# How Much Will a Mortgage Cost Me?

How Much Will a Mortgage Cost Me?
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If you take out a mortgage without thinking through all the consequences, it could be the worst financial decision you ever make. If it goes wrong, it could cost you a lifetime of salaried work to repay and even drive you into bankruptcy.

However, sometimes taking a mortgage could be a lesser evil. If you do decide to take out a mortgage, you will face another important decision: Which type of mortgage to choose? But buyer beware! Bankers don’t tend to hit the headlines for their generosity, so keep in mind that the ‘most convenient’ offers tend to also be the most expensive ones. Their goal is to maximise the amount of money the bank can safely earn from you.

There are three common types of mortgages. I present them below in ascending order of cost, starting with the ‘cheapest’ one. And by cheap I mean least money lost in interest payments, not total cost of homeownership. Again, the formulas will look daunting; don’t let that put you off, just look at the scenarios and the numbers.

Similarly to the computation of compound interest and annuities, it is extremely important that you familiarise yourself with the math behind mortgages so that you can build your own amortization schedule.

When you meet your banker, you should already know exactly what s/he will tell you regarding mortgage costs. Those people are trained to deliver a sales pitch in order to lock you into the most profitable scheme for the bank. You should therefore know beforehand what the best choice for you is, rather than relying on their ‘advice’.

While we are focusing on mortgages, all the formulas I introduce apply to any installment loan or credit card repayment. Mortgages just embody the worst part of all loans because they tend to have the longest payment term.

With a comparatively lower interest rate in relation to a regular loan or a credit card, bankers put your dream of becoming a homeowner within your reach. But the ability to realise your dream comes at a cost! You’ll need to borrow a huge amount for an extremely long time. That ensures their return on investment will be maximal.

When getting a mortgage, you are borrowing an amount of principal. At time $t_0$ when you take out the mortgage, $P_0$ will be the amount of money you have borrowed. The outstanding capital you owe will decrease over time such that at the end, after $n$ periods of repayment, you will have paid back the full amount borrowed, in other words $P_n$ = 0.

The principal repayment $R_i$ refers to the amount of principal being paid back at period $i$. At the end of the mortgage, $\sum_{i=1}^{n}R_i = P_0$. This means that the initial capital borrowed has been fully repaid through principal repayments. The outstanding principal at any period is therefore the previous outstanding principal minus the principal repayment made leading up to that point, or $P_i = P_{i-1} - R_i$.

The dark side of any debt – whether it be a general loan or mortgage – is that at each period $i$, you will need to pay interest $I_i$ on the principal $P_{i-1}$ outstanding over the previous period, at the mortgage rate $r$ (fixed or floating). This means $I_i = rP_{i-1}$. The total cost of a mortgage is therefore the sum of all interest repayments.

Finally, the amount of money periodically paid to the bank, the installment $V_i$, will contain the principal and interest repayments for period $i$, in other words $V_i = R_i + I_i$.

### Constant Principal Repayment

If you have this type of mortgage, you reimburse the same amount of principal at each period. Each principal repayment is one $n^{th}$ of the initial amount borrowed. Hence $R_i = \displaystyle \frac{P_0}{n}$. As a result, the outstanding principal at each period decreases linearly by $\displaystyle \frac{P_0}{n}$. The interest for period $i$ is then computed as a percentage $r$ of the outstanding principal of the previous period $i-1$. This gives the following amortization table for a constant principal repayment mortgage.

