Mortgage, 25 years, interest Cpd s-a @ 6%, monthly payments

Ok I have this question

Matthew is looking to purchase a house that is currently priced at $520,000 and he is expecting to put a 10% down payment. A bank has offered him 6.5% mortgage rate with a 5-year term amortized over 25 years. Interest is compounded semi-annually. Matthew is uncertain of what his monthly payments will be.

a)What are Matthews’s monthly payments?
b)Assume Matthew receives and contributes $2,000 lump-sum from his investments, every 6 months for the first 5 years. Calculate the remaining principle amount on the house after 5 years.
c)Based on part b) how long will it take Matthew to pay off the mortgage?
d)How much interest has Matthew saved by making the extra payments?

The work I have done is:

Picture 1:
http://img393.imageshack.us/img393/448/top020retqd3.png
and
Picture 2:
http://img384.imageshack.us/img384/7530/qwdwfjh3.png

Can someone please check if my monthly payment amount is correct?
My monthly payment amount is $3,314.77.

I also tried part b). I was trying to figure out how much the simple interest and compound interest would be at the 6th month (where the $2,000 lump sum would be paid. In Picture 2, I think I'm trying something that isn't supposed to happen... (I'm thinking that the 12th payment would be more than $3134.7.., is this correct?)

But, it came to an error. I don't know where I am making the mistake.

Thank you beforehand for the help.

a): 3134.775291 is CORRECT. Good work.

b) can't follow what you did!
Easiest way is assume you have 3 accounts at a rate of 6.5% cpd semi-annually:
1: 468000 initial deposit; after 60 months: 468000(1.00534474)^60 = 644,386.5342.....
2: monthly deposits of 3134.775291...; after 60 months: 3134.775291[(1.00534474)^60 - 1] / .00534474 = 221,054.5218....
3: semiannual deposits of 2000 (rate = .065/2 = .0325); after 10 deposits: 2000[(1.0325)^10 - 1] / .0325 = 23,193.4956....

644,386.5342 - 221,054.5218 = 423,332.0124 : that's balance owing if no extra payments.

423,332.0124 - 23,193.4956 = 400,138.5168.... : that's balance owing after the 10 extra payments.

And the interest saved to that point is 23,193.4956 - (2000 * 10) = 3,193.4956

On these:
c)Based on part b) how long will it take Matthew to pay off the mortgage?
d)How much interest has Matthew saved by making the extra payments?

We need to assume rate will remain same; but rate is guaranteed for 5 years only...so ??????
Also, we need to assume the payment will continue @ 3134.77... will it ?????

From when I did part b), I tried to separate the total monthly payment of $3,134.775291 into the amount going towards the principal amount of the actual mortgage, the "simple" interest and the compound interest (which I determine is the "interest on interest").

I tried to do this because in the question of part c), it asks to calculate for the remaining principle amount on the house...

So we don't really have to find the *true* principle amount ( I think I'm confusing myself with this idea of the principle amount too much )..
And thus, just find the balance owing at 5 years time, of the mortgage - monthly payments - semiannual payments.

Wow that part didn't seem hard at all when I look at it now
My teacher says to use it finding the EPR, and EAR first. Does it matter and what are the equivalents of those 2 rates in our example here? I'm not sure with what they are.

Now when I'm solving part c, I'm thinking its safe to assume that the interest rate will still be 6.5%, and the annuity of the mortgage's principle amount has dropped by Matthew making the semiannual payments.

In response to your additional help here:

"We need to assume rate will remain same; but rate is guaranteed for 5 years only...so ??????"
I'm not sure what you mean by this, but the amortization of 5 years, I was told is just Matthew re-negotiating the mortgage with the bank every 5 years, so there's not really much telling to what the interest rate could be, so I would assume the rate remains the same.

"Also, we need to assume the payment will continue @ 3134.77... will it ?????"
I'm assuming the payment will continue at 3134.77, because my teacher has told me that the monthly payments cannot change in class. I don't know why it would change (but would probably change due to a change in the rate, in which case I've assumed the rate will remain the same).

With this being said, I use the following to try to find out the answer to part c)Based on part b) how long will it take Matthew to pay off the mortgage?

Present value of an annuity = C {1-[1/(1+r)^t]/r}
C = amount of monthly payment
r = interest rate
t = time

400,138.5168 = 3134.77291 {1- [1/(1.00534474)^t]/0.00534474}

Myself, not knowing how to solve this, I use a financial calculator and find that t = 215.0688275
Thus, it will take Matthew 215 more months to pay off the mortgage.

Am I at all close to the right answer?
Thanks.

Ok then; from your comments, then the original posted problem really is:
"Matthew obtained a $468,000 mortgage at a rate of 6.5% compounded semi-annually,
amortized over 25 years. What is Matthew's monthly payments ?"
(there is no need for a teacher to complicate it further with a down payment and a 5 year term!)

The formula to calculate n (number of payments) is obtained from the payment formula:
P = Ai / [1 - 1/(1 + i)^n] ; solving for n gives n = LOG[P / (P - Ai)] / LOG(1 + i)
Substitute A = 400138.52, P = 3134.78, i = .005345 to get n = 215.0688; so 215 months plus change.
So, in other words, same as Matthew obtaining a mortgage of $400,138.52 at 6.5% cpd semi-annually,
and offering a monthly payment of $3,134.78; how many payments will pay if off?

The EAR (Effective Annual Rate) and EPR (Effective Periodic Rate) are used: EPR = (1 + .065/2)^1/6 ;
the EAR: (1 + .065/2)^2 - 1 = .06605625... is part of that, though "unseen".

