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I would like to compare mortgages on a $100,000 loan to see which is most economical.

Option 1

5 years at $1.89\%$ then 5 years at $3.78\%$

Option 2

10 years at $2.49\%$

Is it simply a matter of calculating the compounding amount for $\$$50,000 at 1.89, then a further $\$$50,000 at 3.78%? then comparing with the compounding amount for $\$$100,000 at 2.49%?

When I do this, the result is counter-intuitive.

Deepesh Meena
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pingu
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  • Is this a simple interest rate problem or is it truly a mortgage? Do you make regular payments as you go or do you pay at the end? – John Douma Aug 24 '18 at 18:29
  • If this is simple interest rate problem, which I suspect it is, you must apply both interest rates to $100,000 because that is how much you borrowed. In that case you will get the intuitively correct answer that option 1 is more costly because you will pay an effective interest rate of 2.835%. – John Douma Aug 24 '18 at 18:46
  • @johnDouma Regular payments are made. – pingu Aug 24 '18 at 19:26
  • You should fill in the details of the loan. For example, how often is interest compounded? Also, are $50,000 of the loan expected to be paid after the first five years or are you calculating your payment for ten years based on 1.89% and then changing the interest and recalculating the payment after five years? – John Douma Aug 24 '18 at 19:59

2 Answers2

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Using the formula for monthly payment $$M = 100,000\cdot \frac{\frac{i}{12}(1+\frac{i}{12})^n}{(1+\frac{i}{12})^n-1}$$ where $n$ is the number of months and $i$ is the annual interest rate.

$10$ years @ $2.49\%$ is a monthly payment of $\$942.24$.

Rearranging the equation, $5$ years @ $1.89\%$ with a monthly payment of $\$942.24$ will reduce $\$100 000$ to $\$46094.87$

A further $5$ years @ $3.78\%$ with a monthly payment of $\$942.24$ will reduce $\$46094.87$ to $-\$5344.72$.

In other words, option $1$ will save you $\$5344.72$. This is logical as paying off more of the principal for the first $5$ years with a lower interest rate is more beneficial even with a slightly higher average interest.

Phil H
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$$A=\frac{5*50000*1.89}{100}+\frac{5*50000*3.78}{100}=\frac{5*50000*5.67}{100}=$14175$$

$$B=\frac{10*100000*2.49}{100}=$24900$$

clearly 1st choice is better because it results in paying less amount

Deepesh Meena
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  • According to pingu, this is a mortgage, not a simple interest rate problem. Also, you seem to be assuming that 50,000 is being borrowed now and then another 50,000 in 5 years. The loan is for 100,000. – John Douma Aug 24 '18 at 20:01