "In spite of spending so much money, why am I still seeing overhead losses?"

Instead, you can show them how much they save and everyone's happy.