: How do you calculate whether a 4 year mortgage term is cheaper than a 5 year term? For example, assuming a balance of 0k 5 year fixed at 3% 4 year fixed at 2.8% How much would the interest rate have to rise before

Regardless of term, the 3% mortgage costs you per 00 borrowed per year, the 2.8% one, per thousand per year.

The terms impact cash flow, which is why I might choose a 4% 30 year term vs a 3.5% 15 year mortgage, given the difference. But the rate itself is as I described.

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With the four year mortgage, you save 0.2% every year for four years, that's a total of 0.8%. Then you pay X% instead of 3% in the fifth year. If X = 3.8%, you lose exactly what you gained in the first four years. If you predict that X is somewhere between 3.0% and 4.6%, so 3.8% is exactly in the middle, both terms are equally good, but the five year one is more predictable, which is good by itself.

But check carefully if there are any other costs involved than just the interest rate. If you have to pay some fee to get the mortgage, you'd have to take that into account.

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Since you are asking about mortgages let's assume the rates you quote are nominal interest rates compounded monthly and the mortgage repayments are monthly.

The following loan formula can be used:-
www.financeformulas.net/Loan_Payment_Formula.html
pmt = r*pv/(1 - (1 + r)^-n)

where

pmt = periodic payment
pv = present (initial) value of loan
r = rate of interest (as a decimal)
n = number of periods

Taking a monthly rate, mr, the total repayments for the 5 year mortgage are :-

pv5 = 200000
mr5 = 0.03/12 = 0.0025
n5 = 5*12 = 60

pmt5yr = mr5*pv5/(1 - (1 + mr5)^-n5) = 3593.74

total5yr = n5*pmt5yr = 215624

The total repayments for the 4 year mortgage are :-

pv4 = 200000
mr4 = 0.028/12 = 0.002333...
n4 = 4*12 = 48

pmt4yr = mr4*pv4/(1 - (1 + mr4)^-n4) = 4409.21

total4yr = n4*pmt4yr = 211642

To find the monthly rate, mrx4, that makes the 4 year mortgage cost the same as the 5 year mortgage the following equation is solved :-

n4*mrx4*pv4/(1 - (1 + mrx4)^-n4) = total5yr

giving mrx4 = 0.00311287

and a nominal rate compounded monthly : 12*mrx4 = 0.0373545 = 3.73545 %

The 4 year mortgage is less costly than the 5 year 3% mortgage at any nominal rate below 3.73545 %.

Alternative calculation using AER

If the rates you quoted were annual equivalent rates [1.] and repayments are still monthly the calculation is slightly different.

en.wikipedia.org/wiki/Effective_interest_rate#Calculation

Running the 5 year and 4 year calculations :-

pv5 = 200000
mr5 = (1 + 0.03)^(1/12) - 1 = 0.00246627
n5 = 5*12 = 60
pmt5yr = mr5*pv5/(1 - (1 + mr5)^-n5) = 3590.14
total5yr = n5*pmt5yr = 215409

pv4 = 200000
mr4 = (1 + 0.028)^(1/12) - 1 = 0.00230391
n4 = 4*12 = 48
pmt4yr = mr4*pv4/(1 - (1 + mr4)^-n4) = 4406.1
total4yr = n4*pmt4yr = 211493

n4*mrx4*pv4/(1 - (1 + mrxr)^-n4) = total5yr

giving mrx4 = 0.00307086

and an annual equivalent rate : (1 + mrx4)^12 - 1 = 0.0374792 = 3.74792 %

Adjusting for inflation

A fixed payment in the future will effectively cost less due to wage inflation so an adjustment for inflation can be made. Using the mortgage figures calculated for AER rates and taking inflation as 2% AER, then the monthly inflation rate is

mi = (1 + 0.02)^(1/12) - 1 = 0.00165158

and the adjusted totals are :-

So we can see adjusting for inflation closes the gap between the 4 year and the 5 year mortgage from 3915.87 to 1753.08 :-

total5yr - total4yr = 3915.87
adjtotal5yr - adjtotal4yr = 1753.08

Now the break-even rate for the 4 year mortgage can be found by solving

which finds mrx4 = 0.00266243

So the annual equivalent rate is (1 + mrx4)^12 - 1 = 0.0324212 = 3.24212 %

Check

pmt4yr = mrx4*pv4/(1 - (1 + mrx4)^-n4) = 4444.12

4,919 is the same as the inflation-adjusted present cost of the 5 year mortgage.

So, adjusting for wage inflation at 2% AER, the 4 year mortgage is less costly than the 5 year 3% AER mortgage at any rate below 3.24212 % AER.

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