Mortgage Payments, etc.





Kerry Back

Questions about loans

  • I know how much I want to borrow and for how many years and what the rate will be. What will my payment be?
  • I know how much I can afford to pay and for how many years the loan will be and what the rate will be. How much can I borrow?
  • I know how much I want to borrow and what I can afford to pay and for how many years the loan will be. What rate do I need to get?

  • Suppose we borrow for \(T\) years at an annual rate of \(r\) with monthly payments.
  • Our monthly rate will be \(r_m = r/12\).
  • Each month, we incur interest equal to \(r_m\) times the beginning-of-month balance.
  • Our payment goes first to interest, then the remainder to principal, reducing the outstanding balance.
  • Next month’s interest charge is smaller, so more goes towards principal.
  • The last payment is equal to the beginning-of-month balance plus interest on the balance.

  • Call the amount borrowed (principal of the loan) \(B\) and the payment \(P\).
  • What we owe just after the first payment is \((1+r_m)B-P\).
  • What we owe just after the second payment is

\[(1+r_m)((1+r_m)B - P) - P\]

  • This is \((1+r_m)^2 B - (1+r_m)P - P\), which is the future value of the amount borrowed minus the future value of the payments (as of the end of the second month).

Future values

  • The balance after the last payment is zero.
  • So, for future values as of the loan maturity,

\[0 = \text{FV(Principal)} - \text{FV(Payments)}\]

Equivalently,

\[\text{FV(Principal)} = \text{FV(Payments)}\]

Present values

  • Because future values are equal, present values are also equal.
  • The present value of the amount borrowed is just the amount (prinicpal).
  • So loan terms are related by Principal = PV of payments.
  • The PV of payments is

\[P \times \left(\frac{1}{1+r_m} + \frac{1}{(1+r_m)^2} + \cdots + \frac{1}{(1+r_m)^{12 \times T}}\right)\]

Annuity factor

  • The sum of PV factors is called the annuity factor AF.
  • So, Principal = Payment \(\times\) AF.
  • There is a simpler formula for the annuity factor (sum of a finite number of terms of a geometric series):

\[\text{AF} = \frac{1}{r_m}\left(1 - \frac{1}{(1+r_m)^{12 \times T}}\right)\]

  • It is not necessary to remember the AF formula.

Balloon payments

  • A loan might have a balloon payment, meaning a lump sum payment at the end.
  • Loan terms are related by Principal = PV of monthly payments + PV of balloon
  • PV of balloon is

\[\frac{\text{Balloon}}{(1+r_m)^{12 \times T}}\]

Numpy financial

  • You may need to do
    pip install numpy-financial
  • Convention is that payments, including balloon, are negative.
  • Principal received is positive. It is called pv.
  • Ballon is called fv.
  • Functions are rate, pmt, pv, fv.
  • Same functions are in Excel.

Example 

import numpy_financial as npf
 
nper = 30 * 12       # 30 year loan
principal = 400000   # borrow $400,000
pmt = -1500          # pay $1,500 per month
rate = 0.04 / 12     # annual rate is 4%
balloon = 0          # no balloon

We’ll calculate each of the last four, one at a time, given the other inputs.

# Question 1: what will my payment be?
PMT = npf.pmt(rate=rate, nper=nper, pv=principal, fv=balloon)
 
# Question 2: how much can I borrow?
PRINCIPAL = npf.pv(rate=rate, nper=nper, pmt=pmt, fv=balloon)

# Question 3: what will my rate be?
RATE = npf.rate(nper=nper, pv=principal, pmt=pmt, fv=balloon)

# And, how much of a balloon do I need?
BALLOON = npf.fv(rate=rate, nper=nper, pmt=pmt, pv=principal)

If BALLOON>0, you overpaid the bank and should get money back.