Calculate the total cost of a loan with variable interest rate

Question

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.

Answer 1

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.

Answer 2

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

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

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