part a

   You are expecting a tax refund of $5,000 in 10 weeks. A tax preparer offers you an "interest-free" loan of $5,000 for a fee of $50 to be repaid by your refund check when it arrives in 10 weeks. Thinking of the fee as interest, what weekly simple interest rate would you be paying on this loan?

 

weekly simple interest rate ? %

annual simple interest rate ? % 

 

part b

   Meg's pension plan is an annuity with a guaranteed return of 7% per year (compounded quarterly). She would like to retire with a pension of $50,000 per quarter for 10 years. If she works 30 years before retiring, how much money must she and her employer deposit each quarter? (Round your answer to the nearest cent.)

 

$ ?

Please check explanation

Step-by-step explanation

Part A

Given,

Let the weekly simple interest rate be R_w and annual simple interest rate be R

Fees is taken as interest, So Interest, I = $50

Principal amount, P = $5000

Time duration , T = 10 weeks

We know, I = PRT

First lets calculate the weekly interest rate,

I = PR_wT

50 = 5000 x R_w x 10

?Rw?=5000×1050?? = 0.001

= 0.001 x 100% = 0.1 %

To find annual simple interest rate,

I =PRT

50 = 5000 x R x ?5210?? [There are almost 52 weeks in a years, so we provide time period of 10 weeks in terms of years as 10/52]

R = ?5000×1050×52?? = 0.052

= 0.052 x 100% = 5.2%

Part B

Total money that she has to deposit so that she receives $50,000 per quarter for 10 years after she retires, can be calculated as:

PMT =  ?PV1−(1+i)−ni?? ; PV = ?PMTi1−(1+i)−n??

Where,

PMT = Each payment to be made/received = $ 50,000

PV = Present value or principal = To be found

i =  interest rate per period = r/m [r= annual interest rate = 7%; m number of compounds in a year = 4 (4 quarters)]

So i = r/m = 0.07/4 = 0.0175

n = mt is the number of compounding periods; Here t = 10 years; So n = 4 x 10 = 40 periods

Hence, PV = ?50000×0.01751−(1+0.0175)−40?? = 50000 x 28.594 = $1,429,700

This amount is build by the money she and her employer deposit each quarter, for a period of 30 years when she is working. So each deposit she has to make can be found using the same formula as:

PMT =  ?PV1−(1+i)−ni??

Here too, i = 0.0175

PV = $1429700

n = mt, Here t = 30 years. So n = 4 x 30 = 120 periods [The deposit too is compounded quarterly, so m = 4]

So, PMT = 1429700 x ?1−(1+0.0175)−1200.0175?? = 1429700 x 0.01999 = $28584.23157

So quarterly deposits to be made = $28584.23