# Calculating Interest Rate on Regular Savings

J

#### J C

If I know how much I paid in to an investment (£x per week for 10 years) and
what the value is at maturity, how do I work out what the effective rate of
interest was over the period? I want to see just how badly this did compared
to sticking the money in a savings account.

Thanks

R

#### Ronald Raygun

J said:
If I know how much I paid in to an investment (£x per week for 10 years)
and what the value is at maturity, how do I work out what the effective
rate of interest was over the period? I want to see just how badly this
did compared to sticking the money in a savings account.
The value at week 0 is 0, at week 1 £x, at week 2 f*£X + £x, where
f is one plus the weekly rate of interest.

Hence at week 520, the value is £x times the sum of all the powers
of f from 0 to 519. If the value is £y, then the following equality
holds:

£y = £x * (f^520 - 1)/(f-1)

All you need to do is solve for f.

Unfortunately this is not possible algebraically (but if Tim wants to
claim otherwise, I'd be delighted to hear how). So the simplest thing
to do is guess a value for f, compute the right hand side, and compare
it with £y. If the answer is too big, your f was too big, so reduce it
a bit for your next guess.

Having found f, raise it to the power 52 and subtract one, and that's

J

#### J C

Ronald Raygun said:
The value at week 0 is 0, at week 1 £x, at week 2 f*£X + £x, where
f is one plus the weekly rate of interest.

Hence at week 520, the value is £x times the sum of all the powers
of f from 0 to 519. If the value is £y, then the following equality
holds:

£y = £x * (f^520 - 1)/(f-1)

All you need to do is solve for f.

Unfortunately this is not possible algebraically (but if Tim wants to
claim otherwise, I'd be delighted to hear how). So the simplest thing
to do is guess a value for f, compute the right hand side, and compare
it with £y. If the answer is too big, your f was too big, so reduce it
a bit for your next guess.

Having found f, raise it to the power 52 and subtract one, and that's
Tried that and got an effective rate of 0.9%. (0.896701% to be more
precise.)

I used Excel and put the formula in as =5*(A1^520-1)-1/(A1-1) where A1
was used to hold the figure for f.

The payment was £5 per month for 10 years and the payout was £2926. (or £326
more than I paid in!) Does that look about right?

I know it's not a lot of money but I did want to see how badly it had done.

John

R

#### Ronald Raygun

J said:
Tried that and got an effective rate of 0.9%. (0.896701% to be more
precise.)

I used Excel and put the formula in as =5*(A1^520-1)-1/(A1-1) where A1
was used to hold the figure for f.
There is a typo there. You didn't really have the extra -1 between
the ) and / did you?
The payment was £5 per month for 10 years and the payout was £2926. (or
£326 more than I paid in!) Does that look about right?
No, I get f=1.00044665 which makes the annual rate about 2.35%.

gnuplot> f(x)=5*(x**520-1)/(x-1)
gnuplot> print f(1.00044665)
2926.00168976015
gnuplot> print 1.00044665**52
1.02349231203214

J

#### J C

Ronald Raygun said:
There is a typo there. You didn't really have the extra -1 between
the ) and / did you?

No, I get f=1.00044665 which makes the annual rate about 2.35%.

gnuplot> f(x)=5*(x**520-1)/(x-1)
gnuplot> print f(1.00044665)
2926.00168976015
gnuplot> print 1.00044665**52
1.02349231203214
There was a typo, not sure how that got there but it was also present on the
spreadsheet. I think it probably sneaked in using Cut and Paste. Fixed that
and the figures now match yours.

Thanks for all the help, now I know it was as a bad investment and not the