Lottery with two options for the prize HELP

[aisha]

A lottery offer two options for the prize.

Option A $1000 a week for life
Option B $600 000 in one lump sum

The current expected rate of return for large investments is 7%/a compounded weekly

a) which option would the winner choose if he expects to live for another 25 years? For this question I used the PV formula with option A and subbed the values R=1000, n=52*25=1300 and i=0.001346 into the formula and compared my answer $613 660.24 with option B $600 000.00 So the person should choose option A?

b)At what point in time is option A better than Option B? I don't know what to do ? HELP Option A is already better than B in the previous question...im so confused

 

[Caper]

The lump sum prize of $600,00 could also be invested, making it the better paying option.

 

[shmoe]

The lump sum prize of $600,00 could also be invested, making it the better paying option.

Not if you plan to live 25 years it's not (according to aisha's work). The present value formula aisha is referring to is taking the possibility of investment into account.


For part b) I'm not sure exactly what they are asking. My most reasonable interpretation is "Say you plan to snuff it in x years. For what values of x is option A better than option B?" Probably best to think of x in terms of weeks instead of years though. Try to find the value of x that will make the payment plans equal and go from there.

 

[Caper]

The value aisha calculated, $613,660, is how much you would get from option 1, which is $1000 per week for life, and you live for 25 years.

The second option is $600,000 immediately. You could invest that, at 7% interest and get far more than $613,660. When aisha did the calculations they only compared the $613,660 to the $600,000.

Now tell me again where I went wrong?

After you get that, it should be obvious how to get part b.

 

[aisha]

now I am totally confused I only invested Option A I think you guys are trying to say that I should have invested Option B too and then compared them um do I still calculate using the present value formula or future value? How do I answer the second question what do I have to do?

 

[Caper]

Oh i see what shmoe was saying, maybe instead of calculating the PV of option A you should calculate FV of both after 25 years? it's been a few years since I took economics and I got rid of the textbook or I'd plug the numbers in and see if it gives a different answer.

If option A is indeed better, maybe for part b they are just asking like what shmoe said, find the point in time when the two options are equal, then everything after that time it is better to choose option A. Whether it's at 23 years or 24 years etc.

 

[aisha]

How do u figure out when the two are equal?

 

[Andrew Mason]

How do u figure out when the two are equal?

The two are equal for a certain payment period. That payment period is a little less than 25 years. Just set the PV = $600000 for the $1000 per week and work out what the period has to be.

As far as your decision which to take, here is my advice:

1. Take the $1000k a week,
2. Take out a life insured loan for $600k at 7 percent interest or less and pay it off in 25 years or less with the $1000 per week. You then live on/spend/invest/have fun with/do good things with/ the borrowed $600k.

If you live longer than 25 years, you still have the $1000 /week coming in until you die. If you don't, you still get the $600k from the insurance (ok, you will have to use some of the payments to pay the life insurance premiums but you make that back in a few months).

AM

 

[tony873004]

Everything said so far is irrelavant unless you consult a tax attorney. There might be good reasons to spread your income out over several years.

 

[ShawnD]

Option A $1000 a week for life
Option B $600 000 in one lump sum

The money from option B is easy to calculate

bling = (600000)(1 + \frac{0.07}{52})^{(52*25)}

bling = $3,448,700


Option A is incredibly difficult to calculate. The first term would be like this

bling = (1000)(1 + \frac{0.07}{52})

Say that equals X, then the second term would be

bling = (1000 + X)(1 + \frac{0.07}{52})

It will keep going like that every week for 25 years, that's like 1300 terms. You could probably write a computer program to do it, but it would still probably take 30 minutes to crunch that.

 

[shmoe]

Oh i see what shmoe was saying, maybe instead of calculating the PV of option A you should calculate FV of both after 25 years?

This will give the same relative result (both dollar values will be higher though). The whole present value of the annuity thingie for option A takes the 7% value of money into account. For example the 2nd $1000 payment in option A contributes only 1000*(1+.07/52)^(-1)<1000 to the present value total of $613,660. Another way to think of it, if you invested $613,660 today at 7%, then removed $1000 per week, you'd have no money left after 25 years. If you invested only $600,000 today and took out $1000 per week, the well would run dry at an earlier time. If you want to check this on your own, you don't need an econmoics text, you just have to be able to sum a geometric series.


aisha, for the second part, assume your payments will last x weeks (for option A). Find the present value of option A in terms of x. Find out what value of x will make this present value equal to 600,000.


ps. aisha is in Canada so I believe there is no tax on the lottery winnings. Unless it's from an American textbook.

 

[aisha]

Ive read through all your posts I think some of you may be suggesting I do what my teacher said. I emailed her and this is what she said

a) First, calculate FV of the option A. R = 1000,
n = 25 x 52, i = 0.07/52
Then find the FV of the PV = 600 000 (this is just one payment, so dont
need to use the annuity formula). n = 25 x 52, i = 0.07/52.

Then, decide which one is better.

b) without substituting the value of n, set FV of option A = FV of option
B. Then solve for n. That is when both options are the same. Then, answer
the question..I can't give the answer.


