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Math Help - break even point on investment

  1. #1
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    break even point on investment

    What I am trying to figure out is if I do a complete buyout of a particular business then how long would it take in net income for me to break even on my investment assuming a certain rate of income growth each year.

    The variables we know in this equation are 1. the cost to buy the business 2. the current yearly net income of the business 3. the assumed yearly growth rate of net income. The variable we are solving for is the number of years it would take for the accumulated net income to equal the cost of the business.

    A simple compound interest formula won't work on this problem because it will only say when the net income itself for one year is the same as the cost of the business which will obviously take many years. Can anyone help me solve for this?
    Last edited by ohffs; January 12th 2010 at 06:20 AM.
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  2. #2
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    Quote Originally Posted by ohffs View Post
    What I am trying to figure out is if I do a complete buyout of a particular business then how long would it take in net income for me to break even on my investment assuming a certain rate of income growth each year.

    The variables we know in this equation are 1. the cost to buy the business 2. the current yearly net income of the business 3. the assumed yearly growth rate of net income. The variable we are solving for is the number of years it would take for the accumulated net income to equal the cost of the business.

    A simple compound interest formula won't work on this problem because it will only say when the net income itself for one year is the same as the cost of the business which will obviously take many years. Can anyone help me solve for this?
    Have you perhaps an example of one of these questions? It might be easier to explain a direct answer than take a guess at what you're looking for from a general method.
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  3. #3
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    Here's the kicker, I have looked long and hard and I do not believe this equation exists anywhere. I am pretty sure that solution uses logarithms to solve for y. The closest equation I have found to this is:

    Y= (NL(FV/PV)/(NL(1+r))

    The problem with this is that if you assume the future value (FV) is the price, the present value (PV) is the current net income, and the rate (r) is the assumed growth rate of net income then you get a result that tells you when the net income for one year equals the price, but not when the running total of net income equals the price assuming it keeps going up at the same rate. For the equation to work it must lay out the function where the net income for each year, as it goes up by the growth rate is added together until it equals the price. This is the point where assuming you keep your money tied up in the business you earn your principal back in net income. The reason this is important to me is because no metric of intrinsic value makes more sense to me then that. When you get down to the nitty gritty time=money. The P/E model does calculate this, but the problem is that it assumes earnings are static. The P/E/G model includes earnings as well, but displays it's results in a way that to me is not useful.

    There is no example I can show because I choose to think about intrinsic value in my own terms, the ones that make sense to me. I can perform all these operations long hand on paper, but it's a time consuming process and I'd rather copy/paste into a spreadsheet and let that run the figures for me.

    P.S. I know that it's difficult to just 1. take variables 2. ???? 3. profit, but unfortunately that is all the information I can provide.
    Last edited by ohffs; January 12th 2010 at 08:34 AM.
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  4. #4
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    You made a ficticious loan of $10,000 at 12% annual interest rate.
    You will be repaid at $3,000 1 year later; then the annual payment
    will increase by 10% each year
    Code:
    0                                          10000.00
    1      -3000.00         1200.00             8200.00
    2      -3300.00          984.00             5884.00
    3      -3630.00          706.08             2960.08
    4      -3993.00          355.20             -677.72
    Follow?
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  5. #5
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    Quote Originally Posted by ohffs View Post
    The closest equation I have found to this is:
    Y= (NL(FV/PV)/(NL(1+r))
    Can you clarify that thing of yours; half bracket missing;
    do you mean: Y= [NL(FV/PV)] / [NL(1+r)] ?

    Then the "NL"'s cancel out, leaving: Y= (FV/PV) / (1+r)
    Seems to be something not kosher...
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  6. #6
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    Yes, I did leave out a bracket so Y= [NL(FV/PV)] / [NL(1+r)] with NL of course standing for natural logarithm. As I said before though the purpose is to model a formula rather then to plot everything out long hand. The variation of the interest rate formula above will only describe a situation where net income is the same as the cost of buying the business, which I originally thought would work until I started getting results like Y = 65.

    To make it easier to understand if you were to calculate this long hand without using a formula you would take the current net income and multiply it by the assumed rate of earnings growth, write down that total, rinse and repeat until you have about 20 or 30 years of projected net income down. Then you would add the totals together until you reached the desired number, which is the price. The final step then would be to go back and count how many totals you had to add together for this to happen. That would give you the number of years it would take, but obviously that is rather time consuming.
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  7. #7
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    Did you understand what I was doing with this:
    Code:
    0                                          10000.00
    1      -3000.00         1200.00             8200.00
    2      -3300.00          984.00             5884.00
    3      -3630.00          706.08             2960.08
    4      -3993.00          355.20             -677.72
    Is that the way it would work if purchase price = $10,000,
    and income being 3,000, increasing by 10%,
    valued using discount rate of 12%?

    Or are you simply looking at income flow,
    hence 10000 - 3000 - 3300 - 3630 ... to point where zero is left?

    Why don't YOU provide an example: no need for 20 years or so;
    keep it under 5 years; then we'll know what you're after...
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  8. #8
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    I believe the method you showed is very similar to what I explained, but since you were talking about lending I wasn't 100% sure we were on the same page. I'll describe exactly what's on my mind to put things in perspective.

