# 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.
5.00 avg. rating (98% score) - 4 votes

#### Monkey Master

My wife and I are currently living in Sydney, Australia. We plan on becoming financially self-sufficient in 2015 so we can retire at 35. We are regular working people, trying to be smart about saving money and generating passive income. I want to share with you how we reached that decision and how we are planning towards financial independence. Continue Reading.
Contact: monkeymaster@monkeyism.com

### 4 thoughts on “How Much Will a Mortgage Cost Me?”

• 22 May, 2013 at 8:37 pm

It’s idly amusing that a person who claims to know so much about finance has got the best to worst order of these mortgage products inverted. The best mortgage choice is interest only.

The problem is an overly narrow focus on mortgage cost without considering the rest of the picture: the money that can be made by investing the capital repayment money over the years and the risk of liquidity issues due to things like periods of unemployment.

The calculations assume that the repayment money is free, completely ignoring the opportunity cost of using it for mortgage capital payments instead of something else.

Inflation is also a factor, helping to reduce the real (after inflation) cost of deferred debt. When comparing property vs renting it’s also important to consider the investment return of property ownership, which in at least some jurisdictions can be around 6% real a year as a long term average.

Many regulators take a similar silo approach to only considering one aspect of a problem but a sensible consumer shouldn’t be doing that. There’s too much money to be made or saved or made by integrated planning. In my jurisdiction that means things like getting tax relief on pension contributions (instead of making mortgage capital payments) then using a lump sum from the pension, paid out without tax deduction, to clear the mortgage. The result: both free tax relief money to pay off mortgage capital and investment growth that is likely to be at a higher rate than the mortgage interest cost on the money.

The liquidity risk problem is even easier: if you’re investing some of the money in ways that are accessible, you have more money available to pay the mortgage and other bills if unemployed. If the money is instead paid back to the bank it’s not available to do this. And banks are not known for being keen to allow borrowers to switch to interest only payments or defer all repayments when told that the reason is unemployment. Or long term sickness. Or anything else that threatens their money. If you want that flexibility you need to be getting it at the outset.

The final result of these things is that you are telling people to make the choices that are likely to leave them both poorest and at greatest risk. Though the investment risk is lower from the capital repayment approach and if you only consider that, the liquidity risk may seem acceptable, particularly to those with low risk tolerances.

• 23 May, 2013 at 12:59 am

If you read carefully, I’m actually telling people to think twice before taking a mortgage, not to take one. And if they do, they should make a financial plan themselves depending on their personal situation and the market conditions in their country. All I am doing is give people the understanding of what they are getting into.

I am not trying to compare opportunity costs in this post, just giving the formulas for mortgage computations and interest costs. For a proper comparison, you can check my other post on renting versus buying. You can’t just mention missed investment opportunities and tax breaks and at the same time discard all recurrent maintenance costs and taxes that come with being a homeowner.

You sound like you know exactly how to manage your finances. But the reality is that most people taking an interest-only mortgage can’t make higher monthly payments. Hence they can’t make any investments on the side or repay part of the capital in advance to offset the cost of interest. I don’t know if you’ve read the bbc article I put as a link “Shortfall fears for interest-only mortgage holders” but this is not me putting out a personal opinion, the reality is that people get crushed by those types of mortgages.

With respect to illness or unemployment, this is what mandatory mortgage insurance is for. As for liquidity risk, again it assumes people taking a mortgage have a stack of cash on the side to manage. Most people already optimise their installments and can’t spare more. They put all their savings in the downpayment and take the maximum installment they can afford.

In any case, there are arbitrage opportunities depending on each personal situation. I am always pushing for everyone to think for themselves. And I don’t plan on being a personal financial advisor as each situation is different.

• 24 May, 2013 at 12:35 pm

I read the BBC story. I also read both of the FCA reports it was based on and noted that the person who led the study into how people planned to repay said in comments to the press that there wasn’t going to be an exploding time-bomb in the interest only mortgage market. That’s in part because the study found that those with interest only mortgages ending in the next ten years or so generally had other investments that would be used to cover any shortfall in their endowment policies. The average shortfall figures that were widely quoted also weren’t real (inflation-adjusted) values and that significantly distorted them, making the shortfall picture appear far worse than it’s likely to be.

I agree with you that there are costs of home ownership that need to be considered when comparing renting to buying. I’d read your other post before commenting on this one. As you might guess I wasn’t really happy with the opportunity cost and property value change assumptions used in that post. 🙂 But maybe in your location the growth rate on property isn’t anything like the rate here (though in major Australian cities I suspect it’s higher, at least judging from what my co-workers tell me).