Will someone please help me set the question up so i can solve

 

[shmoe]

a) First, calculate FV of the option A.

I say again- there is no difference to the question if you find the future or present value (you should be able to explain why this is though). What you did is fine, though you should have avoided rounding your value for i. It won't hurt to try it again for FV to see the result.

b) without substituting the value of n, set FV of option A = FV of option
B. Then solve for n. That is when both options are the same. Then, answer
the question..I can't give the answer.

Sure you can, I have faith. You were able to find the PV when n=1300. Write down the formula for a general value of n and report back on your efforts to solve for n. Again, you can do this in terms of PV or FV and the result will be the same. The PV way will be slightly cleaner though. (you might want to set up the equations for both to see the difference yourself)

 

[aisha]

a) First, calculate FV of the option A. R = 1000,
n = 25 x 52, i = 0.07/52
Then find the FV of the PV = 600 000 (this is just one payment, so dont
need to use the annuity formula). n = 25 x 52, i = 0.07/52.

Then, decide which one is better.

b) without substituting the value of n, set FV of option A = FV of option
B. Then solve for n. That is when both options are the same. Then, answer
the question..I can't give the answer.

According to what my teacher said I found the Future value of Option A using the FV formula of an annuity to be $3,526,962.78 and She said that I didnt have to use the annuity formula for Option B so I used A=P(1+i)^n and got $34 847.01 If these values are correct then Option A is definately a better choice BUT I THINK I DID SOMETHING WRONG HELP WHAT DO U THINK? I CANT CONTINUE TO PART B UNLESS PART A OF THIS QUESTION IS CORRECT HELP PLZ!

 

[shmoe]

For the FV of option B, you should be using the same values for i and n as you did with option A. The answers should be reasonably close.

An added exercise for you to try. Take your original value for the PV of option A (the 613,660 number) and adjust it forward 25 years using the same equations you used to find the FV of option B (just the usual compound interest formula). Your answer should look familiar.

 

[aisha]

My mistake opps I made option B principal only 6000 ahh lol K now these are the FV they look better now I hope these are right

Option A FV=$3,526,962.78
Option B FV=$3,448,700.70

WHAT DO U THINK?

If these are correct can u please help me set up the question so that I can solve the next part

b) without substituting the value of n, set FV of option A = FV of option
B. Then solve for n. That is when both options are the same. Then, answer
the question.

Where is n in my FV ? I DONT GET IT please set it up so I can solve for n.

 

[shmoe]

Option A FV=$3,526,962.78
Option B FV=$3,448,700.70

WHAT DO U THINK?

Much more reasonable. Did you think about my 'added exercise'? Essentially you should notice 613,660*(1+i)^n=3,526,962, give or take some rounding errors.

Where is n in my FV ? I DONT GET IT please set it up so I can solve for n.

n is the number of payment periods. Part a) was the special case n=1300. Go back to the equation before you put 1300 in. I'm not going to write this out for you, you know what this equation is.

Don't worry, this is the last time I'll say this-life will be simpler if you look at present values here instead.

 

[aisha]

OK here is what I think the formula is

1000((1+.07/52)^(n)-1) = 600 000 (1+0.07/52)^n
-----------------------
0.07/52

Now if this correct someone please help me solve pleasezzzzzz.

 

[shmoe]

That looks good. Your first step is to isolate the (1+.07/52)^n. If this expression looks frightening, replace it with something more friendly, say (1+.07/52)^n= , and your equation becomes (after I've done a step for you):

1000*(-1)=(0.07/52)*600000*

Now you should be able to solve for easily. Once you've done this, replace it with the spooky (1+.07/52)^n and I'll tell you that logarithms will be involved to isolate n after this point.

(yes I'm getting a touch loopyfrom sleep deprevation, someone else will hopefully be able to assist you this evening as I've gone to :zzz:)

 

[aisha]

BUT I DONT KNOW HOW TO USE LOG
ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

 

[aisha]

Great I have done what was said in the last post but am left hanging without a conclusion because I don't know how to use log ahhh

well someone can prob help me I have solved for x and got x=5.199

1.001346^n=5.199 I think this is the equation I have to solve after I have found out what x is can someone help me finish this problem please!

 

[shmoe]

Great! Next you want to get that n out of the exponent. Try taking the log of both sides.

ps. You'll need to remember that \log{a^b}=b\log{a}.

 

[aisha]

1.001346^n=5.199

Ok i emailed my teacher and she showed me a log example from that I did this

log(1+0.07/52)^n= log(1000/192.31) I've put both sides in fractions to get a more accurate answer


0.0005842340426=0.715998132

n=0.715998132/0.0005842340426
n=1226

The original question was at what point in time is Option A better than Option B... I am not sure what to write what does that n mean? Um Option A is better than option B when ... someone help me out please.. OH and is n even correct?

 

[shmoe]

Well you've done some rounding.There probably won't be an integer value for n where the two options are equal, but you should be able to find out what week option A gets better than option B. Compare the two options when n=1225 and n=1226, hopefully this is the point they switch over. It should be clear that if option A is better after at some point in time, then it's better from then on.