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A small business owner obtained two loans from two different banks as follows:

1 - Bank A lent him $15,000 for 5 years at 6% compounded monthly to be repaid in 60 equal monthly payments.

2 - Bank B lent him $10,000 for 3 years at 4% compounded monthly to be paid in 36 equal monthly payments.

3 - He has already made 12 payments on both loans and wishes to consolidate the balance of the two loans in one loan for a period of 4 years to be paid monthly, using the combined monthly payments of the two separate loans.

The questions is: what is the balance of the new consolidated loan, and given the combined monthly payment of the two original loans, what is the effective annual cost to him on the new, consolidated loan?

Any help would be greatly appreciated. I have already asked this question to a number of business people and nobody has been able to answer it correctly, yet! I know the answer, but don't know how to arrive at it.

Thank you very much.

Guest Aug 21, 2018

#2**+1 **

Thanks for your interest. Sorry, I made a small mistake about the term of the new loan. It is actually for** 3 years, NOT 4 **as I stated in the question. I can give you the answer but prefer not to, so that you can come with your own original solution using right financial formulas and/or equations. Thanks again.

Guest Aug 21, 2018

#3**+2 **

OK, young person! I shall attempt to solve it for you but may charge you for it!! Just joking!.

First, you should be familiar with this complicated-looking formula, but in fact, it is just PV and FV formulas combined into one. If you know any 4 of the 5 variables, then you can solve for the fifth, except for interest rate!:

**-P*[(1-(1+R)^-N)/(R)]+FV*(1+(R))^-N+PV=0**, where P = Periodic payment, R= Interest Rate per period, N= Number of periods, FV = Future value, PV = Present value.

So, the first thing we have to do is figure out the original monthly payments for the two separate loans:

1 - From the first loan of $15,000, you enter the following values into the above formula: 6%/12 =0.005 under R, 60 under N, 0 under FV, 15,000 under PV. Then you would solve for P. If you don't make any mistakes, then you should get **P = $289.99**, which is the monthly on the $15,000 loan. Because he has already made 12 monthly payments, we have to find the balance of this particular loan. You would use the above formula again, but this time you would enter 12 under N and $289.99 under P and you would solve for FV, which would be the balance of the loan after 12 payments. If you don't make any mistakes, you should get: **$12,347.98**.

2 - For the second loan of $10,000, you would do exactly the same thing as above entering your numbers accurately. If you don't make any mistakes, you should get the monthly payment to be: **$295.24** and the remaining balance after 12 payments to be: **$6,798.86.**

3 - Now, you must add the balance of the two loans and you get: **$12,347.98 + $6,798.86 =** **$19,146.84 **- which is the balance of the new consolidated loan!. You do the same with the two original monthly payments: **$289.99 + $295.24 = $585.23** - which is the new combined monthly payment on the new consolidated loan for a term of 3 years.

4 - Finally, you have to enter the new loan of $19,146.84 under PV, $585.23 under P, 36 (3 x 12) under N, and 0 under FV, and you would solve for R, or the monthly interest rate. Except, unfortunately, you can't solve for R directly but you have to use iteration and interpolation(basically, trial and error) until you arrive at the right solution.

5 - My computer is programmed to solve such equations and it comes out with the rate of **6.3160745455%** compounded monthly. So, the effective annual cost to him would be **6.502162744568%**. And that is the END!.

Guest Aug 21, 2018

#4**+1 **

Wow, that is amazing! Congrats you got both the loan balance and the annual cost correct! I don't think I have seen that scary formula, but I did try to calculate one of the loan payments and it worked. I will spend some time to see how you solved it in detail. I thank you for the time you took to do this.

Guest Aug 22, 2018

#5**+2 **

One way to directly calculate the new interest rate is to take the __weighted __harmonic mean of the two current rates.

The terms for the 6% loan (compared to the 4% loan) are 50% higher in the amount borrowed and 2/3 longer for the length of the loan. Multiply the 6% by these two dimensions 0.06*1.5*(2/3) = 0.15

Then calculate the harmonic average of 0.04 and 0.15

\(\text {Harmonic mean = } \dfrac {2}{\dfrac {1}{0.04} + \dfrac {1}{0.15}}= 0.06316 \)

GA

GingerAle Aug 22, 2018

#6**+1 **

Notice that he/she changed the term of the new consolidated loan from 4 years to 3 years. Suppose that the new loan's term WAS 4 years, then how would you calculate the new rate? You would still get the same Harmonic average as you calculated it in terms of the original terms and the original amounts. If the new term for the new loan was 4 years, then the interest rate would be **20.25306%** compounded monthly, or **22.242975%** effective annual rate.

Guest Aug 22, 2018

#8**-1 **

Why Mr. BB, do you mean to tell me my answer is ………..**BLARNEY!**!

Blarney answers are acceptable for blarney questions, Mr. BB. It doesn’t matter how absurd the reasoning is, as long as the math is correct, and it is: The weighted harmonic mean is mathematically correct, and the method used to weight it is mathematically correct. The logic is correct in another universe (not this one), but it’s usable here because this question, and the obsequiously (means ass-kisser) polite student who asked it, came from came from the same universe.

In this world, a competent teacher has countless questions and scenarios for assigning practice and to assess a student’s TVM skills. A teacher would only present this question to demonstrate absurdity. No one except a fool would consolidate a loan using this approach, and only an unethical loan company would make such a loan. I can think of one that would: __The First National Vigorish and Loan-Shark Company__; a wholly-owned subsidiary of __Smith, Blarney, Cheatem, and Howl__, your former and principal employer.

The absurdity becomes very clear when you add another year of payments (vigorish) to a fully amortized loan.

Another thing that becomes clear is you are the *Blarney Banker*. … … …

GA

GingerAle Aug 24, 2018