NPV=0 no good, IRR occurs at the payback period

DexterOnline

Junior Member
Joined
Jan 29, 2015
Messages
139
Forget about the interest rate at which discounted benefits equal discounted costs

IRR occurs at the payback period - the breakeven point

Example:

Invest $10 today to receive $1 per year for 10 years

The IRR is 0% and the payback period is 10 years

Code:
-10 1 1 1 1 1 1 1 1 1 1

Invest $10 today to receive $1 per year for 9 years and 90 cents in the 10th year

The IRR is negative and the payback period is a bit short of 10 years

Code:
-10 1 1 1 1 1 1 1 1 1 0.90

Invest $10 today to receive $1 per year for 10 years and 0.10 cents in the 11th year

The IRR is positive and the payback period is 10 years

Code:
-10 1 1 1 1 1 1 1 1 1 1 0.10

Anyone wants to challenge my assertion that IRR is best defined in terms of its correlation with payback period rather than the skewed and often ambigious NPV=0 view of the world

I had plenty of Coke this morning so will have no problem reply to challengers
 
If I am understanding your assertion correctly then invest $10 today and get $10 ten years from now would be the same payback period as Invest $10 today, receive $9.99 instantly and $0.01 ten years from now.

Looking at that, it appears that you are saying the time value of money is zero. I wouldn't mind borrowing that way but you couldn't get me to lend that way. But then maybe I'm not understanding the assertion.
 
You do not follow my argument

I am keeping time value of money in the view with all three examples of mine

Let us use time value of money calculations NPV=0 to find the IRR on the following investment

At time period 0, I invest $10 and receive $1 at the end of Yr1, $2 at the end of Yr2, $3 at the end of Yr3 and $4 at the end of Yr4

Setting net present value of this investment to zero and solving for i, the only valid IRR is 0% the other real valued IRR is -170.87% which doesn't make sense as we can't lose more than -100% and the remaining two IRR values are complex valued whereas the real world has real valued returns

The investment

Code:
-10 1 2 3 4

The four IRR solutions setting NPV=0 and solving for i

Code:
0%
-170.85 %
-109.56+0.75i %
-109.56-0.75i %
 
You do not follow my argument

I am keeping time value of money in the view with all three examples of mine

Let us use time value of money calculations NPV=0 to find the IRR on the following investment ...
But what you said was "Anyone wants to challenge my assertion that IRR is best defined in terms of its correlation with payback period rather than the skewed and often ambigious NPV=0 view of the world".

So which is it, you are keeping the time value of money with the current NPV=0 definition or do you want the correlation with payback period or is NPV=0 the tie breaker or ...? Just what do you mean by 'correlation with payback period'?
 
Hmmm...
k=NPV

x=1 + i

k = 1/x + 2/x^2 + 3/x^3 + 4/x^4
So:
k = (x^3 + 2x^2 + 3x + 4) / x^4

You forgot the -10

k = -10 + 1/x + 2/x^2 + 3/x^3 + 4/x^4

k = -10*x^4 + x^3 + 2*x^2 + 3*x + 4

-10*x^4 + x^3 + 2*x^2 + 3*x + 4 = 0

Sir Wilmer

Try now with the Wolfram Solver

I think 3 cups of coffee weren't enough, another cup was required :)
 
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But what you said was "Anyone wants to challenge my assertion that IRR is best defined in terms of its correlation with payback period rather than the skewed and often ambigious NPV=0 view of the world".

So which is it, you are keeping the time value of money with the current NPV=0 definition or do you want the correlation with payback period or is NPV=0 the tie breaker or ...? Just what do you mean by 'correlation with payback period'?

As we have seen with the example investments so far that when solving for i by setting NPV=0, we get multiple solutions of which there may be one or no appropriate IRR values

As we have noted from the last example and here it is once again, that when we discounted each of the cash flows and solved it for i that it gave us an IRR of 0% which means that the investment yields no gains or losses

Furthermore the investor breakseven at the payback period of t=4 and it is at this time period that the IRR occurs

Below this time period t=4 the investor losses money thus a negative IRR as shown in the second example below and after this time period t=4 the investors makes money thus a positive IRR as shown in the third example below

If these investments were block of weights attached to a scale where the first block weighing 10 lbs was attached at the bottom of the scale and the remaining blocks weighing 1 lbs, 2 lbs, 3 lbs and 4 lbs were attached to the top of the scale then the scale would only stand in balance (0% IRR)

If the weights on top added up short of 10lbs then the scale would tilt to the left (-ve IRR)

If the weights on top added up more than 10lbs then the scale would tilt to the right (+ve IRR)


See this simple diagram
Code:
       01       02       03       [B]04
[/B]_____________________________________
10  [B]^[/B]

Investment with a zero IRR where payback occurs at terminal cash flow

Code:
-10 1 2 3 4

\(\displaystyle -10*x^4 + x^3 + 2*x^2 + 3*x + 4 = 0\)

The only valid IRR is 0% the other real valued IRR does not apply

Code:
0%
-170.87%
-109.57+0.7451652030916903 i%
-109.57-0.7451652030916903 i%

Investment with a negative IRR where payback occurs short of terminal cash flow

Code:
-10 1 2 3 3.90

\(\displaystyle -10*x^4 + x^3 + 2*x^2 + 3*x + 3.9 = 0\)

The only valid IRR is -0.34% the other real valued IRR does not apply

Code:
-0.34%
-170.23%
-109.72+0.740084028215425 i
-109.72-0.740084028215425 i

Investment with a positive IRR where payback occurs at terminal cash flow

Code:
-10 1 2 3 4 0.1

\(\displaystyle -10*x^5 + x^4 + 2*x^3 + 3*x^2 + 4*x + 0.1 = 0\)

The only valid IRR is 0.33% the other real valued IRR does not apply

Code:
0.33%
-102.55%
-169.95%
-108.91+0.7424699578476673 i
-108.91-0.7424699578476673 i
 
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I still don't "get" what you're doing or trying to do.
Let's look at it differently..as future value...
Use 1,000 instead of 10, flows 100, 200, 300, 400;
and assume a rate of 10% annual as cost of money:

So the investor expects 1464.10, but ends up with 1105.10.
That's a rate of ~2.53%; 1000(1.0253^4) = 1105.10

Not what he expected, but didn't "lose", right?

