Annuities Practice Problems

Prepared by Pamela Peterson Drake

  1. Congrats! You just won the $64 million Florida lottery. Now the Surely Company is offering you $30 million in exchange for the 20 installments on your winnings. If your opportunity cost of funds is 8%, should you agree to this deal?

    2. Given:

    CF = $64,000,000 / 20 = $3,200,000

    N = 20

    i = 8%

    Annuity due

    PV = $33,931,517.44

    No: the annuity is worth almost $34 million to you, but Surely is offering only $30.

  2. Carol Calc plans on retiring on her 60th birthday. She wants to put the same amount of funds aside each year for the next twenty years -- starting next year -- so that she will be able to withdraw $50,000 per year for twenty years once she retires, with the first withdrawal on her 61st birthday. Carol is 20 years old today. How much must she set aside each year for her retirement if she can earn 10% on her funds?

    3. PV60 = $50,000 (PV annuity factor for N=20, i=10%)

    PV60 = $50,000 (8.5136)

    PV60 = $425,678.19

    Because she will stop making payments on her 40th birthday (first is on her 21st birthday, last is on her 40th birthday), we must calculate the balance in the account on her 40th birthday:

    PV40 = PV60 / (1 + 0.10)20 = $63,274.35

    Then, we need to calculate the deposits necessary to reach the goal:

    FV40 = PV40 = $63,274.35

    N = 20

    i = 10%

    FV = CF (FV annuity factor for N=20, i=10%)

    $63,274.35 = CF (FV annuity factor for N=20, i=10%)

    $63,274.35 = CF (57.2750)

    CF =payment = $1,104.75 per year

  3. Have I got a deal for you! If you lend me $100,000 today, I promise to pay you back in twenty-five annual installments of $5,000, starting five years from today (that is, my first payment to you is five years from today). You can earn 6% on your investments. Will you lend me the money?

    4. This is a deferred annuity problem

    CF = $5,000

    N = 25

    i = 6%

    PV4 = $5,000 (PV annuity factor for N=25 and i=6%)

    PV4 = $5,000 (12.7834)

    PV4 = $63,916.78

    PV0 = $63,916.78 / (1 + 0.06)4 = $50,628.08

    You probably shouldn't lend the money under these terms. If you lend me $100,000, I am repaying you using terms such that the value of my repayment is $50,628.08.

  4. You have choice when subscribing to our magazine: you can

    1. pay $100 now for a four year subscription,

    2. pay $28 at the beginning of each year for four years, or

    3. pay $54 today and $54 again two years from today.

    Which is the best deal for you, the subscriber, if your opportunity cost of funds is 10%?

    (a): PV = $100

    (b) PV = PV of a 4-payment annuity due = $97.63

    (c) PV = $54 + $54 / (1+0.10)2 = $54 + 44.63 = $98.63

    The best deal is to pay $28 at the beginning of each of the four years.

  5. The Trust Worthy loan company is willing to lend you $10,000 today if you promise to repay the loan in six monthly payments of $2,000 each, beginning today. What is the effective annual interest rate on Trust Worthy's loan terms?

Use the present value of an annuity due to approach this problem (because the first payment is today).

PV = $10,000

CF = $2,000

N = 6

PV annuity due = CF (PV annuity factor for N=6, i=?)(1 + i)

$10,000 = $2,000 (PV annuity factor for N=6, i=?)(1 + i)

5 = (PV annuity factor for N=6, i=?)(1 + i)

Through trial error using the tables for N=6 such that the factor multiplied by 1+ i is equal to 5,

i = 8%

precise answer for i= 7.9308%

EAR = (1 + 0.079308)12 - 1 = 149.89%

Want an easier way to do this problem? OK, if TW lends you $10,000 and you repay $2,000 immediatly, you are really only borrowing $10,000 - 2,000 = $8,000. Therefore, you can use the ordinary annuity approach, modifying the PV and N:

PV = $8,000

CF = $2,000

N = 5

Solve for i for an ordinary annuity:

PV = CF (PV annuity factor for N=5, i = ?)

$8,000 = $2,000 (PV annuity factor for N=5, i = ?)

4.000 = PV annuity factor

Using the tables, i = 8% (factor is 3.9927)

Using a calculator, i = 7.9308%