General Annuity Due

[soulie]

A couple would like to accumulate $20,000 in 3 months as a down payment for a house, by making deposits at the beginning of each week in an account paying interest at 4.2% compounded monthly. Determine the size of the weekly deposit.


I tried solving this using the formula for GAD, and my answer was $1,532.26. However, my prof said that the correct answer is $1,529.80. Im thinking it might be because it's in the case of months and weeks in getting the equivalent rate that made me arrive into the wrong answer. Could you please share your thoughts about this?

 

[Deleted member 4993]

A couple would like to accumulate $20,000 in 3 months as a down payment for a house, by making deposits at the beginning of each week in an account paying interest at 4.2% compounded monthly. Determine the size of the weekly deposit.


I tried solving this using the formula for GAD, and my answer was $1,532.26. However, my prof said that the correct answer is $1,529.80. Im thinking it might be because it's in the case of months and weeks in getting the equivalent rate that made me arrive into the wrong answer. Could you please share your thoughts about this?

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What if there are more than 2 given in normal distribution?

I have this sample problem for a normal distribution here. I know how to get those in (a) and (b). But I'm a bit puzzled on how to get (c). An explanation would be enough. Thank you!

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[JeffM]

A couple would like to accumulate $20,000 in 3 months as a down payment for a house, by making deposits at the beginning of each week in an account paying interest at 4.2% compounded monthly. Determine the size of the weekly deposit.


I tried solving this using the formula for GAD, and my answer was $1,532.26. However, my prof said that the correct answer is $1,529.80. Im thinking it might be because it's in the case of months and weeks in getting the equivalent rate that made me arrive into the wrong answer. Could you please share your thoughts about this?

This is not primarily a math problem.

You need to know exactly how interest is calculated for fractional months. I can tell you how the bank I used to work for did it, but it is only one of several ways. It would divide 4.2% by 365 (or 366 in a leap year) to calculate a daily rate. At the end of each day, the principal in the account would be multiplied by the daily rate and accumulated in an accrual account separate from the principal account. On the last day of the month, the amount in the accrual acount would be truncated to two decimal points. That truncated amount would be added to principal and subtracted from the accrual account.

 

[BigBeachBanana]

This is not primarily a math problem.

You need to know exactly how interest is calculated for fractional months. I can tell you how the bank I used to work for did it, but it is only one of several ways. It would divide 4.2% by 365 (or 366 in a leap year) to calculate a daily rate. At the end of each day, the principal in the account would be multiplied by the daily rate and accumulated in an accrual account separate from the principal account. On the last day of the month, the amount in the accrual acount would be truncated to two decimal points. That truncated amount would be added to principal and subtracted from the accrual account.

Hi Jeff,
The 4.2% is the monthly effective rate and not the annual nominal rate so dividing by 365 wouldn't work. Furthermore, you'd also be compounding daily. I think the question is asking for the weekly effective rate i.e. [imath](1.042)^{12}=(1+r)^{52}[/imath].
Either way, I couldn't match the given answers, but Excel agrees with me.

 

[JeffM]

Hi Jeff,
The 4.2% is the monthly effective rate and not the annual nominal rate so dividing by 365 wouldn't work. Furthermore, you'd also be compounding daily. I think the question is asking for the weekly effective rate i.e. [imath](1.042)^{12}=(1+r)^{52}[/imath].
Either way, I couldn't match the given answers, but Excel agrees with me.

@BigBeachBanana At least in the US, interest rates are almost always quoted at an annual rate. Here it is stipulated that the rate is compounded monthly. Under US law, that would be quoted as an annual yield of 4.33%. Rates and yields are different under US banking law.

And no, you would not be compounding daily. Look carefully at what I said. The accrual amount is not added until after the last multiplication for the month; there is no interest on interest until the accrual is added at the end of the month. All of this must take into account the differences among ledger, available, and collected funds, which I did not bother to go into but are of practical importance.

The problem as given is unanswerable. An account to which amounts can be deposited or withdrawn within a month but with monthly compounding requires some careful handling. All of this is covered for US banks by fairly technical regulations of the Federal Reserve Board and may be supplemented by a bank’s account agreement. Account agreements are not uniform across banks. I have not looked at those regulations in over a decade, but they used to allow for interest to vary depending on what time of the day a deposit was made (if I remember correctly, a bank used to have the right to set a cut-off time after 3 pm on a “banking day,” which was a day the Federal Reserve was open (with a different time for deposits through ATMS and night depositories.) Notice that a banking day differed from a “business day,” which was a day that the specific bank’s physical offices were open to the public.

