Annuity question

[Ronan1990]

Hi,

just wondering could someone help with this question on annuties. Here it is

A loan made now is to be repaid by 99 daily repayments in arrear
using an effective daily rate of 0.01%. The value of the repayments will
alternate between 1,000 and 2,000 with the first of 1,000 to be
paid in exactly one day's time.
Find the current value of the loan.

[NoMoreExams]

Hi,

just wondering could someone help with this question on annuties. Here it is

A loan made now is to be repaid by 99 daily repayments in arrear
using an eective daily rate of 0.01%. The value of the repayments will
alternate between e1,000 and e2,000 with the rst of e1,000 to be
paid in exactly one day's time.
Find the current value of the loan.

So set up 2 annuities? Or one annuity of 1000 and then an incremental annuity every other year of 1000 as well.

[Ronan1990]

So set up 2 annuities? Or one annuity of 1000 and then an incremental annuity every other year of 1000 as well.

i set up 2 annuities and got my answer as 116372.28 but don't know if that's right

[NBran]

I set up the problem two different ways and arrived at an answer different than yours both times (I got the same answer both times, minus rounding errors).

Realize the last payment of 2000 occurs at t=98 and the last payment of 1000 occurs at t=99.

Approach 1: 1000a(angle 99 @0.01%) + 1000a(angle 49 @0.02%)
*Note: the second annuity converts the interest periods to every two days and the new rate, j = 0.02% (solved by 1.0001^2 - 1)

Approach 2: 1000a(angle 99 @0.01%) + 1000[ a(angle 98 @0.01%)/s(angle 2 @ 0.01%) ]
*Note: the second annuity keeps the interest periods at daily

Hope this helps.

[Ronan1990]

I set up the problem two different ways and arrived at an answer different than yours both times (I got the same answer both times, minus rounding errors).

Realize the last payment of 2000 occurs at t=98 and the last payment of 1000 occurs at t=99.

Approach 1: 1000a(angle 99 @0.01%) + 1000a(angle 49 @0.02%)
*Note: the second annuity converts the interest periods to every two days and the new rate, j = 0.02% (solved by 1.0001^2 - 1)

Approach 2: 1000a(angle 99 @0.01%) + 1000[ a(angle 98 @0.01%)/s(angle 2 @ 0.01%) ]
*Note: the second annuity keeps the interest periods at daily

Hope this helps.

Ok thanks for that it does help, i think the two I was setting up were wrong I said the first annuity was 1000a(angle 50 @ 0.01%) and i added this onto 2000a(angle49 @ 0.01%), could you do it this way or is there something wrong with it?

[NBran]

What you have written is a present value of a 49 term annuity with payments of 3000 and an additional payment of 1000 at t=50. So yes, there is something very wrong with it.

[Ronan1990]

What you have written is a present value of a 49 term annuity with payments of 3000 and an additional payment of 1000 at t=50. So yes, there is something very wrong with it.

yeah but if there's 99 repayments in total, and it alternates between 1000 & 2000 should that not mean you're dealing with an annuity of 1000 for 50 time periods and 2000 for 49 time periods??

[NBran]

No. Remember annuities discount the payments to present value. So the payment stream looks like:

PV = 1000v + 2000v^2 + 1000v^3 + ... + 2000v^98 + 1000v^99

What you have written is:

PV= 1000v + 2000v + 1000v^2 + 2000v^2 + ... +1000v^49 + 2000v^49 + 1000v^50
= 3000v + 3000v^2 + .... + 3000v^49 + 1000v^50.


I hope that helps. If you are confused with annuities, write out the payment stream.

[Ronan1990]

No. Remember annuities discount the payments to present value. So the payment stream looks like:

PV = 1000v + 2000v^2 + 1000v^3 + ... + 2000v^98 + 1000v^99

What you have written is:

PV= 1000v + 2000v + 1000v^2 + 2000v^2 + ... +1000v^49 + 2000v^49 + 1000v^50
= 3000v + 3000v^2 + .... + 3000v^49 + 1000v^50.


I hope that helps. If you are confused with annuities, write out the payment stream.

oh right I see what i've been doing wrong now thanks, but one more question. Take your approach 1...are you saying the PV is (1000v +1000v^2+...1000
v^99) + (1000v + 1000v^2....1000v^48)??

[NBran]

No. Note that I converted the interest period to 2 days, instead of 1 day. This means that payments will occur at the end of every even day.

This question tests your ability to calculate the present value of annuity where payments occur less frequent than the interest period. You have two options: (a) convert the interest period to the payment period or (b) use the original interest period. Option (b) requires you to know that if payments occur every k periods over n interest periods, the present value of the annuity immediate evaluates to a(angle n) / [s(angle k).