Question 1196141
Please help with the homework:
Liza purchases an apartment by paying a deposit of R60000, and obtains a 20 year loan for the balance of R120000 at 20% interest rate per annum compounded monthly. After four and half years, the bank adjusts the interest rate to 18% per annum compound monthly. What is the new amount that he must pay if the term of the loan remains the same?
<pre><font color = red><font size = 4><b><u>STEP 1:</font></font></b></u> <font color = blue><font size = 4><b>Determine the monthly payment.</font></font></b> 
<font color = green><font size = 4><b>        However, instead of using an online mortgage calculator, I believe it'd be better to find this 
        using the following formula:</font></font></b>
          {{{highlight(highlight_green(highlight_green(highlight(matrix(1,3, PMT, "=", (PV[oa] * (i/m))/(1 - (1 + i/m)^(- mt)))))))}}}, where: {{{matrix(5,1, matrix(1,8, PMT, "=", Monthly, PAYMENT, "(UNKNOWN,", in, this, "case)"), matrix(1,11, PV[oa], "=", Original, LOAN, "amount/", "Principal/", "Present_Value", "(120,000,", in, this, "case)"), matrix(1,12, i, "=", INTEREST, rate, per, annum, "(20%,", or, .2, in, this, "case)"), matrix(1,14, m, "=", Number, of, COMPOUNDING, periods, per, year, "(monthly,", or, 12, in, this,  "case)"), matrix(1,9, t, "=", Time, in, years, "(20,", in, this, "case)"))}}}
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<font color = red><font size = 4><b><u>STEP 2:</font></font></b></u> <font color = blue><font size = 4><b>Calculate the PRESENT VALUE of an ANNUITY, for 15.5 years, or 15.5(12) = 186 FUTURE monthly payments 
        (calculated in STEP 1), and at the same interest rate (.2, or 20%).
      **Note that 4.5 years' (54 monthly) payments were already made.</font></font></b> 
<font color = green><font size = 4><b>        Again, instead of using an online  calculator, I believe it'd be better to find this 
        using the following formula:</font></font></b>
          {{{highlight(highlight_green(highlight_green(highlight(matrix(1,3, 

PV[oa], "=", PMT * ((1-1/(1+i/m)^(mt))/(i/m))
)))))}}}, where: {{{matrix(5,1, matrix(1,11, PV[oa], "=", "PV/LOAN_BALANCE", after, 54, monthly, payments, "(UNKNOWN,", in, this, "case)"),

matrix(1,10, 
PMT, "=", Monthly, PAYMENT, "(From", STEP, "1,", in, this, "case)"), matrix(1,12, i, "=", INTEREST, rate, per, annum, "(20%,", or, .2, in, this, "case)"), matrix(1,14, m, "=", Number, of, COMPOUNDING, periods, per, year, "(monthly,", or, 12, in, this,  "case)"), matrix(1,9, t, "=", Time, in, years, "(15.5,", in, this, "case)"))}}}
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<font color = red><font size = 4><b><u>STEP 3:</font></font></b></u> <font color = blue><font size = 4><b>Determine the NEW monthly payment.</font></font></b> 
<font color = green><font size = 4><b>        However, instead of using an online mortgage calculator, I believe it'd be better to find this 
        using the following formula:</font></font></b>
          {{{highlight(highlight_green(highlight_green(highlight(matrix(1,3, PMT, "=", (PV[oa] * (i/m))/(1 - (1 + i/m)^(- mt)))))))}}}, where: {{{matrix(5,1, matrix(1,9, PMT, "=", NEW, Monthly, PAYMENT, "(UNKNOWN,", in, this, "case)"), matrix(1,15, PV[oa], "=", NEW, LOAN, "amount/", Balance, after, 4.5, years, "(From", STEP, "2,", in, this, "case)"), matrix(1,12, i, "=", INTEREST, rate, per, annum, "(18%,", or, .18, in, this, "case)"), matrix(1,14, m, "=", Number, of, COMPOUNDING, periods, per, year, "(monthly,", or, 12, in, this,  "case)"), matrix(1,9, t, "=", Time, in, years, "(20,", in, this, "case)"))}}}

When all is "said and done," the NEW, lowered monthly mortgage payment should be <font color = red><font size = 4><b>$1,800.47</font></font></b>, down from $2,038.59.
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