Explain formula for compound interest

Question

I have to solve the following problem:

Mark wants to buy a house for $200000\$$ and decides to take out a twenty five year mortgage to pay for it. Given that the interest on the mortgage is $2\%$ per month, explain why his monthly repayments of $X\$$ will be given by the equation $$ X + X(1,02) + X(1,02)^2 + \dots + X(1,02)^{299}$$ assuming that he makes the first payment one month after taking out the mortgage. Hence, find $X$.

I understand why the final payment will be $200.000 (1,02)^{300}$, but I don't really see why monthly payments will be $X, X(1,02), X(1,02)^2$,up to $X(1,02)^{299}$.

Any help will be appreciated.

Answer 1

The debt repaid in the first payment $X$ would have increased to $X\times 1.02^{299}$ had you left it as debt. Each new payment has spared you interest in one period less.

Sorry if my English is a bit poor. I am not a native speaker.

Answer 2

The equation is describing the mortgage debt and payments, all normalized to dollars at the instant of the last payment, assuming two percent monthly "inflation" to represent the mortgage rate. So for example, the first payment of $X$ will undergo $299$ multiplications by $1.02$ (not $300$ multiplications because there are only $299$ months to go on the mortgage). The debt, on the other hand, has grown by $300$ such steps.

To solve the equation, you need to know how to sum the right hand side, which is a geometric series with starting term $X$ and term ratio $1.02$. That you may remember from high school algebra.

Answer 3

The amount the bank paid you in the beginning of the five-year period is $200K$.

If instead of giving it to you they were to invest it at a rate of $2\%$ per-month for the same period, then they'd have $200K*1.02^{300}$ at the end of the period.

The idea of a loan is the bank giving you $200K$ in the beginning of the period, and bank ending up with the same amount of $200K*1.02^{300}$ at the end of the (five year) period, through the fixed payments you make (and investment the bank makes with the payments).

How is it done ?

Bank takes the first payment (after one month) and invests it for the remaining $299$ months. At the end of the five years period, this gives

$X * 1.02^{299}$.

Now the second payment. This is invested for the remaining $298$ months, so at end of the five-year period this gives

$X*1.02^{298}$,

Continue with all remaining payments, until the last one at the $300$-month (no interest, this is the end of the period).

Total amount they have at the end of the five-year period is then

$X*1.02^{299} + X*1.02^{298} + \dots +X*1.02 + X$.

To find the values of $X$ we need this total amount to be equal to $200K*1.02^{300}$, and that's your equation.