Regular saving effective rate maths challenge.

grumbler

tomstickland wrote:

... in the context of being mathematically correct, no approximated using an expansion to simplify the maths.... 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... the average amount in the account is half the ending amount, so the interet paid will be approx half the gross rate. ie: 5%.

In the context of being mathematically correct your figures are far from being correct. It was told here many times (including this thread above) that average balance of a regular saver is not a half, but about 6.5/12 of the ending amount; the average balance of a standard saving account is about 5.5/12 of the amount.

Dagobert

Milarky wrote:

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.

That is assuming that payments will be carried out using Standing Order (as most people would do). I noticed on this board though that some people do the transfers "by hand" to avoid weekends of lost interest.

tomstickland

grumbler wrote:

In the context of being mathematically correct your figures are far from being correct. It was told here many times (including this thread above) that average balance of a regular saver is not a half, but about 6.5/12 of the ending amount; the average balance of a standard saving account is about 5.5/12 of the amount.

Yes they are mathematically correct for the set up I described. It just so happens that the set up I described is not a regular savings account, but a standard savings account that pays monthly interest. The analysis made no mention of "average balance", it actually added up the interest each month.

Sure, there are many things all being mixed up here, so it is quite a confusing thread. It might appear that I'm confused, but I'm not!

tomstickland

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

Agreed, the "weighted average" formula (5.5/12ths of one account and 6.5/12ths of the other) is easy to understand. What I was saying though, is the lower paying account will normally be a normal savings account that pays interest monthly, so you will actually make slightly more than 5.5/12ths of what comes out of it.

However, lost days etc will pull the overall rate for the year down, as explained.

So, conclusion is, for rule of thumb, average return is going to be approximately the average of the two rates.

Robert_Sterling

I digress


Suppose we have the fraction

26
----
65

and we say that there is a six on the top of the fraction
and a six on the bottom so let us rub them out and get the fraction

2
----
5

is this correct

16
----
64

1
----
4

???

tomstickland

Suppose we have the fraction

26
----
65

and we say that there is a six on the top of the fraction
and a six on the bottom so let us rub them out and get the fraction

2
----
5

No you can't do that. All you can do is multiply or divide the top and bottom by the same number.

is this correct

16
----
64

1
----
4

???

Yes and no. Correct answer but wrong method.

Divide top and bottom by 16:
1/16 * (16 / 64 ) = (16 / 16 ) / (64 / 16) = 1 / 4

Or you can get there quicker by using:
a / b = 1 / ( b / a)

tomstickland

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've just done a mathematical doodle based on what you're saying here. I'm not challenging anything, just seeing what I come up with semi independantly. I'm almost repeating what's already been said, but from my own thought process.
Which is.....

Both accounts pay interest but don't compound
Account a is the account that the money comes out of, paying rate a
Account b is the accout that the money goes into, paying rate b
There are 12 payments made, with the first one at the start of month 1 and the last one at the start of month 12.

Looking at the receiving account (b), it starts life with 1 payment in it and starts the last month with 12 payments in it. The average number of payments in it over the whole year are (12 + 1 ) / 2 which gives you 6.5 in there on average. Or, saying the same thing differently, the first payment sits in there for 12 months and the last one sits in it for 1 month. Hence the money stays in the account for an average of 6.5 months.

The paying account (a) starts life with 11 payments left in it and starts month 12 with no payments left in it. Hence the average number of payments stored in this acccount is (11 + 0 ) / 2 which gives you the 5.5. Or saying the same thing differently, the first payment sits in the account for 0 months and the final payment sits there for 11 months, so the average length of stay is 5.5 months.

So over the year, the money has on average sat in account a for 5.5 months and account b for 6.5 months, giving an blended average of 5.5/12*a + 6.5/12 * b.

This means that the rate of return is about 45% of rate a plus 65% of rate b.
For a = 5% and b=10% this is 7.7%.

If account a was a monthly compounding account then the contribution from that account would be a bit higher because the interest would have compounded whilst waiting for the money to be moved.

If on average 3 days are lost per month, then 1/12th of the total amount is not making any interest for 3 days and this happens 12 times over the year.
The lost amount is 3 * b/365 * 1/12 * 12 = 3/365 * b. For b=10% this is just under 0.1%.

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.

It took me a while to work out your numbers there, but I got it.
At best the rate will be 5.5/12 * a + 6.5/12 * b
At worst the rate will be 6.5/12 * a + 5.5/12 * b {because payment is at end of month, so paying account gains a month and receiving account loses that month)
If paying at middle of month rate will be 6/12 * a + 6/12 b.