Comparing opportunity costs might well make a useful future post if you’re so inclined sometime. It’s quite interesting to see just how expensive (or sometimes cheap) renting can be compared to buying, as well as the flexibility differences between those two choices. If you want a source for investment returns you probably already have some but just in case, the Barclays Capital Equity Gilt study is the one that I seem to see used most and it seems to have quite decent numbers, though not for your current part of the world.

The stack of cash to deal with liquidity risk comes from investing the repayment part of the mortgage payment instead of using it for capital repayments. You don’t need to start with it because by the end you’ll have a stack big enough to repay the mortgage, unless you haven’t been monitoring and adjusting your plans along the way .The FCS study showed that those with interest only mortgages often did adjust, particularly when their investment firm or mortgage lender contacted them with concerns about a shortfall.

Where I am there’s little mandatory mortgage insurance, other than a requirement to have insurance in place that is sufficient to cover the property rebuilding cost. A borrower is free to become unable to work and default if they wish not to provide for cases like that – or if the insurance they have in place doesn’t cover whatever happens to them.

I agree with you that there are people who borrow as much as they can possibly borrow and use interest only to further increase that amount. Wage inflation is what can eventually save those people, the compounding friend of the debtor that’s perhaps doing a really good job in Australia at the moment. But do use a bit of caution in generalising too far from those cases. The FCA study also found lots of high income people who weren’t borrowing excessively. I also have an interest only mortgage and borrowed only a little over one times my income. I simply purchased a low cost property because that is all I need and that greatly lowers my related costs. At a time when I had enough money invested to repay the mortgage if I’d wanted to. That’s not uncommon for people who are older and well into retirement planning, and who are taking that seriously, for it takes quite large amounts of money to provide a good retirement income level. Even more so if planning to retire before state and work pensions can be used. I do wonder if you might be mostly looking at the situations of people of relatively young ages, who may well stretch their finances a lot, particularly in a rising market where there can be an expectation of high capital value growth.

While I’m far from an expert, the total value of my savings and investments is around 97% of my net pay plus gross pension contributions over the last seven or so years, so I’m not doing badly on the financial independence front. Not enough safety margin to stop working yet but I am over my minimum income threshold, without what I consider to be a sufficient safety margin. That’s an expected income equal to the local median pensioner income plus 100% safety margin, sustainable for life. Around £18,000 plus 100%. It’s an interesting quest and the value of even modest amounts of frugality plus investment compounding is a wonderful thing to behold and experience.

We do seem to disagree on some things but it seems that at least on the big financial independence picture we’re on similar courses overall.

• 28 May, 2013 at 10:54 am

Yes we are both looking for financial independence. Our paths may be different but that’s the nature of financial decisions. At any one time, there will be people on the market with different views on where it is heading. One may believe the housing market is going down and sell, the other that real estate is booming and buy. They could both make a profit though if for instance, the seller bought it cheaper than his selling price.

With respect to property growth, I indeed have a personal view. Prices are very expensive in Australia and are increasing regularly. But I personally believe in a housing bubble that could burst in the coming years. The Australian economy was not hit by the global financial crisis but is starting to show signs of weakness. The central bank has been lowering rates to sustain the economy over the last couple of years, the AUD is weakening because demand in China for commodities is slowing down and commodity prices are falling. So applying an increase with the inflation rate seems reasonable.

Another case I have in mind is that we bought a flat in the UK back in 2007 right before the financial crisis and that flat is now worth 25% less. We don’t want to sell it at a loss but it looks like it will take us a long time to break even. So I don’t think housing prices always go up… The only part that saves our investment is that our mortgage was using a floating rate and the crisis brought it down to below 1%. So we lost money on the asset but ended paying less interest on the mortgage (if rates still stay low).

Thanks for sharing your experience, it is very interesting for me to get your vantage point. I write this blog from my own perspective. So indeed with people of my age group in mind. I see a lot of acquaintances rushing into buying their first home with a 7% mortgage on 90% of the property value for 25-30 years, without any further planning. I can’t take into account everyone’s personal situation and opportunity costs but I can raise the issue and let people make up their own mind.

I also think that most people don’t plan for retirement until later in their lives. I personally started realising the potential of financial independence recently and wished I thought about it in my twenties rather than my thirties. I never thought I could stop working before 65 so I never put much thought in early retirement. But now I need to catch up for the time lost.

Good luck with reaching your objectives, you seem to be well on your way. You’ll probably retire before me 🙂