Regular saving effective rate maths challenge.

tomstickland

Since I now pretty much understand gross and AER on a fixed amount in a savings account (thanks for the info everyone), I thought I'd look at another interesting maths challenge today.

If you put £x into a savings account each month with a gross rate of g per year, and they pay interest monthly, what is the effective gain rate over the year. ie: how much more do you end up with as a proportion of the total that you paid in?

Quick recap on definitions:
-g is gross rate. so a 10% reg savers is 0.1, a typical savings acc is 0.05.
-interest is paid monthly
-monthly rate is g/12
-montly multiplier rate, which I'll call r, is 1 + g/12

Account builds up as follows.

End of Month, action
0, £x paid in
1, paid interest on balance, then additional £x added to account
2, paid interest on balance, then additional £x added to account
.....
12, paid interest on balance.

At the end, the first payment has received compound interest 12 times, the next payment 11 times, the next payment 10 times and so on to the last payment which recevied interest on the one time.

So the total amount at the end is:
Total = £x (r^12 + r^11 + r^10.....r^2 + r)
I took a quick look in a maths book and this adds up to:
Total = £x ( r ( r^12 - 1 )/(r - 1)
Over the year then 12 payments of £x are put in. So the ratio of total to what was paid in is:

R = (r/n) * (r^12 - 1) / (r - 1)
For a 5% account this ends up as 2.75%, for a 10% account it's 5.6%. So it's just a bit more than half of the gross rate.

Now I did some mathematical rearranging and once again applied a handy approximation to remove the r^12 term. The answer is then only approximate, but allows an understanding of the solution to be made.

The monthly rate, r, is actually 1 + g/12
If I subsitite this into the equation and do some work on it I end up with:
R = 1 + g/2* 13/12

So, the effective rate over the year is actually 13/12ths of half the gross rate. This correct to within 0.15% for a 10% annual payer.

For example, on a 10% reg savers account, the effective rate will be "exactly" 5.58%, this method says it's 5.4%.
A more exact form includes the next term.

R = 1 + (13/12) * g/2 * + (1/6 * 11/12*10/12) * g^2
approx
R = 1 + (13/12) * g/2 + 0.127 g^2

For a 10% gross rate, this improved version says that the effective rate is 5.54%, now only 0.04% away from the true figure.

Summary
Gross rate, true effective rate, approx rate, approx rate improved method
10%, 5.58%, 5.4%, 5.54%
5%, 2.75%, 2.71%, 2.74%

One other interesting thing is that if all the money was already sat in another account with a gross rate, and all the money was moved over in 12 equal steps, the money in the original account would also earn some effective rate. That's the next mathematical challenge.

None of this has been verified as correct, apart from me running some Excel spreadsheets.

Dagobert

You will find an in-depth discussion of the calculations here: when a regular saver is worth it:.

tomstickland wrote:

At the end, the first payment has received compound interest 12 times

No. Typically, regular savers pay interest at the end of the year and interest is not compounded. You get g/356 for each day.

You also need to consider days of interest lost through transfer (see above-mentioned thread). To work out days of lost interest, you will need to know how many days a transfer between these particular banks takes and whether these banks pay interest on the day of withdrawal and or receipt.

tomstickland

Ah, if interest is calculated daily then the sums are much much simpler, with the effective rate just being the half the gross rate. Well, I'll check that another time.

The above would apply to a normal savings account where you pay in a regular amount though.

Dagobert

tomstickland wrote:

The above would apply to a normal savings account where you pay in a regular amount though.

Only if interest is paid monthly - then it will be compounded every month. Most accounts which pay interest annually calculate the interest earned each day as described above.

tomstickland

Yes, understood. The above would apply to a savings account that paid interest every month. ie: most normal savings accounts that pay monthly interest.

The above is completely wrong for an annual payer. But an annual payer that pays 1/365th of gross per day is much simpler to analyse. The effective rate will be very v ery close to half the gross.

Robert_Sterling

If we assume you contribute the same amount each month on the same day of the month then on average the money will be in the bank for 6.5 months and the interest on the money will be 5.4% of the total capital in the account after one year.

Nothing alters the fact that whatever amount you put in to the regular saver all the the time it is in there it earns 10%.

