# Thread: 1st Payment in a Growth Annuity

1. ## 1st Payment in a Growth Annuity

Hi,

I'm trying to locate a formula for calculating the first payment in a growth annuity where the following is known:

1) The Future Value of the Annuity
2) The growth rate
3) The interest rate.
4) Number of Periods

Can anyone assist with this, It's starting to do my head in.

Thankyou & Regards,
Nick.

PS: I've attached a simple spreadsheet illustrating what I'm trying to do (but it may make things even more confusing!)

2. Originally Posted by nickchandler
Hi,

I'm trying to locate a formula for calculating the first payment in a growth annuity where the following is known:

1) The Future Value of the Annuity
2) The growth rate
3) The interest rate.
4) Number of Periods

Can anyone assist with this, It's starting to do my head in.

Thankyou & Regards,
Nick.

PS: I've attached a simple spreadsheet illustrating what I'm trying to do (but it may make things even more confusing!)
Hi, I think the formula you want can be derived from the formula for the sum of a geometric series:

Sn is the future value. Set the variable r = (1+i)/(1+g), where g is the growth rate and i is the interest rate. Set the variable a = p(1+g)^(n-1), where p is the first payment. The payments in Sn are listed in reverse chronological order. The future value of the first payment in Sn is ar^(n-1) = a(1+i)^(n-1)/(1+g)^(n-1) = p(1+i)^(n-1). The future value of the last payment is a = p(1+g)^(n-1).

There is something in the spreadsheet I found puzzling. The entry FV is the inflation-adjusted value of PV. It is the amount required in the future to equal the purchasing power of PV. These are not the standard meanings of FV (Future Value) and PV (Present Value). The standard definitions of FV and PV involve the time value of money, which is measured by an interest rate, not an inflation rate. FV and PV in the spreadsheet are looking at the purchasing power of money, not the time value of money.

So I find it puzzling that when you accumulate the value of the annuity payment, you do use the time value of money, but then you compare it to FV which uses the purchasing power of money. To me that is inconsistent and so I would like to know what you have in mind when you do that.

3. Originally Posted by JakeD
Hi, I think the formula you want can be derived from the formula for the sum of a geometric series:

Sn is the future value. Set the variable r = (1+i)/(1+g), where g is the growth rate and i is the interest rate. Set the variable a = p(1+g)^(n-1), where p is the first payment. The payments in Sn are listed in reverse chronological order. The future value of the first payment in Sn is ar^(n-1) = a(1+i)^(n-1)/(1+g)^(n-1) = p(1+i)^(n-1). The future value of the last payment is a = p(1+g)^(n-1).

There is something in the spreadsheet I found puzzling. The entry FV is the inflation-adjusted value of PV. It is the amount required in the future to equal the purchasing power of PV. These are not the standard meanings of FV (Future Value) and PV (Present Value). The standard definitions of FV and PV involve the time value of money, which is measured by an interest rate, not an inflation rate. FV and PV in the spreadsheet are looking at the purchasing power of money, not the time value of money.

So I find it puzzling that when you accumulate the value of the annuity payment, you do use the time value of money, but then you compare it to FV which uses the purchasing power of money. To me that is inconsistent and so I would like to know what you have in mind when you do that.

JakeD,

The worksheet is trying to represent a sinking fund problem. If I know (for instance) that a particular piece of equipment needs replacing in 10 years, and that the price of replacing that equipment (in today's dollars is $10,000). Money collected over the 10 years is held in trust and earns interest. In year ten, this price will have increased due to the effects of inflation (and hence the FV). Traditionally, payments to sinking funds appear to be constant (ie$100 in y1, 100 in y2, etc). By my reasoning, this appears to place an undue burden on the contributors at the front end, and eases the burden toward the end ($100 today is worth more than$100 in 10 years).

By using also using inflation as the payment growth factor, the contribution places the same burden on contributors over the whole life of the fund.

I hope this makes some sense, it's the best I can do to explain it.

Regards,
Nick.

4. ## Solutions

Originally Posted by JakeD
Hi, I think the formula you want can be derived from the formula for the sum of a geometric series:

Sn is the future value. Set the variable r = (1+i)/(1+g), where g is the growth rate and i is the interest rate. Set the variable a = p(1+g)^(n-1), where p is the first payment. The payments in Sn are listed in reverse chronological order. The future value of the first payment in Sn is ar^(n-1) = a(1+i)^(n-1)/(1+g)^(n-1) = p(1+i)^(n-1). The future value of the last payment is a = p(1+g)^(n-1).

There is something in the spreadsheet I found puzzling. The entry FV is the inflation-adjusted value of PV. It is the amount required in the future to equal the purchasing power of PV. These are not the standard meanings of FV (Future Value) and PV (Present Value). The standard definitions of FV and PV involve the time value of money, which is measured by an interest rate, not an inflation rate. FV and PV in the spreadsheet are looking at the purchasing power of money, not the time value of money.

So I find it puzzling that when you accumulate the value of the annuity payment, you do use the time value of money, but then you compare it to FV which uses the purchasing power of money. To me that is inconsistent and so I would like to know what you have in mind when you do that.

Just to wrap up, with some assistance from here, and the attached paper i located on the web, I came up with the following formula:

PMT = FV(i-g) / [(1+i)^n - (1+g)^n] (see attached equation.gif)