finding monthly payments for loan with two interest rates

[Henry]

Hi, I want to calculate monthly payment for a loan that has promotional interest rate 0% for part of the loan.
I believe that I have result, but I would like to confirm, that my approach is correct.
What I have:
loan amount is 10000
i (promotional 1-6M)=0%
i (standard 7-12M)=1,606% p.m.
n =12

What I did - I took annuity formula and put it to Excel to find payment (see the print screen).

First columns are equivalent of pmt function in Excel (to confirm that calculation is OK). Second is the result with promotional rate.
I tried another way - amortization table and then by macro and bisection method found the payment that gives 0 for balanace at the end of the period. This gives same result as here 856,41

What I am not sure about:
- why to apply power of 1 to 6 for second half of the year? Why not 7 to 12, it is in fact 7th period. Do I understand it as first period with applied interest?
- is there "single" formula for this? Something like "pmt" in Excel?
Thank you.

 

[Henry]

Thanks for answer, but it does not make it clear to me.

Why to take split 6000/4000, you should be paying same amount for all period so how can I assume that I repay in 6 months 6000?
When I try to put 869,15 to amortization table, then the result at the end is not 0.
Henry

 

[Henry]

obviously to give interest free period make sense for quite a big companies...

What I was expecting is a hint, not finance class. I am glad for any answer and dont take me as complaining!
I am not saying that payment you gave me is not correct.
I just did not get your logic how you split 6000 to 4000 and how you calculated that 6000 will be repaid with 0%. (Like this I would split 700 loan to 420 and 280).

Anyway, thanks for the reply.

 

[Henry]

OK, lets forget this...

I wanted to find equal payments for a loan with specific conditions. It is not about if I want to repay something, but I wanted to understand the way how it is computed in general.
I will stick with my approach that was amortization table similar to the second picture, where I found payment by iterations, seems better than guessing how much would be repaid at 0%.

If I turn it the other way and calculate present value of annuity as sum of c/(1+i)^n where
c=856,41
i for 6M = 0 and following with 1,606%
then I get 10 000 only if I start with n=1 when first applying interest 1,606% = that was originally my question anyway and where I was confused.

Denis, please do not bother answering and consider above as talking to myself - I do not want to read advice, that I should ask car dealer...
Thanks anyway.

 

[jonah2.0]

Hi, I want to calculate monthly payment for a loan that has promotional interest rate 0% for part of the loan.
I believe that I have result, but I would like to confirm, that my approach is correct.
What I have:
loan amount is 10000
i (promotional 1-6M)=0%
i (standard 7-12M)=1,606% p.m.
n =12

What I did - I took annuity formula and put it to Excel to find payment (see the print screen).

First columns are equivalent of pmt function in Excel (to confirm that calculation is OK). Second is the result with promotional rate.
I tried another way - amortization table and then by macro and bisection method found the payment that gives 0 for balanace at the end of the period. This gives same result as here 856,41

What I am not sure about:
- why to apply power of 1 to 6 for second half of the year? Why not 7 to 12, it is in fact 7th period. Do I understand it as first period with applied interest?
- is there "single" formula for this? Something like "pmt" in Excel?
Thank you.

Took a crack at this totally sober and ended up getting puzzled when I wrote an equation of value with the beginning of the 1st month as the comparison date.
Accordingly, I started with
10,000=6R+R*[1-(1.01606)^(-6)]/.01606*1.01606^(-6)
Ended up with R= 896.13
Total nightmare.

Took in some beer and everything seems clearer through my beer goggles. The end of the 6th month is of course the proper comparison date. Thus
10,000-6R=R*[1-(1.01606)^(-6)]/.01606

obviously to give interest free period make sense for quite a big companies...

I can't say I'd agree. You'd forego 6 months of rent?
Unless of course the interest rate is high enough so you'll end up with your desired effective rate to make up for the interest free months.

 

[deg]

Holy of Holies, Henry, no wonder I wasn't making any sense to you:
I took your "1 - 6M" as meaning "1 to $6,000" (M = thousand),
so $6,000 of the loan at 0%!!
We were talking back and forth, each of us talking of something different

I now see what you were doing...0% for 6 months...GOOD WORK; I also get 856.41

Now for the formula you asked about:

The standard payment formula is:
p = ai / (1 - v) where v = 1/(1+i)^n
where:
p = payment
a = amount borrowed
i = interest
n = number of payments

We can adjust that formula by using f = free months.

p = (a - fp)i / (1 - v) where v = 1/(1+i)^(n-f)
Doing the math:
p - pv = ai - fpi
p - pv + fpi = ai
p(1 - v + fi) = ai
p = ai / (1 - v + fi)

With your example, a=10000, i=.01606, n=12, f=6.
Formula will give p = 856.40795.....

Take $30,000 at 1% monthly, for 24 months, 1st 9 months free:
a = 30000, i = .01, n = 24, f = 9
Formula will give p = 1312.04596.....

Sorry for creating the confusion...that'll teach me to read more carefully...

I know this post is a bit old but I am trying to use the formula above and have a question. In the p = (a - fp)i / (1 - v) where do I get the value for p that is multiplied by f in the formula?

In my calculation I did: 30,000-(9*1,250)*.01/1-(1/1+.01)^15 = $1,352.32 slightly different than $1,312.04. The p in my formula was derived by 30,000 / 24 months.

What did I do wrong?

Thanks!