 

[BigBeachBanana]

@bbb At least in the US, interest rates are almost always quoted at an annual rate. Here it is stipulated that the rate is compounded monthly. Under US law, that would be quoted as an annual yield of 4.33%. Rates and yields are different under US banking law.

And no, you would not be compounding daily. Look carefully at what I said. The accrual amount is not added until after the last multiplication for the month; there is no interest on interest until the accrual is added at the end of the month. All of this must take into account the differences among ledger, available, and collected funds, which I did not bother to go into but are of practical importance.

The problem as given is unanswerable. An account to which amounts can be deposited or withdrawn within a month but with monthly compounding require some careful handling. All of this is covered for US banks by fairly technical regulations of the Federal Reserve Board and may be supplemented by a bank’s account agreement. Account agreements are not uniform across banks. I have not looked at those regulations in over a decade, but they used to allow for interest to vary depending on what time of the day a deposit was made (if I remember correctly, a bank used to have the right to set a cut-off time after 3 pm on a “banking day,” which was a day the Federal Reserve was open (with a different time for deposits through ATMS and night depositories.) Notice that a banking day differed from a “business day,” which was a day that the specific bank’s physical offices were open to the public.

I agree with you up to a point, however, given my experience with these type of questions, most of the practical aspects you mentioned like the Fed Reserve, cut off time, US laws, ect…are considered beyond the scope of the curriculum. Often simple questions like this are purely aimed to introduce the concept of annuity, and interest rate conversions. However, I’m still speculating and OP can shine more light on what’s expected.

 

[JeffM]

I agree with you up to a point, however, given my experience with these type of questions, most of the practical as aspects you mentioned like the Fed Reserve, cut off time, US laws, ect…are considered beyond the scope of the curriculum. Often simple questions like this are purely aimed to introduce the concept of annuity, and interest rate conversions. However, I’m still speculating and OP can shine more light on what’s expected.

@BigBeachBanana

I partially agree with you as well. I have no clue what the problem intends or how the teacher interpreted it. I am sure that whoever wrote the problem had no clue as to the practical complexities involved and assumed that its interpretation was self-evident. But it is not.

The most common way, at least in the U.S., to interpret a nominal interest rate is as an annual rate. Furthermore, the number of weeks in a month is not uniform. The number in July, August, and September is 92/7. The number in February, March, and April is 89/7. If I had to guess, it would be that we are to assume 13/3 weeks in a month. But there is no information on how deposits for fractional months are to be handled. It is of course possible that the actual problem has sufficient additional information to reverse-engineer the purported answer, but that our poster did not see its relevance.

 

[Deleted member 4993]

However, I’m still speculating and OP can shine more light on what’s expected.

That was the intent/expectations of the questions posed in response #2.

 

[jonah2.0]

Beer induced opinion follows.

A couple would like to accumulate $20,000 in 3 months as a down payment for a house, by making deposits at the beginning of each week in an account paying interest at 4.2% compounded monthly. Determine the size of the weekly deposit.


I tried solving this using the formula for GAD, and my answer was $1,532.26. However, my prof said that the correct answer is $1,529.80. Im thinking it might be because it's in the case of months and weeks in getting the equivalent rate that made me arrive into the wrong answer. Could you please share your thoughts about this?

I agree with your prof on this one.
Our late friend, the indefatigable math knight-errant Sir Denis will no doubt also agree as I've seen him tackle this type of problem many times.
It's a general annuity due problem.
Simply solve for [imath]j_{52}[/imath] in

[imath]\bigg(1+\frac{0.042}{12}\bigg)^{12}=\bigg(1+\frac{j_{52}}{52}\bigg)^{52}[/imath]

You then solve for R, the weekly deposit, in the version of the future value of an annuity where deposits are made at the beginning of the interval (i.e., the annuity due version). Thus,

[imath]20,000=R\frac{\left(1+\frac{j_{52}}{52}\right)^{\frac{3}{12}\cdot52}-1}{\frac{j_{52}}{52}}\left(1+\frac{j_{52}}{52}\right)[/imath]