May I suggest you switch from "COKE" to "PEPSI" :shock:

You bank schedule is bit messed up
Reinvesment Rate is 10%

Code:
YEAR  WITHDRAWAL  DEPOSIT  INTEREST   BALANCE
 0                                    1000
 1    -100      0        100     1000
 2    -200      0        100     900
 3    -300      0        90      690
 4    -400      0        69      359


Well you have made the assumption that your "Reinvestment Rate" is 10%

If there was someone willing to pay you 10% then that would be correct

What you have calculated is "Net Horizon Value" and Not "Net Future Value"

Net Horizon Value = NPV at 10% * Future value at 10%

Whereas with IRR calculation it is assumed that the Reinvestment rate is the same as the IRR in this case 0%

Thus the Net Horizon Value at 0% reinvestment rateis $0

Reinvesment Rate is 0%
Code:
YEAR  WITHDRAWAL  DEPOSIT  INTEREST   BALANCE
 0                                    1000
 1    -100      0        0       900
 2    -200      0        0       700
 3    -300      0        0       400
 4    -400      0        0       0
 
I still don't "get" what you're doing or trying to do.
May I suggest you switch from "COKE" to "PEPSI" :shock:

The million dollar question is

What is an IRR?

Simply stated it is the profitability rate

It should in theory tell an investor how much money she made or lost from an investment in terms of a percentage rate.

The way IRR is currently defined, in terms of a interest rate that brings net present value of the investment to a naught, makes it very difficult to determine or find the IRR.

Asking the NPV=0 model to give us the IRR is akin to asking a crack cocaine addict to detemine the rate. The NPV=0 model like the junkie on freebase crack cocaine gives us conflicting replies of which one or none may be correct. In my opinion the NPV=0 model has to go through a drug rehab so to remove the toxics that have piled up on its brain.

The key problem with NPV or any other equaiton such as NFV, NIV, NHV, B/C ratio, EAA to find IRR has to do with the discount factor \(\displaystyle 1/(1+i)^n\)

Here a rate of -100% would cause division by zero something not allowed in mathematics.

Using any of the models based on NPV, NFV, NIV, NHV, B/C ratio OR EAA, requires solving these equations with numerical methods (iterative methods)

I have personally researched and developed over 300 iterative or numerical methods to find the IRR

Such methods give one or more real valued IRR, however a number of such methods would give all real and complex valued IRR solutions to a given equation.

Yet just as the case with NPV=0, all these models report IRR values that conflict with each other.

Theorem 1.1
In my theory, there is either 1 or no IRR solution

The no IRR solution only occurs when all cash flows are positive and there is no negative cash flow. Otherwise there is always a single IRR solution.

To find this actual single IRR solution, we must look in other directions to define the IRR and simple payback period is the right candidate.

The very fact that for any investment, at the time period equal to payback period the IRR is 0% offer us a venue to find the IRR of the total investment.

After time period equal to payback period the cash flows will be all positive or zero. Thus the task remaining is to determine how much money was made from the investment.

Same idea holds if the investment didn't reach its payback period thus cash flows to the left are short of any profits and task remaining is to determine the amount of loss incurred from the investment.

Theorem 2.1

Nothing is impossible, the modeller who designed this universe didn't use numerical methods to design and create the model or else the Universe would still be in womb of its mother.

Theorem 2.1a

There is a formula for everything

The Challenge

Is to find the IRR formula

Let us revisit Dexter's Lab in South Essia

https://www.dropbox.com/s/aie86f04l7kycgc/older_and_modern_mathematicians.jpg?dl=0

The top row shows those mathematicians who invented Calculus and extended its reach

The middle row shows those mathematicians from the last 70 years who invented "Theory of finance" and extended its reach.

Harry M. Markowitz, Merton H. Miller, William F. Sharpe, Myron Scholes, Robert C. Merton, Eugene F. Fama, Lars Peter Hansen, and Robert J. Shiller

The models these mathematicians have built help determine term structure of interest rates, a forward looking task.

The last row shows notes of major currencies

Followed at the end by someone who thinks he will join the ranks of Theorists of finance by presenting his own "New Theory of Finance" to determine the term structure of interest rates by looking back at the future that has passed

See this

Code:
(You are here) 2015 2020 2025 2030 2035 2040 2045 (I am here looking back at 2015 so I know the yields on Treasury bills, notes and bonds for the last 30 years)

Sir Wilmer

It's hard to quit Coke and switch to Pepsi, as the company says in it's TV commercial "Coke! You can't beat the feeling"
 
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A kind suggestion, Dexter: get a life :p

I lost my life to power brokers of Washington D.C when I was 24, I will turn times 2 that number by midnight tonight

The last half of my life was spent in a locale that ensured I live a deprived life

As for wife, I had three of those, two in New York the first one colored, the second one white and now the 3rd one desi is unable to spell her own name let alone mine
 
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