Summary of Section 9.1 (Interest)

The rate at which money grows is called the interest rate,
and the interest rate depends on the type of loan involved.
(Generally, lower interest rates are paid for loans where the
lender is more free to demand the money back at an unexpected
time and when the borrower is more likely to pay back.)

Simple Interest

  The first concept is simple interest: if you loan
  out an amount of money P you can expect to be paid
  back the same amount plus an additional amount I
  called interest:
  
    Let P = principal = amount of initial loan,
    let I = interest, and
    let A = amount paid back.
  
  Then
  
    A = P + I.
  
  The interest is proportional to the principal:
  
    I = k*P,
  
  where the proportionality constant k is called the *interest
  rate* for the investment period (the "percent interest").
  That is, k=I/P, i.e., percent interest equals interest per
  principal. So
  
    A = P + k*P, i.e.,
    A = P*(1+k).
  
  That is, the amount paid back is the principal P times
  a growth factor (1+k).  Define
  
    a = growth factor = A/P = 1+k.
  
  Generally the longer you lend out money the more interest
  you expect to earn.
  
    Let t = time of the investment.
  
  For a simple loan the interest rate is proportional to the
  period of time t of the investment:
  
    k = r*t,
  
  where the proportionality constant r is the nominal interest
  rate per time. That is, the nominal interest rate is defined by
  r = k/t = I/(Pt) = interest per principal per time.
  
  So the amount of interest earned is
  
    I = P*r*t
  
  and the growth factor is
  
    a = 1+rt.
  
  In brief:

    P = principal (amount loaned out),
    t = time of loan,
    r = interest rate per time (interest per principal per time) = k/t = I/(P*t)
    k = interest rate per investement period (interest per principal) = I/P,
    I = interest (amount paid back in addition to the principal),
    a = growth factor = A/P = (1+k), and
    A = final amount (amount paid back).

  For example: consider $100 lent at 12% annual interest for 3 months.
  We have:
  
    P = $100,
    t = 3 months,
    r = 12% per year,
    k = r*t = 3%,
    I = P*k = $3,
    a = 103%, and
    A = P + I = $103.

Compound interest:

  Generally when money is paid back all the money including the
  interest is reinvested. Each time the money is invested it gets
  multiplied by a growth factor. The result is that money tends
  to grow exponentially with time.

  An important instance of this is compound interest:
  
    Let t = investment period = time period between "reinvestments",
    let n = total number of time periods,
    let k = interest rate per time period
    let P = P_0 = principal, and
    let P_m = amount after m time periods
      (which becomes the principal of the (m+1)th period).

  Then

    P_1 = P_0 * (1+k),
    P_2 = P_1 * (1+k) = P_0 * (1+k)^2,
    P_3 = P_2 * (1+k) = P_0 * (1+k)^3,
       ...
    P_n = P_0 * (1+k)^n,

  which is the formula on page 415 of the text.
  In other words, the growth factor after n steps is
  the growth factor after 1 step raised to the n-th power.

  To make this formula clear,

    let T = n*t = total time of investment.

  Then the interest rate per period is k = r*t = r*T/n
  and the final amount is

    P_n = P * (1+r*T/n)^n.

  For example: consider $100 lent at 12% annual interest for 12 months
  compounded quarterly (every 3 months).  We have:

    principal:                   P = $100,
    total time interval:         T = 12 months,
    time period per compounding: t = 3 months,
    times compounded:            n = 12/3 = 4 times,
    interest rate per time       r = 12% per year,
    interest rate per period:    k = r*t = 3%,
    growth factor per period:    a = (1+k) = 103%, and
    investment growth factor:    (1+k)^n = 1.03^4 = 1.12550881.

    P_0 = $100
    P_1 = $103        = 103% * $100      = 1.03^1 * $100
    P_2 = $106.09     = 103% * $103      = 1.03^2 * $100
    P_3 = $109.2727   = 103% * $106.09   = 1.03^3 * $100
    P_4 = $112.550881 = 103% * $109.2727 = 1.03^4 * $100

Annual Percentage Yield (APY)

  An interest rate is not fully specified unless you state how
  often the interest is compounded. Interest grows faster if you
  compound more frequently.

  To compare compound interest rates we compare how much the money
  would grow over a standard length of time: one year.
  The percent interest earned over one year is calle the
  Annual Percentage Yield (APY).

  Suppose we know that an investment with principal P yields
  an amount A at the end of a period of T years.  To find
  the annual percentage yield we pretend that the interest
  is compounded annually.  In this case the final amount would be

    A = P*(1+k)^T,
  
  where k is the interest rate for an investment period of one year.
  Solving this equation for k gives:

    k = (A/P)^(1/T)-1.

  The annual percentage yield is k times 100, so

    API = 100*((A/P)^(1/T)-1),

  which is the formula on page 417 of the book.