To call anything other than 10% the "True" effective rate is not a good use of the word true. The instalments not yet paid at anytime may be earning money elsewhere at whatever rate.

tomstickland

It's all definitions. I agree it's not really a rate at all, it's a percentage gain in a fixed amount of money over a year. I'm not coming at this from any other angle than being inquisitve about how how the numbers work. The money in the account is earning 10% gross. However, it's the tapered way that the money is fed in that lowers the overall gain over the year. What interests me is that if you start with £3K at 5% outside the regular saver and move that in so that by the end you have £3K at 10%, then you earn about 7% interest on that £3K over the year.

Of course, if the money is not in my posession before the month that I pay it in then it's not making me any rate until I pay it in...

I'm wondering if you're thinking that I'm out to prove that regular savers don't pay what they say. Well, that's not the case.

I'm still pleased with the initial analysis; if I pay £500 per month into a 5% savings account then I'll end up gaining just over 2.5% of the total that I paid in over the year. That was the original question.

Dagobert

Robert_Sterling wrote:

If we assume you contribute the same amount each month on the same day of the month then on average the money will be in the bank for 6.5 months and the interest on the money will be 5.4% of the total capital in the account after one year.

That wasn't the question Tom posed. What Tom was looking to calculate is the interest rate which you are actually getting for your money during the investment period as it is being drip-fed from a savings account into a regular saver. It is the rate which was referred to as pd in the thread when a regular saver is worth it:.

Assuming that it is fed into the regular saver at regular intervals starting at the beginning of the investment period, it can be calculated as follows:
pd = (6.5*pr+5.5*ps)/12 - (d*ps/365)
pr being the regular saver's interest rate, ps that of the savings account and d the number of days lost through transfer.

If the money was left in the savings account it would attract pr%. If the money is fed into a regular saver, the overall amount yields pd% during that year.

For a 5% savings account and a 10% regular saver it is around 7.6%.

Robert_Sterling wrote:

To call anything other than 10% the "True" effective rate is not a good use of the word true.

It was maybe unfortunate terminology but it is not always easy coining a new phrase.

Robert_Sterling

Quite true.

tomstickland

"True" effective rate

I've just reread the first post and realised what I was saying. I meant "true" in the context of being mathematically correct, no approximated using an expansion to simplify the maths.

Anyway, all good fun this.
I make the numbers similar from a slightly different approach.
The gain from the standard 5% savings account is by my reckoning (using the compounding as shown in the first post) 2.75%.
With the reg savers, ignoring transfer days, then I will receive 365 payments at the daily rate, but the average amount in the account is half the ending amount, so the interet paid will be approx half the gross rate. ie: 5%.

Add those two together and I'll gain 7.75% over the year.

TBH, I need to think through the details a bit more carefully since I note that you have 6.5 vs 5.5 months of each account.

If both the reg savings and the normal savings account compounded monthly then I'd expect to gain 2.75 + 5.58 = 8.3%.

I'm only playing with numbers here; being realistic, essentially you do make in interest somewhere around the average of the rates of the account you're moving from and the account you're moving too.

Milarky

But what regular savings accounts pay other than annual interest [calculated daily and not 'compounded' over any period of less than one year]?

The basic formula [above] is straightfoward to understand:

The receiving account holds the money for an average of 6.5 months. There are 12 months in the year, so the feeder account must hold the funds fort he rest of the time excluding actual days lost due to transfers [now that is a much more interesting and demanding number to estimate or calculate exactly since a payment can fall on any day of seven yet will take an extra two days [at least] when it falls on 4 out of 7 days - Thur, Fri, Sat or Sun. Since there are 12 payments in the series, however, how do you determine how often this will occur - it's all too irregular for a formula, so an approximation is needed.]

I would say the weekends add at least one day on average to the two days the shortest transfers will take. That's 'about' 3 days therefore. And 3 days is 'about' one tenth of the month is it not?

Thus what about [6.4months x 10% + 5.5months x 5%]/12 months = [64%+27.5%]/12 = 91.5%/12 = 7.625%

The approximate loss [of 0.1 months] while the money is in niether account earning interest is 0.1months x 10%/12months or 1/12th of a percent I'd estimate.

It also assumes that the transfer is able to take place at the start of a month - giving it the maximum time before the anniversary arrives.

I sometimes say that this is a '78/66' split - meaning that the money can be in the receiving account for at most 78/144ths of the time and will be in the feeding account the remaining 66/144ths of the time. In the 'worst' case, the money only arrives at the end of the month rather thanthe start and the numbers are exactly reversed - to '66/78'. And at the mid point of the month its '72/72' and you'd get the exact median interest for the two accounts disregarding the lost days involved in the transfer.