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Which bonus to choose?  RSS feed

 
Piet Souris
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Just read a nice puzzle in the newspaper. Can be solved without a computer, just pen and paper or some thinking is all that is required.

You are offered a new job, salary 40.000 eurosper year. And you can choose one of the following bonus arrangements:

1) you get a guaranteed salary increase of 500, at the end of each half year
2) you get a guaranteed salary increase of 2.000, at the end of each year.

Which one would you choose?
 
Campbell Ritchie
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Sounds like the old Arabian Nights‑like story of the king who offered his three sons:-
  • 1: Land two miles square or two square miles.
  • 2: Land one mile square or one square mile.
  • 3: Land half a mile square or half a square mile.
  •  
    Tim Cooke
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    By my calculations, it depends.

    It depends on how long you are intending to stay in that new job?

    Up to the 6th month it makes no difference.
    Between the 7th and 14th month you are better off with the 500 EUR increase every half year option.
    At the 15th month it makes no difference.
    From the 16th month onwards you are better off with the 2000 EUR increase every year option.
     
    Piet Souris
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    hi Tim,

    thanks for your effort! The number of reactions were such that I was starting to fear that no one around here cared for a nice puzzle anymore.

    But, I must disagree with your answer. I won't tell you my outcome, all I can say is that the interesting part of this puzzle is not so much determining the right bonus, but explaining why it is the right bonus.
    The initial reasoning: a raise of 500 per half year is a raise of 1000 per year, turns out to be a bit too simple...
     
    Tim Cooke
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    Here's my workings:

    This table shows the salary, monthly earnings, and total cumulative earnings for the scenario where the person starts at 40,000 and increases by 500 every 6 months.
    MonthSalaryMonthly wageTotal Earned
    140,000.003,333.333,333.33
    240,000.003,333.336,666.67
    340,000.003,333.3310,000.00
    440,000.003,333.3313,333.33
    540,000.003,333.3316,666.67
    640,000.003,333.3320,000.00
    740,500.003,375.0023,375.00
    840,500.003,375.0026,750.00
    940,500.003,375.0030,125.00
    1040,500.003,375.0033,500.00
    1140,500.003,375.0036,875.00
    1240,500.003,375.0040,250.00
    1341,000.003,416.6743,666.67
    1441,000.003,416.6747,083.33
    1541,000.003,416.6750,500.00
    1641,000.003,416.6753,916.67
    1741,000.003,416.6757,333.33
    1841,000.003,416.6760,750.00
    1941,500.003,458.3364,208.33
    2041,500.003,458.3367,666.67
    2141,500.003,458.3371,125.00
    2241,500.003,458.3374,583.33
    2341,500.003,458.3378,041.67
    2441,500.003,458.3381,500.00


    This table shows the salary, monthly earnings, and total cumulative earnings for the scenario where the person starts at 40,000 and increases by 2000 every 12 months.
    MonthSalaryMonthly wageTotal Earned
    140,000.003,333.333,333.33
    240,000.003,333.336,666.67
    340,000.003,333.3310,000.00
    440,000.003,333.3313,333.33
    540,000.003,333.3316,666.67
    640,000.003,333.3320,000.00
    740,000.003,333.3323,333.33
    840,000.003,333.3326,666.67
    940,000.003,333.3330,000.00
    1040,000.003,333.3333,333.33
    1140,000.003,333.3336,666.67
    1240,000.003,333.3340,000.00
    1342,000.003,500.0043,500.00
    1442,000.003,500.0047,000.00
    1542,000.003,500.0050,500.00
    1642,000.003,500.0054,000.00
    1742,000.003,500.0057,500.00
    1842,000.003,500.0061,000.00
    1942,000.003,500.0064,500.00
    2042,000.003,500.0068,000.00
    2142,000.003,500.0071,500.00
    2242,000.003,500.0075,000.00
    2342,000.003,500.0078,500.00
    2442,000.003,500.0082,000.00


    From this data it can be seen that the earning potential is the same for the first 6 months.
    From month 7 the 500 more each 6 months scheme earns more up to and including month 14.
    At month 15 both schemes have earned the same amount.
    From month 16 onwards, the 2000 more each 12 months scheme earns more.

    Does that differ from your analysis?
     
    Piet Souris
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    Slightly!

    For the half year bonus, the half year payments are: 20.000, 20.500 (total first year: 40.500 against your 40.250), 21.000, 21.500 (total second year: 42.500, total first two years: 83.000), while the year bonus gives me a payment over the first two years of: 40.000 + 42.000 = 82.000, et cetera.
    So I guess there is a subtle tiny error in your calculation.
     
    Tim Cooke
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    I maintain that, given the puzzle question as it's worded, my solution is correct.

