Annuities Dilemma

[technokarate]

Ok guys, I’m hoping someone can help me understand this concept a little better. I have a problem regarding TVM when it comes to annuities and annuities due. The issue is regarding an annuity due vs ordinary annuity. From what I understand, as long as FV is 0, you can easily calculate the same result (assuming you’re solving for i) for both types of annuities. You would get the same value for i as you would for annuity due if you took your initial principal and subtracted the first payment (made at time 0) and then solved for i only changing your value from n to n-1. Essentially this solves an annuity due as if it was an ordinary annuity by subtracting the first payment and then solving the rest.
My confusion lies in why the same method does NOT apply when it comes to problems in which FV is =/= 0. For example: if PV = 100, FV = 10, PMT = 20, N =6 (considering this is an annuity due), the result of i is 10.66%. However if I apply the same concept as previous paragraph, I would do the following and expect the same result: PV = 80, FV = 10, PMT = 20, N =5. And yet the resulting value of i is 10.93%, not 10.66%.
Why is this so? I’ve been breaking my head over this. It works for times when FV is 0 but not otherwise.
Any help would be greatly appreciated!!!

 

[Iampossible]

Hi mate,
I assume you are using the TI BA 2 Plus (or TI BA 2 Plus Pro) financial calculator.
So when you input PV = -100, FV = 10, PMT = 20, N =6 (considering this is an annuity due) in the so called “BGN” mode the cashflows that it computes i for are as follows:
@ t= 0 -100 and +20 ;
@ t=1,2,3,4,5 +20
@ t=6 FV= +10
=> this gives i = 10.66%
(you can verify this by pasting these cashflows in Excel and using the IRR function to back out that rate i which equates the PV of future cash inflows to the initial outlay. Note you may have to just input -80 for the cell corresponding to time zero, which is just the net of -100+20)
Now, based on your description, what you want the TI calculator to do is to place the FV=+10 @ t=5 instead of t=6. If it did this, you are absolutely correct in saying that the two series are equivalent. But it won’t do this as N=6 tells it that FV occurs at t=6, not t=5.
For example, with N=6 and the case where FV = 0,the TI realises that the last cashflow occurs @ t=6 and places FV =0 @ t=6.
In practice, you could much more easily use the CF function to get the IRR for these types of problems.

 

[technokarate]

Fantastic explanation! I think I understand. The difference in interest rates arises from the fact that the 10 is at different periods and is therefore worth different in t = 5 than it does in t = 6.
Using the same logic then, the reason why the interest rate (i) produced is the same for an ordinary annuity as it is for an annuity due at FV = 0, is because the FV of 0, whether at t = 5 or t = 6, is still 0 correct?
Thanks!!

 

[Iampossible]

You nailed it - that’s right !
Due to the Time Value of Money, cash flows at different points in time, are not worth the same and cannot be simply added or subtracted (Cashflow Additivity Principle - only cashflows at the same point in time can be added/subtracted). Moreover, differently dated cash flows have equivalence relationships based on the time separating them and the relevant interest rate.
And yes if FV =0, it doesn’t matter whether it is @ t=5 (as is the case for the ordinary annuity with N=5) or t=6 (the case for the annuity due in BGN mode with N=6)
Cheers