Question

Geoffrey decides not to buy the car mentioned earlier. Instead, he is now considering a food delivery service "You, bars, meats" that his friend Gillian has recently started. Gillian has agreed that for a single payment of $78,000 today to help her launch her business, she will provide all the delivery services that Geoffrey needs for his business for the next 5 years. Geoffrey is considering borrowing the full amount from his business account.

Suppose that Geoffrey makes level quarterly repayments over the coming 5 years, the first payment being exactly 3 months from today. Again, the interest rate on Geoffrey's account is 4.2% p.a. effective.

a) Calculate the size of the level quarterly repayment.

b) How much money does Geoffrey owe on this loan after 1 year?

(c) How much interest does Geoffrey pay in the first year?

(d) Geoffrey believes that the overall benefit from this agreement amounts to $369.52448601134 per week in arrears (this would include money he would have spent on alternative delivery services, estimated additional profits from using Gillian's services, etc).

By considering only the initial cost of $78,000 and this weekly benefit of $369.52448601134, calculate the interest rate that represents the return on this investment, expressed as a nominal annual rate compounding weekly

Gillian has entered the agreement with Geoffrey described above. She estimates that the costs of the delivery services she has promised to Geoffrey (petrol, insurance, wear and tear, etc) amount to $1557.7922862927 per month in advance for the coming 5 years.

(a) If Gillian can borrow/invest money at a rate of 3.7% p.a. effective, what is the equivalent amount today of her future liabilities? Note that this calculation should not involve the payment she receives from Geoffrey today.

(b) The money she receives from Geoffrey can be considered a loan, with repayments being the value of the services she provides in return. What is the
interest rate, expressed as an effective annual rate, she is being charged on this "loan"?

Answer 1

A.

Loan amount (P)= 78000

Number of quarterly payments in 5 years (n)= 5*4= 20

Interest rate on loan is 4.2% effective.

It means effective annual rate is 4.2%

Effective Annual rate formula = ((1+quarterly rate)^number of quarter in year))-1

4.2% = ((1+I)^4)-1

1+0.042=( 1+I)^4

(1.042)^(1/4)= 1+i

1.010338563-1= i

I or Quarterly rate = 0.01033856326

Equal level Payment formula = P* i *((1+i)^n)/((1+i)^n-1)

=78000*0.01033856326*((1+0.01033856326)^20)/(((1+0.01033856326)^20)-1)

=4337.143695

So size of level equal Quarterly Payment is $4337.14

B.

Number of quarter left After 1 Year (n)= 4*4=16

Loan value Balance at n time Formula = Periodical Payment*(1-(1/(1+i)^n))/i

=4337.143695*(1-(1/(1+0.01033856326)^16))/00.01033856326

=63656.5271

So after 1 Year Amount owed on loan is $63656.53

C.

Principal repaid = loan balance at Beginning - loan balance at year end

=78000-63656.53

=14343.47

Interest paid formula = (number of payment in year * quarterly payment)- principal repaid

=(4*4337.14)-14343.47

=3005.09

So interest paid in first year is $3005.09

D.

Weekly income or receipt 369.52448601134

Investment = 78000

Rate of Return formula = weekly benefits or Income/Investment

=369.52448601134/74000

=0.004993574135

Annual rate of Return = weekly rate * Number of weeks in Year

=0.004993574135*52

=0.259665855 or 25.97%

So Nominal Annual rate of return earned on investment is 25.97%