=== Summary of Section 9.2 ===

| The essential concept of chapter 9 is that loaned money
| grows with time. This leads to the notion of present
| value.  To compare money at different times you have to
| multiply or divide by the appropriate growth factor.
| This concept is known as the "time value of money" (see
| "http://en.wikipedia.org/wiki/Time_value_of_money"), and it
| is the central concept of everyday mathematical finance. The
| notion of present value requires that you specify an interest
| rate and period. (That is, you must specify how the money grows
| with time.)

Present Value: 

| In general, the *present value* of an amount of money A which is
| to be paid at a time T in the future is the amount of money P
| that would grow to A if we invested it now.  To find this amount
| we have to divide A by the investment growth factor.
  
  For example: assuming 25% annual yield,
  the present value of $100 to be paid one year
  from now is the amount P such that
  
    $100 = P*1.25,
  
  Solving for P, we see that the present value of $100 to be paid
  after one year of 25% annual interest is the final amount divided
  by the growth factor:
  
    P = $100/1.25 = $80.
  
  The present value of an amount A to be paid after n periods of time
  with an interest rate per period k is the principal P such that
  
    A = P*(1+k)^n.
  
  Solving for P shows again that the present value is the final
  amount divided by the growth factor:
  
    P = A/(1+k)^n.

Annuities.

| An *annuity* is a scheduled series of regular equal payments.
| For an ordinary annuity (annuity-immediate) the payments are
| made at the *end* of each payment period (typically a month).
| To find the present value of a series of future payments we
| find the present value of each payment and add them up.

  Let Y = size of payments,
  let n = number of payments,
  let k = interest rate per payment period,
  let V_m = present value of payment m, 
  let V = present value of all the payments, and
  let a = (1+k) = growth factor per payment period.

  The present value of the m-th payment is the amount V_m
  that will grow to become Y in m payment periods:

    Y = V_m*a^m;

  To solve for the present value we divide by the growth factor:

    V_m = Y/a^m.

  The present value of all the payments is

    V = V_1 + V_2 + ... + V_n
      = Y/a + Y/a^2 + ... + Y/a^n.

  To find this sum we first need to study geometric series.

Sum of a Geometric Series

  Let S by the sum of the first n powers of a growth factor a:

    S = 1 + a + a^2 + ... + a^n

  This is called a geometric series.  The trick to find S
  is to multiply by a and subtract the original equation:

    a*S =     a + a^2 + ... + a^n + a^(n+1)
  - [ S = 1 + a + a^2 + ... + a^n  ]

  When you subtract these two equations, all the terms on the right 
  cancel except the first and the last, so you get

    S*(a-1) = a^(n+1) - 1.

  Solving for S gives:

    S = (a^(n+1) - 1)/(a-1).

  The formula on page 426 is incorrect.
  (It has "n-1" instead of "n+1".)  Fix it.

  Observe that the sum

    S = 1 + a + a^2 + ... + a^(n-1)

  is given by the same formula, with n replaced by n-1:

    S = (a^n - 1)/(a-1).

Present Value of an annuity (continued).

  Now resume computing the present value of an annuity.  Recall

    V = Y/a + Y/a^2 + ... + Y/a^n.

  Factoring out Y gives:

    V = Y*(1/a + 1/a^2 + ... + 1/a^n).

  Multiply by a^n.  Get:

    a^n*V = Y*(a^n/a + a^n/a^2 + ... + a^n/a^n).

  Recall the general property of division of powers: a^p/a^q = a^(p-q).
  So we get:

    a^n*V = Y*(a^(n-1) + a^(n-2) + ... + 1).

  We see a geometric series on the right.
  Using the formula for the sum of a geometric series,

    a^n*V = Y*(a^n-1)/(a-1)

  Solving for V, we get

    V = Y*(1 - 1/a^n)/(a-1).

  Recall that a = 1+k. Making this substitution gives the formula
  on page 428 of the book for the present value of an annuity
  with payments Y, n payment dates, and an interest rate per
  period k:

    V = Y*(1 - 1/(1+k)^n)/k.

  For example, the present value of an annuity that pays
  $100 every month for 12 months relative with respect to
  an interest rate of 1% per month is

    V = $100*(1 - 1/1.01^12)/.01
      = $1125.507747348463.

Amount of an annuity.

| The amount of an annuity is the "future value" of
| the payments at the end of the last payment period:
| Again, to determine future value an interest
| rate k and payment period t must be specified.  If the
| payments are put in an account accumulating interest
| at rate k and investment period t, the money acccumulated
| in the account at the end of the last payment period
| is the amount of the annuity.

  In particular, using the same variable definitions that we used
  for the present value of a series of payments,

    A = Y*a^(n-1) + Y*a^(n-2) + ... + Y*a^2 + Y*a + Y
      = Y*(a^n-1)/(a-1)
      = Y*((1+k)^n-1)/k,

  in agreement with equation 9.6 on page 427.

  For example, the amount of an annuity that pays
  $100 every month for 12 months relative with respect to
  an interest rate of 1% per month is

    A = Y*((1+k)^n-1)/k,
      = $100*(1.01^12-1)/.01,
      = $1268.250301319698.

  Recall that we previously calculated the present value
  of this annuity to be

    V = $1125.507747348463.

  Observe that the total of all the payments is $1200.
  Notice that the present value of the total payments is
  less than the total payments, and the amount is more.
  That should make sense to you.

| In general the present value of the annuity is the
| present value of the amount of the annuity.

  Let's check that.  We need that

    A/(1+k)^n = V.

  You can see that this is true in the formulas.
  For our specific case, we have that the growth factor is

    (1+k)^n = 1.01^12 = 1.12682503013197,

  and $1268.250301319698/1.12682503013197 = $1125.507747348463,
  as needed.