Calculating appreciation of investment funds

[Djangaroo]

I need a formula to calculate the appreciation of my pension fund where I have made an initial investment and then additional investments at later times. For example, I start with $10,000 which after five years appreciates to $14,000, a 40% gain. I then invest another $10,000 to make my holding worth $24,000 and after a further five years it appreciates to $36,000, a further 50% gain. If I had not added the second investment after five years my $14,000 would have appreciated to $21,000 after the second five year period, an overall gain over ten years of 110%. But I need to know how I reach that conclusion by using the two separate figures of 40% and 50% gains. Also, is there a way to continue building the formula to make it account for 3^rd, 4^th, 5^th… amounts that I invest?

 

[Deleted member 4993]

I need a formula to calculate the appreciation of my pension fund where I have made an initial investment and then additional investments at later times. For example, I start with $10,000 which after five years appreciates to $14,000, a 40% gain. I then invest another $10,000 to make my holding worth $24,000 and after a further five years it appreciates to $36,000, a further 50% gain. If I had not added the second investment after five years my $14,000 would have appreciated to $21,000 after the second five year period, an overall gain over ten years of 110%. But I need to know how I reach that conclusion by using the two separate figures of 40% and 50% gains. Also, is there a way to continue building the formula to make it account for 3^rd, 4^th, 5^th… amounts that I invest?

What are your thoughts?

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[Djangaroo]

I do not have any workings because this is a real life problem to help me manage my finances.

The data I have are: a) the dates at which I invest the original and additional amounts, and b) the overall value of my investment fund at specific dates, including when the additional amounts are invested. What I want is the formula to calculate is the performance of the fund over the full period (in this example ten years), taking into account the fact that additional amounts will be invested at various points in that ten years.

 

[Ishuda]

I do not have any workings because this is a real life problem to help me manage my finances.

The data I have are: a) the dates at which I invest the original and additional amounts, and b) the overall value of my investment fund at specific dates, including when the additional amounts are invested. What I want is the formula to calculate is the performance of the fund over the full period (in this example ten years), taking into account the fact that additional amounts will be invested at various points in that ten years.

The general rule is a future value [in your case what you have at the end of a time period] is given by
FV = (1+i)ⁿ PV
where i is the interest rate for each period, n is the number of periods, and PV is the present value [in your case what you invested]. If you have more than one PV (more than one investment) then you add them up
(A) FV = (1+i)ⁿ¹ PV₁ + (1+i)ⁿ² PV₂ + ...
if you want the average performance (average interest rate you would earn) over all periods. In your case we stop at the two investments with PV1 equal to PV2 equal to $10000, n1 equal to 10 and n2 equal to 5.
(B)FV = (1+i2)ⁿ² [ (1+i1)ⁿ¹ PV₁ + PV₂ ] + ...
if you have different interest rates for each set of periods. In your case you have the value of (1+i1)ⁿ¹ = 1.40 and the value of (1+i2)ⁿ² = 1.50, along with PV₁ and PV₂, both equal to $10000.

I understand you would like to find the i of (A) using (B). Is that correct?

 

[Ishuda]

1.4 * 1.5 = 2.1
2.1 - 1 = 1.1
1.1 * 100 = 110%

Take 3 period where gains are 36%, 48% and 60%:
1.36 * 1.48 * 1.60 = 3.22048
3.22048 - 1 = 2.22048
2.22048 * 100 = 222% (rounded!)

That would be on a single investment at the beginning. If you wanted multiple investments [of the same amount] you would have 1.5*[1.4+1] for two investments, 1.45*[1.5*[1.4+1]+1] for three investments with the third earning 45%, etc. In the case given, it requires the solution to a quadratic equation and would amount to about 7.89% compounded annually.

 

[Djangaroo]

Ya ya...but his question is: how do I get 110% from 40% and 50%;
that's all I answered...nothing else...

And you answered it beautifully. Thank you, Denis.

 

[Ishuda]

Ya ya...but his question is: how do I get 110% from 40% and 50%;
that's all I answered...nothing else...

Guess I should learn to read the question, huh? Oh well, you can't win them all and some days you can't win any. But then, maybe the one who wants help should ask the same question all the time rather than changing it.