# Annuity : yearly to monthly

Kenton borrows $250,000 on January 1, 2012 to be repaid in 12 annual installments at an effective annual rate of interest of 12%. The first payment is due on January 1, 2013. Instead of annual payment he decides to make monthly payments equal to one-twelfth the annual payment beginning on February 1, 2013. Determine how many months will be needed to pay off the loan. (Ans : 129.381291) Anyone know how to solve this question? ## 2 Answers Following that would be to use the annuity formula$239641 = 3363*\dfrac{(1-\frac{1}{(1+r)^t})}{r}\frac{1}{(1+r)^t} = 0.294671$with r = 0.009489.$(1+r)^t = 3.393617$Taking log$t \times ln(1.009489) = ln(3.393617)t = \frac{ln(3.393617)}{ln(1.009489} = 129.3826$• If the interest become monthly, aren't it suppose 12% / 12 ? how u get the r = 0.009489? Jul 1, 2017 at 16:18 • You should take care of compounding and hence it is not simple division but$(1.12)^{\frac{1}{12}} - 1 = 0.009489$Jul 1, 2017 at 16:26 • from line 2, how is the equation come from? I calculated 1 - (239641/3363 x r) but the answer is not 0.294671. Is it the # of significant problem? Jul 1, 2017 at 16:57 • Yes you are correct, if it is assumed that the first payment is paid on the 1st of Jan 2013, then the number of months is 119. If it is not paid on the 1st of Jan 2013, then use the loan value = 250000 and then you would get 129.38. I believe the text book assumes that the first payment is not paid. Jul 2, 2017 at 1:53 • Thank you very much for taking time to solve my question. :) Jul 2, 2017 at 4:35 I misread the question and have revised He makes the annual payment on 1/1/2013 and then starts paying monthly thereafter. the annual payment:$250,000 = P\sum_\limits{n=1}^{12} (1.12)^{-n}\\ 250,000 = P \frac {(1 - 1.12^{-12})}{0.12}\\ P = 250,000 \frac {0.01}{1-1.12^{-12}}\\ P = 40,359$after making his first months payment he owes$250,000 \frac {1-1.12^{-11}}{1-1.12^{-12}} = 239,640$and he will be paying$\frac {40359}{12} = 3363$per month$239,640 = 3363\sum_\limits{n=1}^k (1.12)^{\frac {k}{12}}\\ 71.25 = \frac {1 - 1.12^{\frac {-k}{12}}}{(1.12)^{\frac 1{12}} - 1}\\ 1-71.25(1.12^\frac{1}{12} - 1) = (1.12)^{\frac {-k}{12}}\\ \ln 0.324 = \frac {-k}{12}\ln 1.12\\ 119.368 = k$But$k$doesn't start counting until one year has passed$131\$ months

• May i know what is the purpose of line 9? Its 250000 x ( a angle n-1 / a angle n). Jun 30, 2017 at 12:18
• The first payment of 40359 has both principal and interest. The interest is (.12*250000) = 30000. So principal component of the first payment is 40359 - 30000 = 10359. For the second annuity the loan value is 250000 - 10359 = 239641 Jun 30, 2017 at 15:42
• Thank you very much for taking time to solve my question. :) Jul 2, 2017 at 4:35