Greetings!
Today's blog entry is about mortgages. When someone takes out a mortgage, or any loan that is to paid over time, the lender charges interest as a price for borrowing the money. Each payment that is made has two components: principal, that actually reduces the loan balance, and interest.
The beginning of the term is where the most amount of interest is paid. If the length of the loan is long enough and the interest rate is high enough, the interest eats more of the payment.
Example 1:
I take a loan for 5%, 3year, $20,000 to finance the purchase of a computer network. Payments are to be made at the end of each month. Assume that there is no balloon amount. I would have a monthly payment of $599.41.
For the first payment I would pay $83.33 in interest ($20,000 × .05 / 12), which results in $516.08 ($599.41  $83.33) going towards the principal. So after the first payment, the balance is $19,483.92 ($20,000.00  $516.08). Clearly, each payment contribute to principal than interest. Happy day.
Unfortunately, mortgages are not so friendly.
Example 2:
A couple finances a purchase of a home for $264,000. They were able to secure a 30 year term at 4%. Payments are due at the end of reach month which come out to $1,260.38 a month.
For the first payment, the couple would pay $880.00 in interest, leaving only $380.38 for the principal.
Fast forward five years, the 61st payment (with balance of $238,781.43), the amount of payment contributing to interest is $795.94, leaving $464.44 for the principal.
Fast forward another five years, the 121st payment (with balance $207,989.65), the amount that contributes towards interest is $693.30, with $567.08 for the principal.
In fact it will take until the 152nd payment is made (12 years, 8 months) that more of the payment is contributed towards principal than interest. I refer to payment 152 as the balance point.
Balance Point: The Math Behind It
To find the balance point, we will take a two step approach:
1. Find the balance of the loan where the amount of interest and principal paid is equal. I call this the magic balance.
2. Use the balance to determine the payment number.
Step 1 is fairly simple.
For any payment:
I = B × R / PY
P = PMT  I = PMT  B × R / PY
Where:
I = portion of the payment that goes towards interest
P = portion of the payment that goes towards principal
PMT = payment amount
B = balance of the previous period
PY = payments per year, usually 12
R = interest rate, in decimal. (i.e. 5% = .05)
To find the required magic balance, as I will name it, equate P and I.
P = I
PMT  B × R / PY = B × R / PY
PMT = 2 × (B × R / PY)
which implies that
B = (PMT × PY) / (2 × R)
Now we can proceed to Step 2, where we can find our balance point.
Let B0 = initial loan amount, B1 = balance after 1 payment, B2 = balance after two payments, and so on.
Simply enough:
B1 = B0  P0
B1 = B0  (PMT  B0 × R / CY)
B1 = B0 × (1 + R / CY)  PMT
And
B2 = B1  P1
B2 = B1  (PMT  B1 × R / CY)
B2 = B1 × (1 + R / CY)  PMT
If we want B2 in terms of B0...
B2 = (B0 × (1 + R / CY)  PMT) × (1 + R / CY)  PMT
B2 = B0 × (1 + R / CY)^2  P × (2 + R / CY)
Quickly we realize if we what to find B3, B4, etc...in terms of B0, the expression will grow complicated very quickly.
Thankfully, our TVM (Time Value of Money) solvers will come to our rescue! We can find the balance point simply by setting our TVM variables as such:
I/YR = R
PV = B0
PMT = PMT
FV = B from Step 1
PY = PY
Solve for N.
The balance point is ceiling(N). The ceiling function of N is N rounded up to the next integer. In essence, if N > 0, ceiling(N) = integerpart(N) + 1.
Interesting...
1. The amount of the loan has no bearing on the balance point.
2. A 5%, monthly payment loan has a balance point of 194 months (16 years, 2 months) in a 30 year term, but a balance point of 314 months (26 years, 2 months) in a 40 year term.
3. Not surprising, the balance point is bigger when the interest rate is higher. At 7%, the balance point is 241 months (20 years, 1 month) for a 30 year term, and 361 months (30 years, 1 month) for a 40 year term.
4. The terms "magic balance" and "balance point" are names I created. I am not aware of any official names used for these values (if you know, I would appreciate it).
Coming up is a program that finds the balance point. I use the TI84+ for this program.
Note that the tvm_Pmt and tvm_N commands are from the Finance App, which is accessed by pressing APPS, 1, then 2 and 5, respectively.
TI84+ Program PMTOFBAL
: Input "LOAN:",P
: Input "I/YR:", I
: Input "NO. PMTS:", N
: Input "P/Y:", C
: tvm_Pmt(N,I,P,0,C,C) →M
: M*C/(.02I) → X
: tvm_N(I,P,M,X,C,C) → O
: iPart(O)+1→O
: Disp "PAYMENT:", M
: Pause
: Disp "MAGIC BALANCE:", X
: Pause
: Disp "BALANCE POINT:",O
: Pause
Input: Loan amount, annual interest rate, number of payments (years × payments per year), payments per year
Output: Periodic payment, magic balance, and balance point. If the balance point is negative, this means for each payment, more of it will contribute towards principal than interest for the entire loan.
Thank you very much. Have a great weekend.
Eddie
This blog is property of Edward Shore. © 2012
A blog is that is all about mathematics and calculators, two of my passions in life.
Friday, July 27, 2012
Finding the Balance Point in a Mortgage
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