Payment Fusion Concept

[dbailey]

I am trying to find the future value of an increasing annuity with the following payment structure:


t=0 payment = 1
t=.5 payment = 1
t=1 payment = 2
t=1.5 payment = 2
t=2 payment = 3
t=2.5 payment = 3
t=3 payment = 4
t=3.5 payment = 4
t=4 payment = 5
t=4.5 payment = 5

I am given an effective annual rate of i.

Let X = payment at the end of year 1

1 x S:angle2 = X (annuity-due)
(where the annuity is calculated using the equivelant i(2) rate.)

So my Future equation of value is X x (IS)_angle5

By looking at the answer that this approach pertained to (ASM page 171 Q29), I have everything correct, except for the fact that my X should be
1 x Sangle2 instead of 1 x S:angle2 (immediate instead of due)

Can anybody help me with my understanding of this.

Thanks,
Dave


The complete question is as follows:

Mary is hired by ABC Company on 1/1/85. She is eligible to qualify for the company's new bonus program. If an employee qualifies at year-end, they will receive a cash bonus payable in two equal installments, half on Jan 1 and half on Jul 1. The first year bonus is 2000 and increases by 2000 each year. Assuming Mary qualified each year. (i.e. she received her first payment on 1/1/86) and deposited her bonus in an account which pays an 8% annual effective rate of interest, what is the accumulated value,A, of her bonus payments through 7/1/90.

[Gandalf]

You are close, but you left out a very key point on your time line.

t = ?, accumulated value needed.

You want the value on 7/1/90, but what value of t is that?

The last payment is also on 7/1/90, so if you assign t's to the payments the way you did, you want the accumulated value at t = 4.5. But if you are using (Is)5 in your formulas, that gives a result 5 years after your process starts, hence at t = 5.0. Something must be done.

One thing that could be done is to realize that your formula (and a fine formula it was) will give the correct value at t=5, so multiply your formula by v^(1/2) and you have a formula that works.

The other thing that could be done is to decide that 7/1/90 should be t=5. If you're calling it t=5, then all your payment times are off. They become t=0.5, t=1, t=1.5,...,t=5. With the payments labeled that way, you'll naturally write the formula with
1 x Sangle2 instead of 1 x S:angle2 (immediate instead of due)

(And, to see that both approaches are equivalent, observe that
Sangle2 = S:angle2 * v^(1/2)

[dbailey]

Thanks Gandalf.
You give the best answers. I was hoping you would reply!!