Question

Aunt Mabel promised to give you $9000 when you successfully complete your freshman year; $3000 when you successfully complete your sophomore year; $5000 when you successfully complete your junior year; and $1000 when you successfully complete your senior year. Aunt Mabel made this promise when you graduated from high school and let's assume you go directly to the U of I and graduate in four years. Aunt Mabel expects the interest rates to be 2.25% during your freshman year, 6.25% during your sophomore year, 7.25% during your junior year, and 9.25% during your senior year. If Aunt Mabel can predict interest rates with accuracy, she will deposit to her bank account (1) __ and after your freshman year her bank balance will be (2) __ and after your sophomore year her bank balance will be (3) __ and after your junior year her bank balance will be (4) __.

Answer 1

Answer:

(1) 16,640.15

(2) 8,014.55

(3) 5,515.46

(4)    915.33

Explanation:

We will construct this backwards using the present vaue of a lump sum:

$\frac{Nominal}{(1 + rate)^{time} } = PV$  

at the end, once you graduate the account will have zero balance.

after junior year it will have the discount value of the 1,000 once senior is complete

so:

Nominal 1,000.00

time   1.00

rate  0.0925

$\frac{1000}{(1 + 0.0925)^{1} } = PV$  

PV   915.33

Then, on sophomore year, you will got this and the amount it gives you at junior discounted for one year:

915.33 + 5,000 = 5915.33

rate  0.0725

$\frac{5915.33}{(1 + 0.0725)^{1} } = PV$  

PV   5,515.46

Then, after the freshman year we discount this and the amount given and sophomore

3,000 + 5,515.46 = 8,515.46

rate  0.0625

$\frac{8515.46}{(1 + 0.0625)^{1} } = PV$  

PV   8,014.55

Lastly, at beginning it will need this and the freshman bonus discounted:

8,014.55 + 9,000 = 17,014.55

rate  0.0225

$\frac{17014.55}{(1 + 0.0225)^{1} } = PV$  

PV   16,640.15