### Linear Mortgage Calculation

One of my clients asked me to do a Linear Mortgage Calculation. Not a big deal were it not that I could not find the mathematical formula for it. There are a number of online calculators, so 'they' know what's going on but 'they' don't want you to know*!

So, what's the deal? With a linear mortgage, also called a straight line mortgage, you pay back the same amount of capital each period and therefore the same 'relative' amount of interest.

This Dutch graph explains it. 'Aflossing' is repayment. 'Rente' is interest.

So, how do you calculate this? It's relatively easy.

Loan (L) is 100.000€

Years (y) is 30

Interest (i) is 2%

Periods (p) is y*12 (months per year) is 360

Repayment (r) = L/p = 100.000/360 = 277,78€/month

The total amount of interest (t) = L*i*y = 100.000*0,02*30 = 60.000€

The first month, the interest you pay is t/p = 60.000/360 = 166.67€

Now we can solve the following equation.

r + first month interest = r + p * x

<==> first month interest = p * x

<==> x = first month interest / p = 166,67/360 = 0,463

f(x) = r + (p+1-x)*0,463

f(1) = 277,78 + (360+1-1)*0,463 = 277,78+166,67 = 444,45€

f(360) = 277,78 + (360+1-360)*0,463= 277,78+0,463 = 278,23€

f(x) = L/p + (p+1-x) * (L*i*y) / (p*p)

There you have it. Replace the numbers with your own and Bob's your uncle.

* Just kidding

So, what's the deal? With a linear mortgage, also called a straight line mortgage, you pay back the same amount of capital each period and therefore the same 'relative' amount of interest.

This Dutch graph explains it. 'Aflossing' is repayment. 'Rente' is interest.

Source: De Hypotheker (Dutch) |

Loan (L) is 100.000€

Years (y) is 30

Interest (i) is 2%

Periods (p) is y*12 (months per year) is 360

Repayment (r) = L/p = 100.000/360 = 277,78€/month

The total amount of interest (t) = L*i*y = 100.000*0,02*30 = 60.000€

The first month, the interest you pay is t/p = 60.000/360 = 166.67€

Now we can solve the following equation.

r + first month interest = r + p * x

<==> first month interest = p * x

<==> x = first month interest / p = 166,67/360 = 0,463

f(x) = r + (p+1-x)*0,463

f(1) = 277,78 + (360+1-1)*0,463 = 277,78+166,67 = 444,45€

f(360) = 277,78 + (360+1-360)*0,463= 277,78+0,463 = 278,23€

f(x) = L/p + (p+1-x) * (L*i*y) / (p*p)

There you have it. Replace the numbers with your own and Bob's your uncle.

* Just kidding

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