Question

(Related to Checkpoint​ 6.1) ​ (Annuity payments) Mr. Bill S.​ Preston, Esq., purchased a new house for ​$80 comma 000. He paid ​$20 comma 000 upfront and agreed to pay the rest over the next 25 years in 25 equal annual payments that include principal payments plus 9 percent compound interest on the unpaid balance. What will these equal payments​ be?

a.  Mr. Bill S.​ Preston, Esq., purchased a new house for ​$80 comma 000 and paid ​$20 comma 000 upfront. How much does he need to borrow to purchase the​ house? ​$ nothing  ​(Round to the nearest​ dollar.)

b.  If Bill agrees to pay the loan over the next 25 years in 25 equal​ end-of-year payments plus 9 percent compound interest on the unpaid​ balance, what will these equal payments​ be? ​$ nothing ​ (Round to the nearest​ cent.)

Answer 1

(a) Value of the house = $80000

Upfront Payment = $20000

Hence, amount to be borrowed = P = 80000 - 20000 = $60000

(b) Rate of interest = r = 9% or 0.09

Number of payment periods = N = 25 years

Loan Amount = P = $60000

Let the annual end of year payments be X

Hence, X/(1+r) + X/(1+r)² +....+ X/(1+r)²⁵ = Loan Amount (P)

=> X[1- (1+r)^-25]/r = 60000

=> X[1- (1+0.09)^-25]/0.09 = 60000

=> X = $6108.38

Hence, Annual Payments = $6108.38

This can also be calculated using the formula, Annual Payment =  rX(1+r)^N/[(1+r)^N-1]

Hence, Annual Payments = rX(1+r)^N/[(1+r)^N-1] =  (0.09)*60000*(1+0.09)²⁵/[(1+0.09)²⁵-1] = $6108.38