Question

Albert and Allison borrow exactly the same amount from Liberty Financial. Albert will repay his loan with 20 end of year annual payments. Albert's first payment will be 950, and each of his successive payments will be 950 greater than the one before. Allison will make level payments of 50000 at times T; 2T; 3T; 4T. Both Albert and Allison's loans are subject to the same annual effective rate of 5%. Determine the time of Allison's first payment.

(a) 2.7-2.9 years

(b) 3-3.2 years

(c) 3.5-3.7 years

(d) 5-5.2 years

(e) 5.5-5.7 years

Answer 1

Firstly, we need to calculate how much Albert paid over the period of 20 years:

Applying the formula for arithmetic progression:

Formula :

Sum of Arithmetic Progression = (n/2)*[2a + {(n-1)*d}]

where; n = number of payments

a = first payment amount

d = difference between first payment and second payment

Putting the values in place:

(20/2)*[2*950 + {(20-1)*950}]

=> 10*(1900+(19*950)

=> 10*(1900+18050)

=> 10*19950

=> 199,500 ...........................................................................(i)

Therefore, Albert paid 199,500 totally over a period of 20 years.

We will attempt to figure out how much Albert borrowed with the above information:

Applying Interest Formula for Principal borrowed:

P= A/[1+(r*t)]

where; P = Principal borrowed

A = Total Payment made = 199500 (from (i))

r = Annual effective rate = 5% (from the question)

t = Total years (from the question)

Putting values in the formula:

P = 199500/[1+(0.05)*20]

=> P = 199500/[1+1]

=> P = 199500/2

=> P = 99,750

Now, since the problem states, that Allison made 4 equal payments of 50,000 each, therefore, Allison made payments ammounting to 50,000*4 = 200,000

Now since we know that Allison and Albert borrowed the same amount (i.e. 99,750), we need to find out how many years Allison took to make all the payments.

Applying interest formula to find out time period for Allison:

t = [(A/P) - 1]/r

where; t = time in years

A = total amount paid

P = Principal borrowed

r = Annual effective rate

t = [(200000/99750)-1]/0.05

=> t = [2.005-1]/0.05

=> t = 1.005/0.05

=> t = 20.1 years

However, in the problem, it is mentioned that Allison made 4 payments of t, 2t, 3t and 4t

Since from our above calculations it is established that: 4t = 20.1

Therefore, t = 20.1/4

t = 5.025 years

Hence, in conclusion, Allison made the first payment between 5-5.2 years, which is option (D)