Interest compounding with a twist

**swe_dev1@hotmail.com** · February 3rd 2009, 06:22 AM

There are heaps of webpages with formulas for calculating compounding interest, but none (that I have found) that explains the formula to use when the deposit frequency does NOT necessarily match the compounding frequency.

In other words:

§1: equal sized deposits are made regularly at the beginning of each month
§2: compounding of interest m times a year
§3: interest rate i
§4: t months to grow

What is the value of the principal at the end of each MONTH?

**jonah** · February 3rd 2009, 10:48 AM

Originally Posted by swe_dev1@hotmail.com

There are heaps of webpages with formulas for calculating compounding interest, but none (that I have found) that explains the formula to use when the deposit frequency does NOT necessarily match the compounding frequency.

In other words:

§1: equal sized deposits are made regularly at the beginning of each month
§2: compounding of interest m times a year
§3: interest rate i
§4: t months to grow

What is the value of the principal at the end of each MONTH?

Go here.

**swe_dev1@hotmail.com** · February 5th 2009, 05:47 AM

Jonah,

The page you referred to contains a formula (your posting at 01-06-2009, 11:13 AM, third from bottom) that is great, however:

1) It is important for me to know what the total amount is at the end of every MONTH. The formula only tells me what the total amount is at the end of each year. E.g, if we assume that

R = 100
r = 10%
m = 12 payments/year
c = 1 compound per year

, what will be the total amount after y = 18 months? Note that deposits are made regularly at the beginning of every month (i.e. m = 12) for the entire duration (i.e. y = 18 months).

2) When I do a manual control calculation in Excel, all is fine when c=12 (other values same as above, y = 1). However, when c=1, I get 1265,45833 manually, whereas the formula returns 1264,05366. I use R*r/12/100 + 2R*r/12/100 + 3R*r/12/100 ... + 12R*r/12/100 Is the formula really valid when c=1 and m=12?

**jonah** · February 5th 2009, 09:59 AM

Originally Posted by swe_dev1@hotmail.com

1) It is important for me to know what the total amount is at the end of every MONTH. The formula only tells me what the total amount is at the end of each year. E.g, if we assume that

R = 100
r = 10%
m = 12 payments/year
c = 1 compound per year

, what will be the total amount after y = 18 months? Note that deposits are made regularly at the beginning of every month (i.e. m = 12) for the entire duration (i.e. y = 18 months).

Ans.
R = 100
r = 10%
m = 12 payments/year
c = 1 compound per year
y= (18 months)/( 12 months) = 1.5 year(s)
S” ≈ $1,942.72067…

You just need to plug the proper equivalent number for y. Thus,
for 1 month, y = 1/12 ≈ .08333…, and S” ≈ $100.797414…
for 2 months, y = 2/12 ≈ .1666…, and S” ≈ $202.3986008…
.
.
for 12 months or 1 year, y = 12/12 = 1, and S” ≈ $1264.053661…
.
.
for 18 months, y = 18/12 = 1.5, and S” ≈ $1,942.72067…
etc.

Originally Posted by swe_dev1@hotmail.com

2) When I do a manual control calculation in Excel, all is fine when c=12 (other values same as above, y = 1). However, when c=1, I get 1265,45833 manually, whereas the formula returns 1264,05366. I use R*r/12/100 + 2R*r/12/100 + 3R*r/12/100 ... + 12R*r/12/100 Is the formula really valid when c=1 and m=12?

The formula is indeed quite valid. Your “manual control calculation in Excel” must be the one that needs a closer scrutiny. If you still can’t reconcile your “manual control calculation in Excel” with the formula, make the necessary arrangement in attaching the Excel document in question in this thread and I’ll take a look at it. Hopefully, I’ll be able to make the necessary corrections on it for you.

**swe_dev1@hotmail.com** · February 6th 2009, 04:56 AM

Jonah,

1. Does that mean that I could use the formula below for calculating cases when only ONE single deposit is made at the beginning of the period and I want to know the total sum at the end of each MONTH (e.g. y = 1.5 for 18 months and c = number of coumpounds per year)?

$ \ddot S = R \cdot {\left( {1 + {\textstyle{r \over c}}} \right)^{cy}}$

2. Please see the attached Excel worksheet with my manual calculations. I think that the problem is that I multiply the monthly interest with the current balance instead of raising it, but then I dont understand why that is wrong when no compounding is made that month.

