Savings Account

Question
All savings accounts offer some small rate of return, usually between .25% and 1.5%. Is this rate insignificant, or is it worth caring about? The answer: probably insignificant.

First some definitions:

• Monthly compounding interest: This is usually stated as 1.5% per year compounding monthly. What this really means is each month, the average daily amount in your account times one twelfth (because there are twelve months per year) of 1.5% is added in as your interest for that month. Obviously by adding in that amount each month, the second month contains a little interest on the first month's interest, and in the third month, the interest that was accrued in months one and two are both included in the amount for which the interest is calculated.

Math Time

Let's start by defining some variables:

• S = Starting amount
• F = Final amount
• p = number of periods in a year (p = 12 if we're dealing with monthly compounding)
• r = annual interest rate (.0025 <= r <= .015, or r is between .25% and 1.5%)
• y = the number of years we let it continue to get interest (note: we can use whole years, or we can use parts of year, but only in denominations of p. In other words we can have 1/12, 2/12, 3/12, and so on as values for y if we're dealing with months, i.e. p = 12.)

1. From Interest Math, we know that when p = 1, meaning we are dealing with annually compounding interest, that we get S(1+r)^y = F. Let's follow the same logic to get our periodically compounding interest.
2. What happens after the first period (after the first month if p = 12)? S + S*(r/p) = F. Here we are adding the fractional interest accrued over the first period. Factoring out the S gives us S(1+r/p) = F
3. For the second month, S + S*(r/p) + (S + S*r/p) * r/p = F. Let's distribute the r/p: S + S*r/p + S*r/p +S*(r/p)^2 = F. Let's combine terms and factor out an S: S(1 + 2(r/p) + (r/p)^2). This looks just like the annually compounding interest! Clever mathematicians will realize that 1 + 2(r/p) + (r/p)^2 = (1 + r/p)^2
4. This same idea holds going forward, so we get S(1+r/p)^(p*y) = F. Again, this can be proven rigorously using Mathematical Induction.

We know F in terms of S, r, p, and y. What about the other forms?

Final Answers

Using basic algebra (a little more for logs) we get:

• F = S(1+r/p)^(p*y)
• S = F/(1+r/p)^(p*y)
• r = ((F/S)^(1/(p*y)) - 1) * p
• p is not nice .. leave a comment if you can figure this one out
• y = log(F/S) / (p*log(1+r/p))

Lets make a table showing for $1,000, $5,000, and $20,000, how much you would make the first year with .25% and 1.5% interest:

• For $1,000:
• At .25%, 1000(1 + .0025/12)^12 = $1,002.50 or $2.50 in interest
• At 1.5%, 1000(1 + .015/12)^12 = $1015.10 or $15.10 in interest
• For $5,000:
• At .25%, $5,012.51, or $12.51 in interest
• At 1.5%, $5075.52, or $75.52 in interest
• For $20,000:
• At .25%, $20,050.06 or $50.06 in interest
• At 1.5%, $20,302.07 or 302.07 in interest

If you don't have much money, then you should be looking at this and thinking "15 bucks a year ... for saving $1000 ... that's pretty skimpy". If you have $20,000 saved up, then you probably won't notice 50 to 300 bucks a year, or between $4 and $25 a month.

At the best rate banks will offer for savings, you might be getting more squat but it still amounts to diddly.