Find out monthly growth based on yearly growth

Question

Let's say I have a number, 10, and over a period of a year that number has increased by 100% to 20. I would like to figure out what the monthly increase should be for 10 to reach 20 over 12 months.

If I convert 100% to a coefficient I get 2. Now by trying several numbers manually I figured the monthly coefficient should be around 1.065:

M.  NUMBER          COEFF
1   10              1.065
2   10.65           1.065
3   11.34225        1.065
4   12.07949625     1.065
5   12.8646635063   1.065
6   13.7008666342   1.065
7   14.5914229654   1.065
8   15.5398654581   1.065
9   16.5499567129   1.065
10  17.6257038992   1.065
11  18.7713746527   1.065
12  19.9915140051   1.065

Q: How can I obtain the monthly coefficient (something around 1.065) from 2 and 12?

Answer 1

Actually, your exponent is off by one: if the table represents the number at the end of each month, it should start at 0. Otherwise, if it represents the number at the beginning of the month, it should go to 13.

Since you're arriving at 20 by multiplying repeatedly by $x_{\mathrm{month}}$, we have $$x_{\mathrm{year}}=(x_{\mathrm{month}})^{12}$$ so $$x_{\mathrm{month}}=(x_{\mathrm{year}})^{1/12}=\sqrt[12]{x_{\mathrm{year}}}$$

Numerically, we arrive at ca. $1.059$, or $5.9\%$.

Answer 2

Ansgar had the solution, although with an imperfection.

If you use 12 months for the root calculation you get this

COEFF = (1 + 100%/100) ^ (1/12)
M.  NUMBER          COEFF
1   10  
2   10.5946309436   1.0594630944
3   11.2246204831   1.0594630944
4   11.89207115     1.0594630944
5   12.5992104989   1.0594630944
6   13.3483985417   1.0594630944
7   14.1421356237   1.0594630944
8   14.9830707688   1.0594630944
9   15.8740105197   1.0594630944
10  16.8179283051   1.0594630944
11  17.8179743628   1.0594630944
12  18.8774862536   1.0594630944

The good way of doing it is use 11, because there are only 11 increases.

COEFF = (1 + 100%/100) ^ (1 / (12-1) )
M.  NUMBER          COEFF
1   10  
2   10.6504108944   1.0650410894
3   11.343125222    1.0650410894
4   12.080894444    1.0650410894
5   12.8666489801   1.0650410894
6   13.7035098472   1.0650410894
7   14.5948010568   1.0650410894
8   15.5440628177   1.0650410894
9   16.5550655977   1.0650410894
10  17.6318250999   1.0650410894
11  18.7786182132   1.0650410894
12  20              1.0650410894

Answer 3

$$10 (1 + j)^{12} = 20$$

is equivalent to

$$(1 + j)^{12} = 2,$$

which is equivalent to

$$(1 + j) = \sqrt[12]{2}.$$

Answer 4

Please see @Graphth's post for the math. This is just to clarify the crucial point @user359650 made, but I fear a bit unclearly.

If there are $12$ monthly compounding periods, then there are $13$ time points, since each period is an interval with an adjacent starting and ending time point. It is customary to call the first left endpoint $t=0$. (How many tick marks to you see on the $x$ axis below, and how many intervals between tick marks?)

plot(10*2^(x/12),(x,0,12)).show(figsize=(4,1),ymin=0,ymax=20)

In sage, we calculate:

def grow(ratio, periods):
    r = ratio^(1/periods); print 'growth per period =',r.n()
    print '%2s %-12s' % ('Period', 'Ending Balance')
    for n in range(periods+1):
        print '%5d %12.6f' % (n, 10*r^n)
grow(2,12)

to get (values commensurate with the graph above):

growth ratio = 1.05946309435930
Month Ending Balance
    0    10.000000
    1    10.594631
    2    11.224620
    3    11.892071
    4    12.599210
    5    13.348399
    6    14.142136
    7    14.983071
    8    15.874011
    9    16.817928
   10    17.817974
   11    18.877486
   12    20.000000

or (erroneously, if you want a period of one year with a monthly-based calendar system):

grow(2,11)

to get:

growth per period = 1.06504108943996
Period Ending Balance
    0    10.000000
    1    10.650411
    2    11.343125
    3    12.080894
    4    12.866649
    5    13.703510
    6    14.594801
    7    15.544063
    8    16.555066
    9    17.631825
   10    18.778618
   11    20.000000