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Math Help - Economics homework help

  1. #1
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    Economics homework help

    Hello all,
    Im having some difficulty with the following problems. Can someone please guide me on how to solve them?

    1) Suppose Sean invests P dollars today in a new company. In return, the company will pay him 185.65 dollars per year, starting with year 19 and ending with year 45 . In addition, in the last year (i.e., year 45) the company will pay Sean a lump-sum (i.e., a single) amount equal to 804.29 dollars. What is the P amount such that Sean achieves an annual rate of return equal to 185.65/804.29 per year ?


    For this problem i think the part im stuck on is how to handle the rate or return. I know how to manage the shifted annuity.




    2) Suppose interest is 17.13% per year, compounded daily. Assume there are 364 days in one year.
    Find the present value of the uniform series given by
    At=236.28, t=1,...,50
    given the payment period length is "one week" , that is, 1/52 of a a year.



    For this one im confused by the first sentence, how can interest be 17.13% per year compounded daily?





    3) Pat is an excelent tennis player. Her friend Mat estimated that if she becomes a professional player today, her income will be 9 dollars in year 1 and then will continue to increase by 25.78% each year until year 8. According to Mat's estimates, Pat will be at her peak in year 8. After that point in time, her income in each period will be only 1/7 of her income in the previous period. Suppose Mat's predictions are correct. If Pat plans to retire at the end of year 14 and to save 1/7 of her income each year in an account paying interest at 21% per year, how much will she accumulate by the end of 14?

    For this one i have tried to sole several times, id like it if someone could provide the steps, i cant seem to do this one correctly and have difficult with finding the 1/7th of the income. do i have to do this for each year or can i find the present value of the series and multiply by 1/7?
    In short this one will need to be spelled out for me im new at these problems.



    Thank you for your help! I hope i didnt break any rules by posting several questions in on thread, i just didnt want to clutter up the forum.
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  2. #2
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    May I ask you a question: did your teacher write up these problems after drinking
    a 40-ouncer of tequila straight ?
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  3. #3
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    Quote Originally Posted by testing12 View Post
    1) Suppose Sean invests P dollars today in a new company. In return, the company will pay him 185.65 dollars per year, starting with year 19 and ending with year 45 . In addition, in the last year (i.e., year 45) the company will pay Sean a lump-sum (i.e., a single) amount equal to 804.29 dollars. What is the P amount such that Sean achieves an annual rate of return equal to 185.65/804.29 per year ?


    For this problem i think the part im stuck on is how to handle the rate or return. I know how to manage the shifted annuity.
    I suggest:

    rate of return (over the whole period of investment)=total return/total investment, where investment is P
    annual rate of return = total rate of return as above/'relevant' number of years

    I think here the 'relevant' number of years is from the initial investment to the last payout, ie 45 (or may be even 46, if he invests in the beginnign of Year1, and the last payout is in the end of Year 45)

    So once you add up all incomes (annual and the last payout), substitute it into the equations above and solve for P.
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  4. #4
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    Quote Originally Posted by testing12 View Post
    3) Pat is an excelent tennis player. Her friend Mat estimated that if she becomes a professional player today, her income will be 9 dollars in year 1 and then will continue to increase by 25.78% each year until year 8. According to Mat's estimates, Pat will be at her peak in year 8. After that point in time, her income in each period will be only 1/7 of her income in the previous period. Suppose Mat's predictions are correct. If Pat plans to retire at the end of year 14 and to save 1/7 of her income each year in an account paying interest at 21% per year, how much will she accumulate by the end of 14?

    For this one i have tried to sole several times, id like it if someone could provide the steps, i cant seem to do this one correctly and have difficult with finding the 1/7th of the income. do i have to do this for each year or can i find the present value of the series and multiply by 1/7?
    In short this one will need to be spelled out for me im new at these problems.
    I would consider two periods separately: year 1 to 8, and year 9 to 14. Each period has its own income pattern, and the year 8 income is used in calculating the Year 9 income. Once you know her income each year/period, figure out how much she saves each year, then compound at the given interest rate. No need to mention that say income from year 10 gets compounded for 4 years and income from year 11 gets compounded for 3 years etc etc.
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  5. #5
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    Quote Originally Posted by testing12 View Post
    1) Suppose Sean invests P dollars today in a new company. In return, the company will pay him 185.65 dollars per year, starting with year 19 and ending with year 45 . In addition, in the last year (i.e., year 45) the company will pay Sean a lump-sum (i.e., a single) amount equal to 804.29 dollars. What is the P amount such that Sean achieves an annual rate of return equal to 185.65/804.29 per year ?
    > an annual rate of return equal to 185.65/804.29 per year ?
    That "seems" to represent an annual rate of ~23.08%, since 185.65/804.29 = .2308....

    The problem "seems" to be equivalent to:
    A company borrows $P from Sean at an annual rate of 23.08%, repayment terms
    being $185.65 annually from year 19 to 45, at which point a final (balloon) payment
    of $804.29 will be due. What is P ?

    Well, .07 (7 cents) will accumulate to ~$800 at that rate, over 45 years;
    hence my confusion. Plus whole thing is made worse with the annual payments...
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  6. #6
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    You are right, it's more twised than I though initially. I would expect to see some discount rate, as a start. Not sure if annual rate of return can be approximated for the discount rate. And 7 cents will accumulate to that much only if any annual income from this P is reinvested down to the last cent, and I don't see anything in the question about him doing that.

    The model looks like purchasing a life insurance policy.
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  7. #7
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    Quote Originally Posted by testing12 View Post
    2) Suppose interest is 17.13% per year, compounded daily. Assume there are 364 days in one year.
    Find the present value of the uniform series given by
    At=236.28, t=1,...,50
    given the payment period length is "one week" , that is, 1/52 of a a year.
    For this one im confused by the first sentence, how can interest be 17.13% per year compounded daily?
    17.13% per year compounded daily (364day basis) means effective annual rate of:
    (1 + .1713/364)^364 - 1 = .186798... hence ~18.68%

    The rate compounded weekly to achieve this effective rate is:
    (1 + r)^52 = 1.186798...
    r = .003298..., hence annual of .003298 * 52 = .171542... or ~17.15%

    Hope that helps...disregard if I've "missed the point"!
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  8. #8
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    the way of approach of wilmer is just perfect. you can go with it and considering it as 2 periods is the easiest way.
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