# weighted average of interest rate

Let say that I have 2 loans. One loan has an interest rate of 5% and the other of 10%. I calculate the weighted average both loans and it is 8% (This is an example). I don't know how can you prove that the total interest yield of both loans with thier original interest rates is the same as if both loans interest rate was 8%. I don't know how to prove it. I would like some intuition.

• How do you calculate that the "weighted average of both loans" is 8^%? The only way I know to find such a weighted average is use the fact that "the total interest yield of both loans with their original interest rates is the same as if both loans interest rate was 8%." You don't need to prove it- that is the definition of "weighted average". – user247327 Sep 25 '18 at 23:30
• my confusion is for example, let say that loan 1 is a loan with a term of 60 month at 5% and that at the end the loan will yield 2000 in interest. Also Loan 2 is with term of 60 month at 7% and the loan will yield 2500. The balance of the loan does not matter my confusion y how can i be sure that using the weight average will yiled the same 4500 in interest. you know that the interest of a loan is not just multiplying the balance by the rate, there is an amortization process – kprincipe Sep 26 '18 at 0:22

Set the interests equal to each other to figure the relative size of the loans.

$$.08x + .08y = .05x + .10y$$

$$.03x = .02y$$

$$x = \frac{2}{3}y$$

Therefore the $$x$$ loan is $$\frac{2}{3}$$ of the $$y$$ loan.

Example:

$$x = 600; y = 900$$

$$.08x + .08y = .05x + .10y$$

$$.08(600) + .08(900) = .05(600) + .10(900)$$

$$48 + 72 = 30 + 90$$

$$120 = 120$$