    You are offered a new job, salary 40.000 eurosper year
    That's 20,000 for the first 6 months.
    you get a guaranteed salary increase of 500, at the end of each half year

    That increases your base salary to 40,500 per year, which is 20,250 per 6 months bringing the total over 12 months to 40,250.
     
    Piet Souris
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    Ah, I see...

    well, the raise is 500 per half year, not 500 per year. Hmm. perhaps I should have made that more clear.
     
    Tim Cooke
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    In which case the bonus options are:

    1) a salary increase of 1000 per annum every 6 months
    2) a salary increase of 2000 per annum every 12 months

    And your analysis holds true.

    YearSalary (1)Income (1)Earned (1)Salary (2)Income (2)Earned (2)
    0.5€ 40,000.00€ 20,000.00€ 20,000.00€ 40,000.00€ 20,000.00€ 20,000.00
    1€ 41,000.00€ 20,500.00€ 40,500.00€ 40,000.00€ 20,000.00€ 40,000.00
    1.5€ 42,000.00€ 21,000.00€ 61,500.00€ 42,000.00€ 21,000.00€ 61,000.00
    2€ 43,000.00€ 21,500.00€ 83,000.00€ 42,000.00€ 21,000.00€ 82,000.00
    2.5€ 44,000.00€ 22,000.00€ 105,000.00€ 44,000.00€ 22,000.00€ 104,000.00
    3€ 45,000.00€ 22,500.00€ 127,500.00€ 44,000.00€ 22,000.00€ 126,000.00
    3.5€ 46,000.00€ 23,000.00€ 150,500.00€ 46,000.00€ 23,000.00€ 149,000.00
    4€ 47,000.00€ 23,500.00€ 174,000.00€ 46,000.00€ 23,000.00€ 172,000.00
    4.5€ 48,000.00€ 24,000.00€ 198,000.00€ 48,000.00€ 24,000.00€ 196,000.00
    5€ 49,000.00€ 24,500.00€ 222,500.00€ 48,000.00€ 24,000.00€ 220,000.00
    5.5€ 50,000.00€ 25,000.00€ 247,500.00€ 50,000.00€ 25,000.00€ 245,000.00
    6€ 51,000.00€ 25,500.00€ 273,000.00€ 50,000.00€ 25,000.00€ 270,000.00
    6.5€ 52,000.00€ 26,000.00€ 299,000.00€ 52,000.00€ 26,000.00€ 296,000.00
    7€ 53,000.00€ 26,500.00€ 325,500.00€ 52,000.00€ 26,000.00€ 322,000.00
    7.5€ 54,000.00€ 27,000.00€ 352,500.00€ 54,000.00€ 27,000.00€ 349,000.00
    8€ 55,000.00€ 27,500.00€ 380,000.00€ 54,000.00€ 27,000.00€ 376,000.00
    8.5€ 56,000.00€ 28,000.00€ 408,000.00€ 56,000.00€ 28,000.00€ 404,000.00
    9€ 57,000.00€ 28,500.00€ 436,500.00€ 56,000.00€ 28,000.00€ 432,000.00
    9.5€ 58,000.00€ 29,000.00€ 465,500.00€ 58,000.00€ 29,000.00€ 461,000.00
    10€ 59,000.00€ 29,500.00€ 495,000.00€ 58,000.00€ 29,000.00€ 490,000.00
     
    Piet Souris
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    Correct, Tim.

    The actual question is: a raise of 500 per half year equals a raise of 1000 per year. Now, bonus B gives me a raise of 2.000 per year. How come bonus A is better, nevertheless? But I have the impression you already figured that out   
     
    Tim Cooke
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    This was fun Thanks for sharing Piet.

    I actually quite like the idea of a bonus scheme that differs in benefit based on how long you intend to stay in the position. It's like some sort of sly corporate game to gauge employee loyalty.
     
    Campbell Ritchie
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    As opposed to the blatant scheme to gauge loyalty whereby 99% of employees approach their line manager for references forty‑eight hours after the bonus scheme is withdrawn
     
    Liutauras Vilda
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    And if ever bonus gets paid, no need to be so happy, because bonuses are tend to be taxed as twice as regular salary, so, seem to be both of the options manager offered are not optimal
     
    Piet Souris
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    My boss offered me a 0 increase per day every day, or 0 per year, every year. Now, I did not know of this puzzle then, so stupid me chose the second one....

    And Tim, glad you liked it!
     
    Liutauras Vilda
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    Piet Souris wrote:My boss offered me a 0 increase per day every day, or 0 per year, every year. Now, I did not know of this puzzle then, so stupid me chose the second one....

     
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