To calculate "interest saved", simply subtract the actual payments made from 300*3134.78.
Actual will be: (60 + 215)*3134.78 + 10*2000; result will be ~$58,370

By the way (in case helpful) using the "5 year term" as the guaranteed period at 6.5%,
you can look at this mortgage as being the Bank purchasing a 5 year bond from Matthew:
1: Bank purchases bond at selling price of $468,000 ; bond specifies:
2: 60 monthly coupons of $3,134.78
3: additional 60th month coupon of $423,332.01

"To calculate "interest saved", simply subtract the actual payments made from 300*3134.78.
Actual will be: (60 + 215)*3134.78 + 10*2000; result will be ~$58,370"

Is there supposed to be interest added on the 20,000?
I don't think those 2 amounts can be subtracted like that
This is what you are saying:
940,432.5873-862,063.205 = 78,369.3823 - 20,000 ~ 58,370

I'm thinking you can;t do 940k-862k yet, because those amounts are in different points in time, and this shouldn't be allowed due to the concept of "time value of money".

I.e, the 940k is after 25 years or 300 months have passed and the 862k is after about 23 years or 275 months have passed

So I'm thinking of either bringing the 862k amount to the point of where 25 years have passed or bringing the 940k to where 275 months have passed...
Is this what is supposed to be done?

Case#1: no extra payments
the $468,000 is repaid with 300 payments of $3,134.78

Case#2: extra 10 semi-annual payments of $2,000
the $468,000 is repaid with 275 payments of $3,134.78 plus 10 payments of $2,000

In both cases, $468,000 is borrowed.
So the interest saved can ONLY be the difference of total payments.

Regarding your questions/comments:
> Is there supposed to be interest added on the 20,000?
No idea what you mean.

> This is what you are saying:
> 940,432.5873-862,063.205 = 78,369.3823 - 20,000 ~ 58,370
Yes, that's what I'm saying: 940,432.5873 - (862,063.205 + 20,000) = ~ 58,370

> I'm thinking you can;t do 940k-862k yet, because those amounts are in different points in time,
> and this shouldn't be allowed due to the concept of "time value of money".
That has nothing to do with the question: how much interest did Matthew save?
That question simply means: Matthew's extra 10 payments reduced the interest he paid to the Bank by how much?

We can get "cute" by bringing in stuff like:
1: well, Matthew really saved less than $58,370 since he has to borrow these extra $2,000 payments
2: well, Matthew really saved more than $55,370 since he was able to invest payments #276 to 300
But that has nothing to do with the original question.

Hey TK, can you make this clearer; the one finger I type with is getting tired :x

Denis said:

And the interest saved to that point is 23,193.4956 - (2000 * 10) = 3,193.4956

Click to expand...

Great insight!!! annuity_man, if you ever manage to duplicate the process that went into this innocent looking statement, as I’m sure you will in the near future, you’d probably go down on your knees and say “I’m not worthy, I’m not worthy”, in imitation of Wayne and Garth from the movie Wayne’s world.

annuity_man said:

"To calculate "interest saved", simply subtract the actual payments made from 300*3134.78.
Actual will be: (60 + 215)*3134.78 + 10*2000; result will be ~$58,370"

Is there supposed to be interest added on the 20,000?
I don't think those 2 amounts can be subtracted like that
This is what you are saying:
940,432.5873-862,063.205 = 78,369.3823 - 20,000 ~ 58,370

Click to expand...

I think you need to sleep on this and let subconscious assimilation kick in. Since you didn’t have any difficulty dealing with the concept of a general annuity, this should give you no trouble. To give you another perspective, consider the following:

With R = 3,134.78, and bearing in mind that Total Interest Paid = n*(Regular Payment) - Principal

“interest saved” =
(Total interest paid for 300 months with no extra payment of $2000 at the end of six months for the first 5 years) minus (Total interest paid for 275 months with extra payment of $2000 at the end of six months for the first 5 years)
?
“interest saved” =
(300*R – 468,000) –
{[5*R + (R+2,000) + 5*R + (R+2,000) +5*R + (R+2,000) +5*R + (R+2,000) +
5*R + (R+2,000) +5*R + (R+2,000) +5*R + (R+2,000) +5*R + (R+2,000) +5*R + (R+2,000) +
5*R + (R+2,000) + 215*R] – 468,000}
?
“interest saved” =
(300*R – 468,000) – [(60*R + 10*2,000 + 215*R) - 468,000]
?
“interest saved” =
(300*R – 468,000) – [(275*R + 10*2,000) - 468,000]
?
“interest saved” =
300*R - 275*R - 10*2,000

Note: If I may add on the work supplied by Denis, the smaller payment at the end of the 216th month should be included in the actual payments for a better approximation of the “interest saved”. Accordingly,
The 216th payment is

{400,138.5168(1.00534474)^215 - 3134.775291[(1.00534474)^215 - 1] / .00534474}(1.00534474) =
$216.29.

Thus,
“interest saved” = 300*R - 275*R - 10*2,000 - 216.2918836 ? 58,153.09

jonah said:

Note: If I may add on the work supplied by Denis, the smaller payment at the end of the 216th month should be
included in the actual payments for a better approximation of the “interest saved”.
Accordingly, The 216th payment is
{400,138.5168(1.00534474)^215 - 3134.775291[(1.00534474)^215 - 1] / .00534474}(1.00534474) = $216.29.
Thus,
“interest saved” = 300*R - 275*R - 10*2,000 - 216.2918836 ? 58,153.09

Click to expand...

Right on, jonah; with no rounding of interest and payments: 58,153.0910780435414596248.... :wink:

I rounded the thing out to 215 payments (skipped the small 216th) because of (you guessed it) laziness!!