# How do you calculate whether a 4 year mortgage term is cheaper than a 5 year term?

For example, assuming a balance of \$200k

• 5 year fixed at 3%
• 4 year fixed at 2.8%

How much would the interest rate have to rise before you lose out if you take the 4 year term?

• Most banks I know of have mortgage computation/comparison tools on their websites. While it is worth understanding how to do the math, in practice it's faster and easier to use one of these many, many, many free tools. – keshlam Jul 10 '16 at 5:27

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:-

http://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.

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 %`

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
``````

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
`````` \$204,919 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.

Regardless of term, the 3% mortgage costs you \$30 per \$1000 borrowed per year, the 2.8% one, \$28 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.

• Say I had 5 years amortization left though so I might have a choice between 5 years at 3% or 4 years at 2.8% + 1 year at n% – David Hayes Jul 3 '14 at 22:09
• I'd take the 4, and make extra payments so n doesn't matter. i.e. no balance left after year 4. – JTP - Apologise to Monica Jul 3 '14 at 22:11
• But in the general case any ideas how to work it out? I actually have more than 5 years remaining but for the purposes of the decision that's irrelevant. The rates will be what they are after 5 years – David Hayes Jul 3 '14 at 23:18
• You calculate the balance after the initial X years, and see what difference in the new rate is breakeven. This is how to calculate worst case scenario for adjustable rate loans. – JTP - Apologise to Monica Jul 3 '14 at 23:26
• Ah ok that makes sense, I'll fire up Excel later tonight – David Hayes Jul 3 '14 at 23:59

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.

• That was the answer I just came up with too based on JoeTaxPayer's answer. I did it slightly differently though I used an online calculator to calculate the interest cost of each and then work out what interest rate would generate that cost in a 1 year term (your way is much simpler) – David Hayes Jul 4 '14 at 0:35
• Does the answer change if I make additional principal payments (I'd have a 20% regular increase I could use) – David Hayes Jul 4 '14 at 0:36
• Yes, if you pay off a lot of money then it makes a difference. Let's say you pay off 20K of 100K after four years, then you saved 0.8% of 100K, and lose X% of 80K, so you would be equal if the interest rate goes up to 4%. If you pay off say 2K early, it doesn't make much difference. – gnasher729 Jul 4 '14 at 14:14