Doubts

Exercise Set 2 - question 5.c)

Exercise Set 2 - question 5.c)

by Matilde P. T. Ferreira da Fonseca -
Number of replies: 5

Hello professor,

In the exercise 5.c), my final result is the same as in the solutions. Although, the PV that I got is different from the one in the solutions and I don't understand why for the present value the solution multiply the value by (1+r). For me it just make sense to do when we are computing its FV.

Thank you in advance,

Matilde Fonseca

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In reply to Matilde P. T. Ferreira da Fonseca

Re: Exercise Set 2 - question 5.c)

by Julio Crego -
Dear Matilde,

It is just a different way of defining things. In the Excel sheet, PV means the value today. Note that in an annuity formula, the first payment is in the next month; therefore, your present value calculation obtains the value in the previous month (not today). To move it to today, you need to multiply by (1+r).

You get the right result because then you use 13 months instead of 12 when computing the FV. I suspect that you took into account that you were starting from one period before.
In reply to Julio Crego

Re: Exercise Set 2 - question 5.c)

by Matilde P. T. Ferreira da Fonseca -
But what differs from the exercise b)?
In that exercise I got the same PV as in the solutions and I used T=11, which should mean that in exercise c) I would use T=13, right?

Thank you in advance,
Matilde Fonseca
In reply to Matilde P. T. Ferreira da Fonseca

Re: Exercise Set 2 - question 5.c)

by Julio Crego -
So in exercise b, PV refers to the value next month (#1) and not this month. If you use the quantity to the right (which is the value today) and capitalize by 12 months, you get the result.

Note that there is a big difference. In b, we start 2 months from now, so the annuity computes the PV next month. If we want it this month, we divide by (1+r)
Instead, in c, the annuity provides the present value last month. Then we need to multiply by (1+r) to move it to today.

Once we have the PV today, we always multiply by \((1+r)^{12}\)
In reply to Julio Crego

Re: Exercise Set 2 - question 5.c)

by Matilde P. T. Ferreira da Fonseca -
Maybe this way us better for me to explain my doubt:

b)
me: PV=(200/0,4%)*(1-1/(1+0,4%))^11*1) => FV= PV*(1+0,4%)^11
solutions: PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)=> FV= PV*(1+0,4%)^11 => FV=PV*(1+0,4%)^11

Here we are doing the same thing with PV, I use T=11 (already accounting for the "start 2 months from now")

c)
me: PV= 200/(0,4%-0,2%)*(1-((1+0,2%)/(1+0,4%))^13*1) => FV =PV*(1+0,4%)^13
solutions: PV= 200/(0,4%-0,2%)*(1-((1+0,2%)/(1+0,4%))^13*1)*(1+0,4%) => FV= PV*(1+0,4%)^12

Here I would be expecting the same logic: since saving are starting today, T=13
As we already account for starting now (As we use T=13),, I don't understand why we multiply the value by (1+0,4%). In my point of view, that would be done only when computing FV.
In reply to Matilde P. T. Ferreira da Fonseca

Re: Exercise Set 2 - question 5.c)

by Julio Crego -
If you find your method more intuitive, it is perfectly fine. Keep in mind that, in the solutions, for b) it says PV(month#1) and for c) it says PV because the first one is the present value one month from now, and the second is the present value today. You are using PV in b) for the present value next month and in c) for the present value the month before today. 

If you ask my favorite method, I would do

b) PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)/(1+0.4%) --> This way it is today's value because the annuity formula gives me the value one month before the first payment (month 1)
FV= PV*(1+0,4%)^12 -> because it is 12 months from today

c) PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)*(1+0.4%) --> This way it is today's value because the annuity formula gives me the value one month before the first payment (month -1)
FV= PV*(1+0,4%)^12 -> because it is 12 months from today

Instead of moving the PV to today every time and capitalizing 12 months, you preferred leaving the value where the annuity formula computes it and capitalizing accordingly. It will work every time as long as you keep in mind that the annuity formula always provides the value in the month before