Period $i$
Outstanding Principal $P_i$
Interest Repayment $I_i$
Principal Repayment $R_i$
Installment $V_i$
1
$P_0 \left(1 - \displaystyle \frac{1}{n} \right)$
$rP_0$
$\displaystyle \frac{P_0}{n}$
$\displaystyle \frac{P_0}{n} (1+nr)$
2
$P_0 \left(1 - \displaystyle \frac{2}{n} \right)$
$rP_0 \left(1 - \displaystyle \frac{1}{n} \right)$
$\displaystyle \frac{P_0}{n}$
$\displaystyle \frac{P_0}{n} (1+(n-1)r)$
3
$P_0 \left(1 - \displaystyle \frac{3}{n} \right)$
$rP_0 \left(1 - \displaystyle \frac{2}{n} \right)$
$\displaystyle \frac{P_0}{n}$
$\displaystyle \frac{P_0}{n} (1+(n-2)r)$
...
...
...
...
...
$n-1$
$P_0 \left(1 - \displaystyle \frac{n-1}{n} \right)$
$rP_0 \left(1 - \displaystyle \frac{n-2}{n} \right)$
$\displaystyle \frac{P_0}{n}$
$\displaystyle \frac{P_0}{n} (1+2r)$
$n$
$0$
$rP_0 \left(1 - \displaystyle \frac{n-1}{n} \right)$
$\displaystyle \frac{P_0}{n}$
$\displaystyle \frac{P_0}{n} (1+r)$
Total
$rP_0 \displaystyle \frac{n+1}{2}$
$P_0$
$P_0 \left(1+\displaystyle \frac{n+1}{2}r \right)$

Assuming a $n$ = 25-year mortgage on $P_0$ = $500,000 with a fixed rate $r$ = 5% per year, this would result in the following: Period $i$ Outstanding Principal $P_i$ Interest Repayment $I_i$ Principal Repayment $R_i$ Installment $V_i$ Total$325,000
$500,000$825,000
1
$480,000$25,000
$20,000$45,000
2
$460,000$24,000
$20,000$44,000
3
$440,000$23,000
$20,000$43,000
...
...
...
...
...
24
$20,000$2,000
$20,000$22,000
25
$0$1,000
$20,000$21,000

### Constant Installment

This form of mortgage is particularly attractive to home buyers, as the installment is constant over time.  It is easier to plan for because the same amount is debited at each period.

In this situation, $P_0$ is the present value of a constant annuity stream. As a result, the installment $V$ is equal to $\displaystyle \frac{rP_0}{1-(1+r)^{-n}}$. From there, it is easy to find out the formulas for $P_i$, $I_i$ and $R_i$. These are summarised in the below table.

Period $i$
Outstanding Principal $P_i$
Interest Repayment $I_i$
Principal Repayment $R_i$
Installment $V_i$
1
$P_0 \displaystyle \frac{1-(1+r)^{-(n-1)}}{1-(1+r)^{-n}}$
$rP_0$
$\displaystyle \frac{rP_0}{(1+r)^n-1}$
$\displaystyle \frac{rP_0}{1-(1+r)^{-n}}$
2
$P_0 \displaystyle \frac{1-(1+r)^{-(n-2)}}{1-(1+r)^{-n}}$
$rP_0 \displaystyle \frac{1-(1+r)^{-(n-1)}}{1-(1+r)^{-n}}$
$\displaystyle \frac{rP_0(1+r)}{(1+r)^n-1}$
$\displaystyle \frac{rP_0}{1-(1+r)^{-n}}$
3
$P_0 \displaystyle \frac{1-(1+r)^{-(n-3)}}{1-(1+r)^{-n}}$
$rP_0 \displaystyle \frac{1-(1+r)^{-(n-2)}}{1-(1+r)^{-n}}$
$\displaystyle \frac{rP_0(1+r)^2}{(1+r)^n-1}$
$\displaystyle \frac{rP_0}{1-(1+r)^{-n}}$
...
...
...
...
...
$n-1$
$P_0 \displaystyle \frac{1-(1+r)^{-1}}{1-(1+r)^{-n}}$
$rP_0 \displaystyle \frac{1-(1+r)^{-2}}{1-(1+r)^{-n}}$
$\displaystyle \frac{rP_0(1+r)^{n-2}}{(1+r)^n-1}$
$\displaystyle \frac{rP_0}{1-(1+r)^{-n}}$
$n$
$0$
$rP_0 \displaystyle \frac{1-(1+r)^{-1}}{1-(1+r)^{-n}}$
$\displaystyle \frac{rP_0(1+r)^{n-1}}{(1+r)^n-1}$
$\displaystyle \frac{rP_0}{1-(1+r)^{-n}}$
Total
$P_0 \left(\displaystyle \frac{nr}{1-(1+r)^{-n}}-1 \right)$
$P_0$
$\displaystyle \frac{nrP_0}{1-(1+r)^{-n}}$

