Doubts

Review Quiz 2 - exercise 5

Review Quiz 2 - exercise 5

by Matilde P. T. Ferreira da Fonseca -
Number of replies: 2

When I was solving the problem I did:

2,500,000=𝑆/(0.05)*[1−(1/(1.05)^36)]

Then, as I didn't achieve none of the result of the multiple choices I went through approach 2 which I believe is pretty similar to my resolution:

"You use the annuity to compute the PV of the 36 payments starting today, which will give you the PV of all payments at t=-1 and then bring that value forward one year:

2,500,000/(1.05)^35=𝑆/(0.05)*[1−(1/(1.05)^36)]×1.05

And solving for S we get S=26,086."

Although I don't understand the highlighted parts.


In reply to Matilde P. T. Ferreira da Fonseca

Re: Review Quiz 2 - exercise 5

by Julio Crego -
Thanks for sharing your answer, I am sure it helps many.

There are two minor mistakes in your answer.

The first one is that you are comparing a monetary quantity on his 65th birthday (2,500,000) with a monetary quantity today (discounted annuity). We cannot do that. We need to decide whether we want to put every quantity in the future (65th birthday) or today (30th birthday). I will assume we want everything today just to align with the solutions but both are correct.

What is the value of 2,500,000 on my 65th birthday as of today:

\(V = \dfrac{2,500,000}{(1+r)^{35}}\)

We use 35 because there are 35 years from now to our birthday. 

The second mistake is an extremely common one. The annuity and perpetuity formulas always assume that the payment is in the next time period. Therefore, if we use a formula like:

 \( V = \dfrac{𝑆}{0.05} \left(1−\dfrac{1}{1.05^{36}}\right) \)

we are computing the value as of our 34th birthday of an annuity that has the first cashflow in the 35 birthday and ends in the 65 birthday. Because we want the value on our 35th birthday, we need to move the value forward one period and multiply it by 1.05.

There is another way to face a problem like this one; that is, an annuity or perpetuity that starts today (maybe it is more intuitive to you). Note that we can separate the problem into two parts:

Today -> I save S
From next year to 65th-> I save S

The second part is a standard annuity that starts next period and lasts for 35 periods. Meanwhile, the first part is today so it does not need discounting. Overall the total present value is:

\(V=S + \dfrac{𝑆}{0.05}\left(1−\dfrac{1}{1.05^{35}}\right) \)

The result is exactly the same