MHB Calculating Your $10,000 Loan Payment

[Wilmer]

Methinks ya'll will have fun with this one!

A loan of $10,000 is set up this way:
3 years: 36 monthly payments of SAME amount
year#1 rate: 12% APR compounded monthly
year#2 rate: 10% APR compounded monthly
year#3 rate: 8% APR compounded monthly

What's the monthly payment?

 

[I like Serena]

Methinks ya'll will have fun with this one!

A loan of $10,000 is set up this way:
3 years: 36 monthly payments of SAME amount
year#1 rate: 12% APR compounded monthly
year#2 rate: 10% APR compounded monthly
year#3 rate: 8% APR compounded monthly

What's the monthly payment?

Just to clarify, does '12% APR compounded monthly' mean that we pay a monthly interest on the remaining loan of 12% / 12?
And is the remaining part of the fixed monthly payment the reduction of the loan?

If so, then this looks like an Excel excercise, for which I get that the monthly payment is \$326,97.
That means that in total we pay \$32,697 instead of \$10,000, which suggests we're talking to a loan shark.

 

[Wilmer]

Just to clarify, does '12% APR compounded monthly' mean that we pay a monthly interest on the remaining loan of 12% / 12?
And is the remaining part of a the fixed monthly payment the reduction of the loan?

If so, then this looks like an Excel excercise, for which I get that the monthly payment is \$326,97.
That means that in total we pay \$32,697 instead of \$10,000, which suggests we're talking to a loan shark.

Huh? 326.97 * 36 = 11770.92 : so total interest of 11770.92 - 10000.00 = 1770.92

326.97 is correct as monthly payment.
What do you mean with "Excel exercise"? Guess and check?

The payment can be precisely calculated. No Excel required :)

.12 / 12 = .01 would be monthly rate during 1st year.
Owing after 1st payment: 10000.00 + 100.00 - 326.97 = 9773.03

Similarly .10/12 during year2 and .08/12 during year3

 

[Jameson]

I find this problem interesting.

You can clearly solve for the IPMT and PPMT components each month using Excel for all 36 months then solve for the total interest owed and get a payment, but I don't think that's a very elegant solution. Do you have another way to solve for this without doing this recursion?

Also this sounds like a challenge problem rather than you asking for help, right?

 

[Wilmer]

Do you have another way to solve for this without doing this recursion?

Also this sounds like a challenge problem rather than you asking for help, right?

Yes to both questions :)

 

[tkhunny]

Methinks ya'll will have fun with this one!

A loan of $10,000 is set up this way:
3 years: 36 monthly payments of SAME amount
year#1 rate: 12% APR compounded monthly
year#2 rate: 10% APR compounded monthly
year#3 rate: 8% APR compounded monthly

What's the monthly payment?

I've always been a fan of just writing it out.

$v_{1} = \dfrac{1}{1+0.12/12}$
$v_{2} = \dfrac{1}{1+0.10/12}$
$v_{3} = \dfrac{1}{1+0.08/12}$

$P = Level\;Payment$

$A_{3} = P\cdot (v_{3} + v_{3}^{2} + ... + v_{3}^{12}) = P\cdot \dfrac{v_{3} - v_{3}^{13}}{1-v_{3}}$

$A_{2} = P\cdot (v_{2} + v_{2}^{2} + ... + v_{2}^{12}) + v_{2}^{12}\cdot A_{3} = P\cdot \dfrac{v_{2} - v_{2}^{13}}{1-v_{2}} + v_{2}^{12}\cdot A_{3}$

$A_{1} = P\cdot (v_{1} + v_{1}^{2} + ... + v_{1}^{12}) + v_{1}^{12}\cdot A_{2} = P\cdot \dfrac{v_{1} - v_{1}^{13}}{1-v_{1}} + v_{1}^{12}\cdot A_{2}$

$A_{1} = 10000$

$P = 326.9653$

That's equivalent to a level interest rate of 10.9108857% -- No need to call the Consumer Protection Bureau. It HAD to be between 8% and 12%.

 

[Wilmer]

Yepper Halls!

FV of loan amount = FV of the payment stream
Condensed:

a=10000
r1=12/1200 : u = (1 + r1)^12
r2 = 10/1200 : v = (1 + r2)^12
r3 = 8/1200 : w = (1 + r3)^12

f = a*u*v*w

p = (u - 1)/r1 * v * w + (v - 1)/r2 * w + (w - 1)/r3
p = f/p = 326.96532...

 

[tkhunny]

Yepper Halls!

FV of loan amount = FV of the payment stream
Condensed:

a=10000
r1=12/1200 : u = (1 + r1)^12
r2 = 10/1200 : v = (1 + r2)^12
r3 = 8/1200 : w = (1 + r3)^12

f = a*u*v*w

p = (u - 1)/r1 * v * w + (v - 1)/r2 * w + (w - 1)/r3
p = f/p = 326.96532...

Mine's prettier.

 

[Wilmer]

Hate to agree :)