As before, assuming a $n$ = 25-year mortgage on $P_0$ = $500,000 with a fixed rate $r$ = 5% per year, this would provide the following results: Period $i$ Outstanding Principal $P_i$ Interest Repayment $I_i$ Principal Repayment $R_i$ Installment $V_i$ Total$386,906
$500,000$886,906
1
$489,524$25,000
$10,476$35,476
2
$478,524$24,476
$11,000$35,476
3
$466,974$23,926
$11,550$35,476
...
...
...
...
...
24
$33,787$3,298
$32,178$35,476
25
$0$1,689
$33,787$35,476

### Interest-Only

Interest-only mortgages are the worst mortgage of all. They create the illusion of being a low-rate mortgage, because you do not have to make big payments during the mortgage period, in fact you are only paying off the interest.

The principal is actually paid as a lump sum at the end of the mortgage. However, as you will see, this is the most expensive way to run a mortgage because it maximizes interest paid by ensuring that the outstanding capital remains the same during the life of the mortgage. Clever!

Unfortunately, many people do not realise this and their financial plans are inadequate to make the final payment; this leaves them with a shortfall they can’t pay and could put them in a very uncomfortable position. The amortization table for this type of mortgages is straightforward and depressing:

Period $i$
Outstanding Principal $P_i$
Interest Repayment $I_i$
Principal Repayment $R_i$
Installment $V_i$
1
$P_0$
$rP_0$
$0$
$rP_0$
2
$P_0$
$rP_0$
$0$
$rP_0$
3
$P_0$
$rP_0$
$0$
$rP_0$
...
...
...
...
...
$n-1$
$P_0$
$rP_0$
$0$
$rP_0$
$n$
$0$
$rP_0$
$P_0$
$P_0(1+r)$
Total
$nrP_0$
$P_0$
$P_0(1+nr)$

Assuming a $n$ = 25-year mortgage on $P_0$ = $500,000 with a fixed rate $r$ = 5% per year, this would give the following results: Period $i$ Outstanding Principal $P_i$ Interest Repayment $I_i$ Principal Repayment $R_i$ Installment $V_i$ Total$625,000
$500,000$1,125,000
1
$500,000$25,000
$0$25,000
2
$500,000$25,000
$0$25,000
3
$500,000$25,000
$0$25,000
...
...
...
...
...
24
$500,000$25,000
$0$25,000
25
$0$25,000
$500,000$525,000

### Summary

When it comes to mortgages, don’t do it! But if you do decide to take one, at least be clever about it. The total cost for each of the mortgage types we studied are summarised below (principal of $500,000 reimbursed over 25 years with a mortgage rate of 5% per annum): Constant Principal Repayment Constant Installment Interest Only Total Cost$325,000
$386,906$625,000

Learning to think as a banker is a necessity when making financial decisions, particularly if those decisions involve borrowing money. Your personal banker will always guide you towards the seemingly most convenient option; but remember that this also tends to be the most profitable scheme for the bank. Doing your homework puts you in a better position to make an informed choice and lower the costs.

The main lessons to draw from the figures in those tables are:

1. Given the costs associated with a mortgage, are you sure it is the best financial decision you can make? Have you done your maths? Are you better off renting?
2. Always choose the shortest mortgage term you can afford. Compound interest becomes unbearable with longer durations. Bankers will always push you for the longer options with arguments about flexibility and lower installments. Just remember that you pay for that flexibility!
3. Always ask to repay as much principal as early as possible. This will automatically lower the amount of interest you pay.