=== Summary of Chapter 9 ===

Okay folks, here's the essence of chapter 9.
Somebody is making a series of payments at regular intervals, and
the money is growing with a specified interest rate. You want to
relate the payments to the value of the money at the beginning
of the payments (the "present value") or the end of the payments
(the final amount).  Here are the variable names that the book uses:

  k = interest rate per period
  n = number of payments
  Y = payment per period
  V = present value (at beginning of payments)
  S = final amount (at end of payments)

Section 9.2 discusses annuities, where you are given the payments
and you are supposed to find the present value or the final
amount. Section 9.3 discusses loans (where you are given the
present value and are supposed to find the payments) and sinking
funds (where you are given the final amount and are supposed to
find the payments).

In summary there are four problems we might be presented with:

  section  problem                    Y      V      S
  =======  ========================   =====  ====   =====
  9.2      present value of annuity   given  find
  9.2      amount of annuity          given         find
  9.3      amortization of loan       find   given
  9.3      sinking fund               find          given

Here's how these quantities are related:
Let a = (1+k) = growth factor per period.

  S = Y*(1+a+a2+...+a(n-1))
  V = Y*(1/a + 1/a2 + ... + 1/an)
    = S/an

To sum such geometric series you multiply
or divide by the growth factor a and subtract:

  a*S-S = Y*(an-1).

So
  
  S = Y*(an-1)/(a-1) and
  V = Y*(1-a(-n))/(a-1).

That is (using that a = 1+k),

  S = Y*((1+k)n-1)/k and
  V = Y*(1-(1+k)(-n))/k


So here is all of chapter 9 in a nutshell (which might be all the
financial math that 99% of the people out there will ever use):

  (A) compound interest (9.1):
  
    P_n = P*(1+k)n, where
  
    P = principal
    n = number of compoundings
    k = interest rate per compounding
        (Recall that k=r*t, where r is the nominal interest rate
        and t is the time per compounding.)
  
  (B) regular payments (9.2 through 9.3):
  
    section  problem                    formula
    =======  ========================   ======================
    9.2      present value of annuity   V = Y*(1-(1+k)(-n))/k
    9.2      amount of annuity          S = Y*((1+k)n-1)/k
    9.3      amortization of loan       Y = V*k/(1-(1+k)(-n))
    9.3      sinking fund               Y = S*k/((1+k)n-1)

    where
    
      Y = payment size,
      n = number of payments,
      k = interest rate per payment (nominal interest rate times time),
      V = present value of payments, and
      S = final value of payments.

That's all!