Hopefully someone can clear this up for me because it is causing me quite a good deal of confusion. Here is a question similar to what I found in the book (I’m not sure if posting questions directly is forbidden so I changes it around a bit)
"You need to make five annual $1,000 payments, the first one starting at the beginning of year four (end of year 3). To get the necessary money to make these payments, you will make three equal payments into an investment account, the first payment to be made one year from today. Assuming you earn 10%, how much will these three payments be?"
The annuity starts at the beginning of year four (i.e. annuity due). You make three annual payments into this savings account, the first one starting “one year from today” (i.e. it sounds like this is an annuity due, also) and if that is the case, wouldn’t the last payment into the savings account be made on the same day that the annuity is made? I know the procedure for solving (mostly), I:
I. Figured out the PV of the annuity at the start of year 4 ($4,169.87). I solved this by treating it as an annuity due.
II. Figured out how much I would need to save for the three periods to be able to make these payments.
However, the book is telling me that the amount I need to save is an ordinary annuity. It seems to me that because the first three payments are made at the beginning of each of the three years, that this is an annuity due. Can someone clear this up for me?
The only difference between an ordinary annuity and an annuity due is where you set zero on your timeline; i.e., you can treat a series of payments as an ordinary annuity or as an annuity due as you wish. Most people make the choice based on the date as of which the present value or future value will need to be calculated.
If you want the future value as of the date of the last payment, or the present value one period before the first payment, it’s easier to treat it as an ordinary annuity.
If you want the present value as of the date of the first payment, or the future value one period after the last payment, it’s easier to treat it as an annuity due.
Here, you have five $1,000 payments from the account – beginning of year 4, beginning of year 5, . . . beginning of year 8 – and you’d like to know their present value as of the beginning of year 4; it’s easier to treat this as an annuity due.
To fund the account, you’ll have three payments – end of year 1, end of year 2, and end of year 3 – and you’d like to know the value in the account at the end of year 3; it’s easier to treat this as an ordinary annuity.
The book states that the three payments are to be made “one year from today”. My reasoning is that “one year from today” is, for example, Jan 1, 2013 - Jan 1, 2014 as opposed to Jan 1, 2013 - Dec 31, 2013. The book seemed to put emphasis on payments made at the beginning of the year vs. at the end of the year. For example, in this problem, they say to solve the PV of the 5 year annuity, we set the calculator to [BEG] mode. Given that the three payments into the savings account too, are made at the beginning of the year, I would imagine the same logic applies.
There’s no difference between 13/12/31 and 14/01/01: the end of the day New Year’s Eve is the same as the beginning of the day New Year’s Day (i.e., you don’t earn any additional interest).
I find it easiest to draw a timeline for these questions, write down the cash flows, then figure out the PV or FV or whatever at the appropriate dates. When I have a moment I’ll write an article on this that will (I hope) help make it clearer.
But the PV of an annuity due would be worth more than the PV of an ordinary annuity, all other things being equal, correct? I’m working on another problem where we have to fund an annuity comprised of three uneven CF’s with a saving account into which a series of equal payments are made. The depositis into the account and the withdrawls are both made “at the beginning of the period.” It comes out to be something similar to: |________|________|________|________|________|________|________|________|________|________|
T=0 T=9 ($19,609)
Where $19,609 is the PV of three uneven cash flows at T=9, T=10 and T=11. You are suposed to make 10 equal deposits starting at T=0 then making the last at T=9 (the same day you make your first withdrawl). I got the right answer by computing the PV of the three uneven series of CF’s at T=9. I used the CF worksheet on my TI with the first flow being at CO0 (i.e. an annuity due) then, I entered the following: [N]=10
[I/Y]=8
[PV]=0
[FV]=19604
[CPT] [PMT] and got the correct answer of 1353.
However, computing the annuity as an annuity due and the savings as an ordinary annuity was pure trial and error. If I had solved for the annuity as an ordinary annuity, I would have gotten a wrong answer. I can’t seem to get a grasp as to what I should calculate for and when.
Correct, by a factor of (1 + r) where r is the interest rate for one period.
Assuming that you have quoted the problem correctly, you did not get the correct answer of $1,353; you got the incorrect (but deceptive) answer of $1, 2 53 ($1,253.01, if we’re counting pennies). The correct answer is $1,353.25 (if we’re counting pennies).
I like this problem! It takes some thinking.
First, because you’re making the payments at the beginning of the period, starting today, it’s easier to do this problem as an annuity due. The rub is that when you calculate the future value of an annuity due that has 10 annual payments starting today, the future value it will give you will be the future value 10 years from today: one year after the final payment. You want the future value to be $19,604 9 years from today : on the day of the final payment.
What do you do?
Simply put: you cheat. If you know that it will give you a future value at the end of 10 years, and you want the future value after 9 years, you lie to it about the future value: you don’t tell it that the future value is $19,604; you tell it that the future value is $19,604 × 1.08. (There’s that factor of (1 + r): the same as in your first question and my first answer.) If the annuity due is worth $19,604 at the end of year 9 (beginning of year 10), when you make the last payment, then in another year is will grow by 8%, to $21,172.
If you calculate the payment for an annuity due with:
Allow me to post one more similar problem to make sure I am getting it.
Bob wants to reture 15 years from today. He currently has 121,000 dollars. He thinks he will need #37,000 at the beginning of each year for 25 years of retirement, making the first withdrawal on the day he retires. Bob believes that he will earn 8% over this period. The amount he needs to deposit at the beginning of this year and each subsequent year (for 14 years, 15 deposits in all) is:
So Bob makes his 15th payment at the beginning of the 15th year but retires at the end of the 15th/ beginning of the 16th. We need to know how much money he needs at the end of that 15th year and FV of an annuity due is calculated one period AFTER the final payment is made. In this case, the 15th payment is made at T=14 and we need to figure out how much Bob needs at T=15 (end of 15th year/beginning of 16th year)
So I calculated the PV of the annuity which he starts drawing upon at T=15. I set the calculator to beginning mode and got 426,564. I leave the calculator in beginning mode and enter:
[FV]=-426564
[N]=15 (the number of compounding periods fom T=14 to T=0)
[I/Y]=8
[PV]=121000
[CPT][PMT] = 1,456 (the answer in the book is 1,450, but it asks for closest to, so I should be okay)
