Jane wants to take out a 20 yr mortage. the interest rate is 8% compounded semi-annually. jane can afford monthly payments of 0. what is the largest mortage that she can afford?
My attempt at a solution ( using a graphing calculator):
N= 20 x 12 = 240
I%= 8
*PV= 102 612.70???
PMT= -850
FV = 0
P/Y = 12
C/Y = 2
PMT = END
but im suppose to solve this without a graphing calculator.
PV = (R(1-(1/1+i)^n)/i
R = 850
n = 240
i = ????
HELP!!!!!!
Consider:
Say the amount of the largest mortgage Jane can afford is A. If Jane makes payments monthly for 20 years she makes 240 payments. If the interest is 8% compounded semiannually then it earns 4% at each compounding, and is compounded 39 times (I assume Jane makes the last payment before the last interest compounding). So start out by considering Jane’s last six payments. These payments total 6*850 = 5100, and they pay off some amount of the original mortgage A(39). This amount will be given by
A(39)*(1.04)^{39} = 5100
so A(39) = 5100*(1.04)^{-39}
This was the last set of payments, but you can use the same reasoning to say that the nth set of six payment pays off some amount A(n) of the original mortgage, and A(n) is given by
A(n) = 5100*(1.04)^{-n}.
Thus, the amount Jane can pay off is the sum of all the A(n):
A = Sum_{n=0}^{39} A(n) = 850*Sum_{n=0}^{39} (1/1.04)^n
This gives A = 5100*(1-(1/1.04)^40)/(1 – 1/1.04).