Finding the Balance Point in a Mortgage

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%, 3-year, $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 TI-84+ 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.

TI-84+ 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

Financial Calculators: Finding a Balance

Blog Entry #101

The Balance of an Annuity

If you have a financial calculator, chances are that you have a Amortization function, which assists you in finding the interest and principal paid and the balance after a set of payments.

Another quick way way to find the balance is to calculate the future value (FV).

Example:

A 20-year loan for $100,000 is issued at 5% annual interest. Find the monthly payment.

Known:
PV (Present Value) = 100,000
I/YR (Annual Interest) = 5%
Periodic Interest = 5/12 %
N (number of payments) = 20 * 12 = 240
FV (Future Value) = 0

Computed (HP 10bII+):
PMT (Payment) = -659.955739217 (-659.96 rounded to 2 places)

Two Ways to Find the Balance

Amortization Function

Enter the range of payments you want to amortize and execute the amortization function/spreadsheet.

For the HP 10bII+, to amortize the first k payments, press:

1 [INPUT] k [Down shift key] [FV] (AMORT)

The HP 10bII+ will display the range of payments to be amortized. Confirm this by pressing the equals key. [ = ].

The total principal paid is given. Press [ = ] to get the total interest paid. Press [ = ] one more time to get the balance.

In short, for the balance of the first n payments:
1 [INPUT] k [Down shift key] [FV] (AMORT) [ = ] [ = ] [ = ]

Future Value Method

To find the balance after the first k payments, enter k as N and compute future value (FV).

For example, if I want to find the balance of a loan after 12 payments, I enter 12 as N, and compute FV.

This is an alternate method, and a possible work-around for anyone having a finance calculator sans an amortization feature.

Remember: The TVM functions uses the cash-flow convention: positive numbers for receipts and negative numbers for payments.

A Comparison of Methods

A recap of the data in our example:

Known:
PV (Present Value) = 100,000
I/YR (Annual Interest) = 5%
Periodic Interest = 5/12 %
N (number of payments) = 20 * 12 = 240
FV (Future Value) = 0

Computed (HP 10bII+):
PMT (Payment) = -659.955739217 (-659.96 rounded to 2 places)

*Note: Signs will be ignored.

Balance after 60 payments:
Amortization Method: 83,454.57
FV Method: 83,454.86

Balance after 120 payments:
Amortization Method: 62,220.85
FV Method: 62,221.52

Balance after 180 payments:
Amortization Method: 34,970.35
FV Method: 34,971.35

I find similar results by using the Hewlett Packard HP 10bII+, Texas Instruments BA II Plus, and the Casio FC-200V.

Why is there a difference?

The TVM module uses a master equation involving N, I/YR, P/Y, PMT, PV, and FV and the financial calculator calls on this master equation to solve for the desired variable.

When using the amortization method, the financial calculator uses a different set of formulas. The formulas have this format:

Interest for the Period = Previous Balance * I/YR% * P/Y^-1

Principal for the Period = Payment - Interest for the Period

New Balance = Previous Balance - Principal for the Period

Prior to calculation, payment and interest are rounded.

For the exact formulas, consult the calculator financial manuals. Casio, Texas Instruments, and Hewlett Packard all provide formulas for TVM and amortization.

Link to the HP 10BII+ Manual: Check the Appendix

Thank you and take care!

Eddie


2012