Question

A ​$92,000 mortgage is to be repaid over a ten​-year period by monthly payments rounded up to the​ next-higher $100. Interest is 4.6% compounded semi-annually.

​(a)	Determine the number of rounded payments required to repay the mortgage.
​(b)	Determine the size of the last payment.
​(c)	Calculate the amount of interest saved by rounding the payment up to the next higher​ $100 versus rounding the payment to the nearest cent.

Answer 1

Annual interest rate of 4.6% compounded semi annually is equivalent to:

Monthly rate of 0.379711% and

Yearly rate (monthly compounded) 4.556526%

Calculation as follows:

8 с 1 L =((1+C5)^(1/C8))-1 A B 1 2 Yearly Nominal Rate (R) Given 3 Compounding frequency Given 4 Times compounded a year (t) 5 EAR=((1+R/t)^t)-1 ((1+C2/C4)^C4)-1 6 7 Frequency 8 Times compounded a year (t) 9 Monthly rate ((1+C5)^1/C8)-1 10 Yearly rate (Monthly compounding) C9*C8 4.6% Semi annually 2 4.6529% Monthly 12 0.379711% 4.556526%

Part (a): Number of rounded payments required to pay off the loan= 113

Part (b):

Size of the last payment= Balance at the 113^th payment plus interest for last month

= $394.22 + $1.50 = $395.71

Relevant portion of amortization schedule below:

к L M N Amortization Schedule Month Beginning Monthly Monthly Interest Principal End balance interest Payments component Component balance rate =Col (2)*(3) =Col (4)-(5) =Col (2)-(6) (1) (2) (3) (4) (5) (6) (7) 1 92000 0.003797 1000.00 349.33 650.67 91349.33 2 91349.33 0.003797 1000.00 346.86 653.14 90696.20 111 112 113 3367.12 0.00379711 2379.91 0.00379711 1388.94 0.00379711 394.22 0.00379711 1000.00 1000.00 1000.00 395.71 12.79 9.04 5.27 1.50 987.21 990.96 994.73 394.22 2379.91 1388.94 394.22 0.00 114

Part (c ): Interest saved by rounding to next higher $100= $1,322.15

Details of calculation as follow:

G A B С D E F 1 Monthly payments of fixed rate loan. Payments at the end of each month 2 Monthly payments (PMT) is calculated using the formula PMT=[P*r*(1+r)^n]/[(1+r)^n)-1 3 Where P= Principal (loan amount), 4 n= Number of installments. ie if N= number of years, n=N*12 5 r= Rate of interest per month in decimals ie. If R= Yearly rate, r=R/(100*12) 6 All amounts in $ 7 Given that 8 Principal (loan amount) (P)= 92,000 No. of years (N)= 10 9 Rate of interest (% p.a)= R= 4.556526 No. of months (n)= 120 10 Calculations: Formula Cell reference Value 11 Interest rate per month in decimals (r )= R/(100*12) D9/(100*12) 0.003797 12 No. of months (n)= N*12 G8*12 120 13 (1+r)= 1+G11 1.003797 14 (1+r)^n= G13469 1.575842 15 [(1+r)^n]-1= G14-1 0.575842 16 P*r*(1+r)^n= D9*G14*G17 550.494701 17 Monthly payments (PMT)= [P*r*(1+r)^n]/[(1+r)^n)-1 G16/G15 955.9821805 18 Rounded to $ 955.98 19 Total payment if rounded to nearest cent Monthly payment*120 G18*69 $ 114,717.86 20 Interest if rounded to nearest cent Total-Principal G19-68 $ 22,717.86

A B с D E F G 21 22 Monthly payment if rounded to next higher $100= $1,000 23 Number of full monthly payments 113 24 Last month payment (as per amortization schedule)= $395.71 25 Total payment if rounded to next higher $100= $1000*113+$395.71 = 622*G23+G24 $113,395.71 26 Total interest if rounded to next higher $100= Total-Loan G25-D8 $21,395.71 27 28 Interest saved by rounding to next higher $100= G20-626 $ 1,322.15