# How long does it take to double my savings

We save money for later so that we have an extra perk. It can be used to cope with financial bad luck but also to have fun or to buy something special. At this moment you already have put aside some money on your savings account, how long will it take before it is doubled and how can you calculate it conveniently?

## Why save money?

Having some money aside means that you have a financial reserve. It can be used as a buffer but you also can use it to build up some capital. Saving money you do for later, for which you can put aside some money every month. This can be the remaining money of your salary after one month of usage or a structural inlay in your savings account. How much to you need to lay in to build up a certain amount of money en how long does it take before the initial amount has been doubled?

## Hoe does interest buildup work?

If you have a certain amount on your account then it will grow with time. In one year your amount grows with the following factor: (1 + r)n. If you have 1.200 dollar with an interest of 3% then you will have after 1 year 1.200 * 1,031 = 1.236 dollar.  After 2 years that amount will grow to 1.200 * 1,032 = 1.273 dollar and after 5 years 1.200 * 1,035 = 1.391 dollar. This is called the compound interest over the initial amount resulting in capital growth due to interest on interest. How can you calculate your savings for the future and how long does it take to get a certain amount?

## Year or monthly interest

Normally when you choose an interest rate it will be shown as a percentage per year. But they don’t calculate with this factor as the year percentage is converted into the interest rate per month. This can be done with the following formula:

If you have a loan against 5% interest then that is 1,05(1 / 12)-1 = 0,4074 % per month. The outline of a loan will be given in yearly interest rates but for payments it is calculated accurately per month.

## Doubling savings through interest

If you put only once a certain amount in your account it can grow with time. If you want to change 1.000 dollar into 2.000 dollar how long does this taken? This depends purely on the interest rate. You can calculate it by doing the following:

• saved amount = a * (1 + r)n = b with a = inlay, b = result, r = interest rate, n = how long it takes;
• a = 1, b = 2;
• saved amount = (1 + r)n = 2;
• n = log (2) / log (1 + r).

Calculation example time interest rate build up

With previous formula for every interest rate you directly can calculate how long it takes for it to double. For instance how long does it take with an interest rate of 3, 4 and 5%:

• n(3%) = log (2) / log (1,03) = 23,4 periods;
• n(4%) = log (2) / log (1,04) = 17,6 periods;
• n(5%) = log (2) / log (1,05) = 14,2 periods.

This method of calculation is independent of how much you lay in your account. If you want to know how long it takes until the amount reaches one and a half times as much then you replace 2 (b) with 1,5 (this you can do for any magnification factor.

## Calculating saving with monthly inlay

If the interest rate is low it will take a long time before you have saved a certain amount of money. If you lay in a fixed amount per month, how long will it take before you reach a specific goal? You need to do the following:

• c * [ r / (1 + r)] / [(1 + r)n -1] = d with c = needed amount, d = inlay per month. This can be rewritten as the following;
• (1 + r) ^ n = (c / d) * [ r / (1+r)] + 1 and can be solved also with the log-relation;
• n = log {[(c / d) * [ r / (1+r)] + 1} / log (1 + r).

Calculation example time saving with inlay

You need 12.500 dollar and can lay in 250 dollar a month with a yearly interest rate of 3,5%, how long do you need to save?

• interest rate per month = 1,035(1/12) – 1 = 0,287%;
• n = log {[(12.500/250) * [ 0,00287 / 1,00287] + 1} / log (1,00287) = 46,7 months.

This means after 3 years and 11 months you will have 12.500 dollar on your savings account.

## Doubling savings with initial amount and monthly inlay

If you already have an amount on your bank account, how long does it take to double when you also lay in a monthly amount? You need to use the following:

• [c – e * (1 + r)n] * [ r / (1 + r)] / [(1 + r) ^ n -1] = d with e = initial amount;
• (1 + r) ^ n = {c * [ r / (1 + r)] + d) / (d + e * [ r / (1 + r)]} = {c/d * [ r / (1 + r)] + 1} / {e/d * [ r / (1 + r)] + 1};
• n = log {c/d * [ r / (1 + r)] + 1} / {e/d * [ r / (1 + r)] + 1} / log (1 + r).

Calculation example time initial amount, saving and inlay

You have 5.000 dollar on your savings account, how long does it take to double that amount (10.000 dollar) with interest rate of 3,5% and monthly inlay of 50 dollar?

• log {(10.000 / 50 * [0,00287 / 1,00287] + 1)/(5.000 / 50 * [0,00287 / 1,00287] + 1} / log (1,00287) = 70,1 months.

After 5 years and 10 months your amount has been doubled. If you safe 100 dollar per month then it is the following:

• log {(10.000 / 100 * [0,00287 / 1,00287] + 1)/(5.000 / 100 * [0,00287 / 1,00287] + 1} / log (1,00287) = 41,2 months.

## Interest versus inflation

An aspect with which every saver needs to reckon with is the ratio between interest and inflation. When time goes by money gets less worth, and so you will need more money to buy the same things. This is called inflation. If the interest is 2,5% but the inflation is 0,9% how long do you have to save then?

• Real interest rate = (1,025-1,009) / 1,009 = 1,025 / 1,009 - 1 = 1,58% (aka Fisher effect);
• Time for doubling = n(1,58%) = log (2) / log (1,0158) = 44,2 periods.

This also means that you will have to put aside money longer (within previous examples) having the same purchasing power. How long does it take then?

• Real interest rate = (1,035-1,009) / 1,009 = 1,035 / 1,009 - 1 = 2,57% (Fisher effect);
• Real interest rate per month = 1,0257(1/12) – 1 = 0,212%;
• n = log {[(12.500/250) * [ 0,00212 / 1,00212] + 1} / log (1,00212) = 47,5 months are needed with 250 dollar inlay to reach 12.500 dollar with the same purchasing power;
• log {(10.000 / 50 * [0,00212 / 1,00212] + 1)/(5.000 / 50 * [0,00212 / 1,00212] + 1} / log (1,00212) = 76,0 months with 50 dollar monthly inlay to double the initial amount of 5.000 dollar to 10.000 dollar corrected with inflation.

You save money for later. What if you already have some money, how long does it take to double? How can you calculate this?

Source: http://financieel.infonu.nl/sparen/139309-bereken-hoe-lang-het-duurt-om-mijn-spaargeld-te-verdubbelen.html