    I'll pick this example because it's in my spreadsheet right now and is convenient. Let's say you really like Faygo moonmist and are thinking about buying the company that makes Faygo moonmist, which would be National Beverage Corp. (ticker symbol: FIZZ). Fizz has been hovering around 12 a share lately and has 46 million shares outstanding. If you bought all the shares of fizz you would pay 552 million dollars for the entire company. There's the first variable.

    Second variable I get from going to msn money, looking at their ten year income summary, and importing it into an excel spreadsheet. The cell at the bottom of net income column uses the format =(LN(F2)-LN(F11))*10 to get the net income growth rate. Basically, the natural log of the current income minus the natural log of ten years ago's income times ten for a result of 5.88%.

    Third variable is just the current net income which is 24.74 million.

    (When I calculate all this I assume the growth rate and assets are all overstated to give myself a margin of safety. I also discount the difference between assets and liability from the price before running all these calculations, but for the sake of this exercise I'll use just these three numbers.)

    When I think about stocks I look at it like this: if I had 552 million to drop on fizz would the company be a good buy? Forget "momentum", volatility, asset allocation models and all that other nonsense. I pretend like I am a Warren Buffett or Peter Lynch buying the company outright when I consider a stock.

    In business the first thing you want to know is when is the investment going to start making a profit because if you were to buy fizz today as an actual owner it would take quite awhile before you got your 552 million dollar investment back and started making a profit. Like Warren Buffett I don't think in terms of "owning stocks" I think in terms of owning companies, which is why my methodology might seem odd at first, but makes perfect sense from my perspective.

    If Peter Lynch was handling these numbers he'd take the price, divide it by the earnings, and then divide that by the growth rate. In theory if that number is less than 1 then the stock is cheap and vice versa. That works for him, but to me that number doesn't show anything useful. For fizz the peg is around 3.79, which would suggest it is overpriced. That is fine, but perhaps for the sake of this exercise I'm am considering fizz for a relatively defensive dividend play (they might be a poorly run company, but 59 cent Faygo cola is recession proof as it gets). In that case peg = 3.79 does not provide any useful information to comparing it to say the return of a US treasury bond. At 5% simple interest a bond will take 20 years to pay for itself and start making a profit assuming you keep rolling over that note after it expires. The problem is business profits accumulate year to year and if you assume growth then the profit won't be the same every year either (this is what p/e assumes). Using the P/E model fizz would take 22 years to pay for itself, which would also suggest that you'd be better off just buying the 5% bond then buying this stock, but maybe accounting for growth you would be better off...

    Start with 24.74 million in earnings for 2008 and assume that is current. Based on 5.88% growth rate in 2009 it should be 26.19 million, in 2010 27.73 million, in 2011 29.37 million (rounded off) so on an so forth. I cannot predict when I'll reach 552 million so I just project about 20 years like this, then go back and add it up until I get 552 million like this:

    2009 - 26.19
    2010 - 27.73
    2011 - 29.37
    2012 - 31.09
    2013 - 32.92
    2014 - 34.86
    2015 - 36.91
    2016 - 39.08
    2017 - 41.37
    2018 - 43.81
    2019 - 46.38
    2020 - 49.11
    2021 - 52.00
    2022 - 55.05
    2023 - 58.29
    2024 - 61.72

    Assuming I made no arithmetic errors around the year 2022 is the point where the accumulated net income matches the price so a buyout in fizz would pay itself back in around 14 years. In comparison to our simple interest investment in the theoretical 5% US treasury bond fizz all of a sudden isn't looking too bad when you account for it's projected earnings. The company may be a laggard in comparison to coca cola, but perhaps it's returns are reliable enough (projecting does no good if you can't be certain the returns will be there) that it is justifiable as a bargain issue investment. Now on the other hand how does this boring stock compare with a high flier like China Mobile growing at an impressive 18.63% growth rate? Well, it's difficult to tell without some kind of process to level the playing field considering that I trust a company like National Beverage Corp. much more then I trust a communist run business like China Mobile; however, the earnings are good enough where it might be worth assuming the extra risk. See my dilemma here?
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  9. #9
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    Quote Originally Posted by ohffs View Post
    I
    2009 - 26.19
    2010 - 27.73
    2011 - 29.37
    2012 - 31.09
    2013 - 32.92
    2014 - 34.86
    2015 - 36.91
    2016 - 39.08
    2017 - 41.37
    2018 - 43.81
    2019 - 46.38
    2020 - 49.11
    2021 - 52.00
    2022 - 55.05
    2023 - 58.29
    OK; clear now! With accumulation:
    2022 (14 years) - 55.05 : 545.77
    2023 (15 years) - 58.29 : 604.06

    So the 552 million is reached a bit over 14 years...

    This formula will give the total reached after Y years:
    F[(1 + r)^Y - 1] / r
    F = 1st year Flow (26.19)
    r = increase rate (.0588)

    So, for Y = 14 years as example:
    26.19(1.0588^14 - 1) / .0588 = 545.77

    You want this in terms of Y; let P = purchase price (552 million).
    Formula:
    Y = log(Pr/F + 1) / log(1 + r)

    With your example:
    Y = log(552*.0588/26.19 + 1) / log(1 + .0588) = 14.1096.....

    Ya'll ok now?
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  10. #10
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    Ok, I am going to test this formula out and see if it works. I believe that is it.
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  11. #11
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    Quote Originally Posted by ohffs View Post
    Ok, I am going to test this formula out and see if it works. I believe that is it.
    Oh, it'll work for what YOU're after...
    However, I can't see WHY you're looking at this without using a rate of return.
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