Help defining and refining a logarithmic function: Say I want to save money over the period of 60 months with a final target: say $50,000

[SirLouen]

This is the case:
Say I want to save money over the period of 60 months with a final target: say $50,000

Let's say that now I know 100% for sure that I will be able to save a lot of money because I'm in an excellent job, but the job is very unstable, and I don't really know when it will end and if I will be able to find another job in the future.

So the idea is that in the first month I would like to save a good amount of money, but by the end of the process I would like to save way less

Basically, the target saving graph would look like this (I'm making up the numbers, just to show the idea). Basically, in the first months I would be saving like 3 times more than the average, but in the last months I would be saving like 0.8 or 0.7x of the average.

The idea here is to figure out a formula, which clearly involves a logarithmic curve, and then refine it and make some adjustments, with Excel for example.

But I'm not 100% sure where to start from. If anyone could give me some guidance, I would be grateful.

 

[SirLouen]

I've been thinking for a while and still don't love it but my first iteration is

i = number of the month (1 first month)
T = total money amount

(Log_i+2(T))*T) / (Sum(i,n)(Log_i+2(T)))

I added +2 to remove Log_2 because it's too step, even thinking on removing Log_3 to make it a little less step.

Plotted, this is the result



Not bad, but not best. I would be open for further ideas. The idea is that in the beginning there are more bigger results and in the end, more smaller results, so the curve goes a little more in the middle.

 

[Dr.Peterson]

The idea here is to figure out a formula, which clearly involves a logarithmic curve

Why do you say that? Your first drawing looks more like exponential decay. And even that isn't necessarily what you want.

The finance people here may have better ideas about your goal.

 

[BigBeachBanana]

This is the case:
Say I want to save money over the period of 60 months with a final target: say $50,000

Let's say that now I know 100% for sure that I will be able to save a lot of money because I'm in an excellent job, but the job is very unstable, and I don't really know when it will end and if I will be able to find another job in the future.

So the idea is that in the first month I would like to save a good amount of money, but by the end of the process I would like to save way less

Basically, the target saving graph would look like this (I'm making up the numbers, just to show the idea). Basically, in the first months I would be saving like 3 times more than the average, but in the last months I would be saving like 0.8 or 0.7x of the average.

The idea here is to figure out a formula, which clearly involves a logarithmic curve, and then refine it and make some adjustments, with Excel for example.

But I'm not 100% sure where to start from. If anyone could give me some guidance, I would be grateful.

View attachment 37357

You might be looking for a fixed-term geometrically decreasing payment annuity.

[math]PV = x\left( \dfrac{1-\left(\dfrac{1-k}{1+i}\right)^n}{1-\left(\dfrac{1-k}{1+i}\right)} \right)[/math][math]FV = PV\left(1+i\right)^n[/math]
where:
[imath]x[/imath] is the initial deposit
[imath]n[/imath] is the number of period
[imath]k[/imath] is the % each subsequent deposits of the previous
[imath]i[/imath] is the effective interest rate
[imath][/imath]

Here's an illustrative cash flow with the assumptions.
[imath]x = 4047.19[/imath] (Calculated using the formulas above)
[imath]n=60[/imath]
[imath]k = 90\%[/imath]
[imath]i = 5\% \text{ per annum or } \dfrac{5\%}{12} \text{ effective}[/imath]
[imath]FV = 50,000[/imath]




Some data points.

Time	Deposit	Ending Balance
0​	$ 4,047	$ 4,047
1​	$ 3,642	$ 7,707
2​	$ 3,278	$ 11,017
3​	$ 2,950	$ 14,013
4​	$ 2,655	$ 16,727
5​	$ 2,390	$ 19,186
...	...	...
55​	$ 12	$ 48,933
56​	$ 11	$ 49,148
57​	$ 10	$ 49,363
58​	$ 9	$ 49,578
59​	$ 8	$ 49,793
60​	$ 7	$ 50,007