Geometric Progression: Present value of winning lottery ticket

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lemunsips

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Hello! I am not very sure if i title this correctly or if it under the correct topic but i really need help with this question.


The present value (PV) of a stream of income is \(\displaystyle D_t,\, t\, =\, 0,\, 1,\, ...,\, T\) is defined as

. . . . .\(\displaystyle \displaystyle PV\, =\, \sum_{i=0}^T\, \dfrac{D_t}{(1\, +\, r)^t}\)

where \(\displaystyle t\, =\ 0\) is the current year and \(\displaystyle r\) is the rate of interest.

i. Compute the present value of a winning lottery ticket that will pay $200,000 per year for 20 years starting in the present year. Assume an interest rate of 12%. Solve the same problem assuming interest rates of 15% and 20%. [Hint: For instance rate of 12%, \(\displaystyle r\, =\, 0.12.\)]

ii. Compute the value of a government bond that pays one dollar every year in perpetuity (i.e., forever) given the interest rate of \(\displaystyle r.\)


[Removed imgur links, my bad]

The answer for 12% is apparently 1,673,155.37

From what I understand about the question is that, every year, the $200 000 drops to a lower value because of the interest rate (12%).
What i have tried was to find the first and last element using the equation given.
Then i tried to put it into the equation needed to find the sum of the geometric series (i also tried arithmetic)

Thank you!!
 

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lemunsips said:
Hello! I am not very sure if i title this correctly or if it under the correct topic but i really need help with this question.


The present value (PV) of a stream of income is \(\displaystyle D_t,\, t\, =\, 0,\, 1,\, ...,\, T\) is defined as

. . . . .\(\displaystyle \displaystyle PV\, =\, \sum_{i=0}^T\, \dfrac{D_t}{(1\, +\, r)^t}\)

where \(\displaystyle t\, =\ 0\) is the current year and \(\displaystyle r\) is the rate of interest.

i. Compute the present value of a winning lottery ticket that will pay $200,000 per year for 20 years starting in the present year. Assume an interest rate of 12%. Solve the same problem assuming interest rates of 15% and 20%. [Hint: For instance rate of 12%, \(\displaystyle r\, =\, 0.12.\)]

ii. Compute the value of a government bond that pays one dollar every year in perpetuity (i.e., forever) given the interest rate of \(\displaystyle r.\)


[Removed imgur links, my bad]

The answer for 12% is apparently 1,673,155.37

From what I understand about the question is that, every year, the $200 000 drops to a lower value because of the interest rate (12%).
What i have tried was to find the first and last element using the equation given.
Then i tried to put it into the equation needed to find the sum of the geometric series (i also tried arithmetic)

Thank you!!
Click to expand...

Please don't link to imgur; due to its ads, it is not acceptable here.

Please show your actual work -- what did you find for those elements, and what did you put into the formula for a geometric series? In fact, what formula did you use? (It is not an arithmetic series, so you can't use that. It is a geometric series, because Dt is a constant in this case.)

I do get their answer, so that is not the problem. Possibly, however, you are using the wrong value for a or r in the formula.
 
Last edited by a moderator:
lemunsips:

As I understand the problem, it is from a (relatively advanced) algebra class, in which no knowledge of finance is assumed, but the topic is geometric series. So although you may gain some insight from reading about annuities, your main focus should probably be on the formula for the geometric series.

But we'll know better when you reply, showing the formula you used and how you used it.
 
Dr.Peterson said:
lemunsips:

As I understand the problem, it is from a (relatively advanced) algebra class, in which no knowledge of finance is assumed, but the topic is geometric series. So although you may gain some insight from reading about annuities, your main focus should probably be on the formula for the geometric series.

But we'll know better when you reply, showing the formula you used and how you used it.
Click to expand...

Hello, thank you. Sorry, ive removed the imgur link.
Here is what i've worked out:
My take on the question is that each year, the price decreases by 12% from 200 000 each year. Then i would have to add the total amount of elements together. I can't seem to work out the answer or maybe I just got the context completely wrong.
78c9dc2f-ee89-4804-a6fc-324ab17daef0.jpg
Thank you!
 
lemunsips said:
Hello, thank you. Sorry, ive removed the imgur link.
Here is what i've worked out:
My take on the question is that each year, the price decreases by 12% from 200 000 each year. Then i would have to add the total amount of elements together. I can't seem to work out the answer or maybe I just got the context completely wrong.
View attachment 9848
Thank you!
Click to expand...

You have not correctly identified "r" in the formula. If "i" is the annual interest rate, then "r" is the annual discount factor, \(\displaystyle r = \dfrac{1}{1+i}\)

It's probably important to point out that if \(\displaystyle D_{t}\) doesn't cooperate, then it isn't a Geometric Progression or Geometric Series. In this case, \(\displaystyle D_{t}\) is constant. That is good. There are other things that would be okay, such as decreasing 4% every year - something that is, by itself, a geometric sequence. However, if it is 200,000, 190,000, 180,000, etc ... decreasing 10,000 each year, it is NOT a Geometric Progression and you'll have to find some other way to add them up. It can certainly be done. It just takes more work.
 
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Are you still unsure of 1st problem?
Are you also asking about 2nd problem?

Next time, please post ONE PROBLEM per thread; thank you.
 
lemunsips said:
Hello, thank you. Sorry, ive removed the imgur link.
Here is what i've worked out:
My take on the question is that each year, the price decreases by 12% from 200 000 each year. Then i would have to add the total amount of elements together. I can't seem to work out the answer or maybe I just got the context completely wrong.
Click to expand...

The problem is that you are applying the formula without thinking about what r means in the formula -- it's not the same as the r in the problem!

The formula is for the sum of \(\displaystyle a r^{i-1}\); to avoid confusion with r being used two ways, let's rewrite it as

\(\displaystyle \displaystyle \sum_{i=1}^N A R^{i-1} = \frac{A(1 - R^N)}{1 - R}\)
​

You want to find

\(\displaystyle \displaystyle \sum_{t=0}^t \frac{D_t}{(1 + r)^t}\)
​

What are the values of A and R here? I think you got everything right except for R.
 
Dr.Peterson said:
The problem is that you are applying the formula without thinking about what r means in the formula -- it's not the same as the r in the problem!

The formula is for the sum of \(\displaystyle a r^{i-1}\); to avoid confusion with r being used two ways, let's rewrite it as
\(\displaystyle \displaystyle \sum_{i=1}^N A R^{i-1} = \frac{A(1 - R^N)}{1 - R}\)
​

You want to find
\(\displaystyle \displaystyle \sum_{t=0}^t \frac{D_t}{(1 + r)^t}\)
​

What are the values of A and R here? I think you got everything right except for R.
Click to expand...

Okay thank you.
It was because I thought that each year I have to x12%, the geometric sequence common ratio was 0.12.
 
lemunsips said:
It was because I thought that each year I have to x12%, the geometric sequence common ratio was 0.12.
Click to expand...
PV of 1st 200000 = 200000/1.12^0
PV of 2nd 200000 = 200000/1.12^1
PV of 3rd 200000 = 200000/1.12^2
....and so on
 
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