**jonah** · February 7th 2009, 10:01 AM

Originally Posted by swe_dev1@hotmail.com

Jonah,

1. Does that mean that I could use the formula below for calculating cases when only ONE single deposit is made at the beginning of the period and I want to know the total sum at the end of each MONTH (e.g. y = 1.5 for 18 months and c = number of coumpounds per year)?

$ \ddot S = R \cdot {\left( {1 + {\textstyle{r \over c}}} \right)^{cy}}$

Fundamental compound interest formula. I was under the impression that you're already familiar with it due to the nature of your original question. Go here for further explanation. Read the preview of chapter 4, page 41 to 69.

Originally Posted by swe_dev1@hotmail.com

2. Please see the attached Excel worksheet with my manual calculations. I think that the problem is that I multiply the monthly interest with the current balance instead of raising it, but then I dont understand why that is wrong when no compounding is made that month.

What you need is a sinking fund schedule. See mine below.
Your original annuity problem falls under the category of general annuities where the periodic deposits/payments of an annuity are made more or less frequently than the interest is compounded. A general annuity may be transformed into an equivalent simple annuity, i.e. the periodic deposits/payments of an annuity is the same as the interest compounding/conversion period, in two ways:
(i) By changing the given interest rate to an equivalent one for which the new interest conversion/compounding period is the same as the deposit/payment period.
(ii) Go here. See preview of page 100.

Method (i):
To understand how the formula for the account balance at the end of time y was derived if the deposits were made at the beginning of each period, we take note that that the effective rate of r whose compounding frequency is c is given by

$ w_{c} = \left( {1 + \frac{r}{c}} \right)^c - 1 $

Similarly the effective rate of j whose compounding frequency is m is given by

$ w_{m} = \left( {1 + \frac{j}{m}} \right)^m - 1 $

Equating these two effective rates and solving for j, we have

$ \left( {1 + \frac{j}{m}} \right)^m - 1 = \left( {1 + \frac{r}{c}} \right)^c - 1 \Leftrightarrow j = m\left[ {\left( {1 + \frac{r}{c}} \right)^{{\textstyle{c \over m}}} - 1} \right] $

Accordingly, the formula for the future value of an annuity due, i.e. deposits/payments at the beginning of each interval, when the deposit/payment interval is the same as the interest conversion/compounding period is given by

$ \ddot S = R \cdot \frac{{\left( {1 + {\textstyle{j \over m}}} \right)^{ym} - 1}}{{{\textstyle{j \over m}}}} \cdot \left( {1 + {\textstyle{j \over m}}} \right) $

Since the given nominal rate r has a compounding frequency of c, which is not the same as the deposit/payment interval, we convert said nominal rate r into its equivalent nominal rate j which has a compounding frequency of m; the compounding frequency of the deposit/ payment interval. We then plug the value of j into the formula for the future value of an annuity due whose deposit/payment interval is the same as the interest conversion/compounding period as just cited. This is basically the essence of method (i).

To bypass this time consuming method, we simply derive the calculation-ready (but not so intuitive) form. Thus,

$ \ddot S = R \cdot \frac{{\left( {1 + {\textstyle{{m\left[ {\left( {1 + \frac{r}{c}} \right)^{{\textstyle{c \over m}}} - 1} \right]} \over m}}} \right)^{ym} - 1}}{{{\textstyle{{m\left[ {\left( {1 + \frac{r}{c}} \right)^{{\textstyle{c \over m}}} - 1} \right]} \over m}}}} \cdot \left( {1 + {\textstyle{{m\left[ {\left( {1 + \frac{r}{c}} \right)^{{\textstyle{c \over m}}} - 1} \right]} \over m}}} \right) $
$ \Leftrightarrow $
$ \ddot S = R \cdot \frac{{\left( {1 + \left[ {\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1} \right]} \right)^{ym} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} \cdot \left( {1 + \left[ {\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1} \right]} \right) $
$ \Leftrightarrow $
$ \ddot S = R \cdot \frac{{\left[ {\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} } \right]^{ym} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} \cdot \left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} $
$ \Leftrightarrow $
$ \ddot S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} \cdot \left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} $

**swe_dev1@hotmail.com** · February 10th 2009, 03:49 AM

Thank you for your help Jonah!

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