A grauate student has an outstanding student loan of $100,000 @ 5% compounded monthly. He recently landed a job on Wall Street with a six-figure salary. He decided to pay off his student loan over a period of 10 years as follows: a) He will pay fixed monthly payments, at the end of each month, for the first 5 years or 60 payments. b) He will pay the balance of his student loan for the last 5 years or 60 months, by doubling the amount of his monthly payments, compared to the payments he made in the first 5 years. What would his monthly payments be for the first 5 years? What is the total amount of interest that he will have paid in that 10-year period?
This is a very tough problem for me. Any help would be greatly appreciated. Thanks a lot.
You may have different formulas to work with than what I know but this is how I did it. I have not checked my answer but it should be ok.
Let him pay C dollars each month for 60 months and then 2C dollars each month for another 60 months.
X = 100000 = initial loan
n=5*12 = 60 (used in both halves)
i =0.05/12 = 0.0041666666666666 repeater
L =1+i = 1.0041666666666666 repeater I usually use capital I but it looks like a 1 here :(
PV60 refers to value owing after 60 months
\(PV_{60} = 2C * \left[ \frac{1-L^{-n}}{i} \right]\)
Now I want to solve for C and there are two unknowns namely PV60 and C so I need another equation so that I can solve them simultaneously.
I do not know such an equation so I will need to derive one.
The initial amount owing is X and each month for the first 5 years a payment of C is made at the end of each month.
So
After one month the amount owing will be
\( PV_1 = XL - C\\ then\\ PV_2 = (XL-C)L - C\\ PV_2 = XL^2-CL - C\\ PV_2 = XL^2-C(L+1)\\ then\\ PV_3 = [ XL^2-C(L+1)]L-C\\ PV_3 = XL^3-CL(L+1)-C\\ PV_3 = XL^3-C(L^2+L+1)\\ \mbox{I can already see the pattern}\\ PV_n = XL^n-C(L^{n-1}+L^{n-2}+\ldots L^2+L+1)\\ \mbox{In the bracket is the sum of a GP}\\ PV_n = XL^n-C\left(\frac{1(L^n-1)}{L-1}\right)\\ PV_{60} = XL^{60}-C\left(\frac{L^{60}-1}{L-1}\right)\\\)
Ok now I have two equations for PV^60
\(PV_{60} = 2C * \left[ \frac{1-L^{-n}}{i} \right]\\~\\ PV_{60} = XL^{60}-C\left(\frac{L^{60}-1}{L-1}\right)\\~\\ so\\ 2C * \left[ \frac{1-L^{-n}}{i} \right]=XL^{60}-C\left(\frac{L^{60}-1}{L-1}\right)\\ 2C * \left[ \frac{1-L^{-n}}{i} \right]+C\left(\frac{L^{60}-1}{i}\right)=XL^{60}\\ \frac{C}{i} * \left[ 2-2L^{-n} +L^{60}-1 \right]=XL^{60}\\ \frac{C}{i} * \left[ 1-2L^{-n} +L^{60} \right]=XL^{60}\\ C=\frac{iXL^{60}}{1-2L^{-n} +L^{60}}\\ C=\frac{0.0041\dot6 *100000*(1.0041\dot6 )^{60}}{1-2(1.0041\dot6 )^{-60} +(1.0041\dot6 )^{60}}\\ \)
(100000*0.00416666666666*1.00416666666666^60)/(1-2*1.004166666666666^-60+1.00416666666666666^60) = 737.615467649757885214385162703476071476657715295737666683233409836
So $737.62 is paid at then end of each month for the first five years and then $1475.24 is paid at the end of each month for the next 5 years and then the loan is all paid back.
