appendix Appendix 1 Appendix – Compound Interest by fanzhongqing



             Appendix – Compound Interest:
               Concepts and Applications

                      METHODS, AND USES
                         10th Edition

              Clyde P. Stickney and Roman L. Weil
Appendix-2            Learning Objectives

 1. Begin to master compound interest concepts of future
    value, present value, present discounted value of single
    sums and annuities, discount rates, and internal rates of
    return on cash flows.
 2. Apply those concepts to problems of finding the single
    payment or, for annuities, the amount for a series of
    payments required to meet a specific objective.
 3. Begin using perpetuity growth models in valuation
 4. Learn how to find the interest rate to satisfy a stated set of
 5. Begin to learn how to construct a problem from a
    description of a business situation.
                  Appendix Outline
1. Compound interest concepts
2. Future value concepts
3. Present value concepts
4. Nominal and effective rates
5. Annuities
    a. Ordinary annuities or annuities in arrears
    b. Annuities due
6. Perpetuities
7. Implicit interest rates: finding internal rates
   of return
Appendix Summary
               1. Compound interest concepts
   A dollar to be received in the future is not the same as
    a dollar presently held, because of:
      Risk -- you may not get paid
      Inflation -- the purchasing power of money may
      Opportunity cost -- cash received today can be
       invested and turn a positive return
   Compound interest concepts (a.k.a. present value or
    time value of money or discounted cash flow) are
    mathematical methods of ascribing value to a future
    cash flow recognizing that a future cash payment is
    not as valuable as the same amount received earlier.
              1.a. Compound interest concepts
   All three factors--risk, inflation and opportunity cost--
    can be captured in a single number, the interest rate.
   That is, the interest rate contains a component to
    compensate for risk and inflation and alternative
    opportunities for investment.
   The time value of future cash flows can be measured
    by the amount of interest the cash flow would earn at
    an appropriate rate of interest.
   Also, interest earned accumulates and earns interest
    itself -- this is called compound interest.
               1.b. Compound interest example
   Consider a $10,000 loan at 12% interest for one year.
   Simple interest would be 12% divided by 12 months or
    1% per month. One percent of $10,000 is $100 so the
    interest would be $100 per month or $1,200 for the year or
    $11,200 in total including the principle.
   If you were entitled to the interest each month, you could
    withdraw the interest. If you did not, then that interest
    itself has time value and should earn further interest. The
    total cost of the loan under monthly compounding of the
    interest is $11,268.25. (You will learn how to compute this
    value in the next section).
   This is a small increase, $68.25, but it is an increase and
    could be significant for long periods or high interest rates.
                2. Future value concepts

   The future value of one dollar is the amount to
    which it will grow at a given interest rate
    compounded for a specified number of periods.
   The future value of P dollars is P time the future
    value of one dollar.
   The future value F is considered the equivalent
    in value of the present value P because the F
    will not be received until some time in the
             2. Future value concepts (cont)
   P dollars invested at r percent interest will grow to
    P(1+r) at the end of the first period.
   If this amount, P(1+r), continues to earn r percent
    interest, the at the end of the second period it will
    be P(1+r)(1+r) which is P(1+r)2.
   In like manner, it will grow to P(1+r)3 in 3 periods
   And in general it will grow after n periods to the
    future value Fn given by:
    Fn = P(1+r)n where P is the principle
                            r is the rate of interest and
                          n is the number of periods
                2. Future value (example)
   Consider a certificate of deposit, CD, which pays a
    nominal rate of 6% per year compounded monthly.
   You invest $5,000. How much will your CD be
    worth when it matures in one year?
     Fn = P(1+r)n
     since the CD compounds monthly,
         r = 6% /12months = 0.5 % per month
         n = 1 year * 12months = 12 periods
    Fn = $5000(1.005)12 = 5000(1.061678) = $5308.39
    which is a little better than 6% simple interest.
   Thus, your CD will earn $308.39 on $5000.
                  3. Present value concepts
    Present value is the reverse of future value.
    If the future value of x dollars is y; then the present
     value of y dollars is x.
    Present value answers the question, how much must
     be invested to grow at r rate of interest
     compounded for n periods.
    The present value P is considered the equivalent
     in value of the future value F because the F will
     not be received until some time in the future.
                  3. Present value concepts

    P dollars invested at r percent interest for n periods
     will grow to P(1+r) n.
    Recall that the future value is given by:
                     Fn = P(1+r)n
    Solving for P gives the equation for the present

           Fn            where P is the principle
        (1  r ) n       r is the rate of interest and
                         n is the number of periods
                3. Present value example
   You hold a bond which pays no interest but will
    pay $10,000 upon maturity in three years. You
    need cash now, so you try to sell the bond. A bank
    says that they can’t pay you $10,000 because money
    has time value, but that they will pay you the
    present value discounted at 9% compounded
  How much is the bank offering you?

 P = Fn (1+r)-n = 10000 (1.09)-3 = 10000/1.295 = $7,722
  Thus, the bank will pay you $7,772 for your $10,000
    bond. Is this a good price?
               4. Nominal and effective rates

   By convention and subject to some federal regulations,
    many interest rates are stated as an annual rate and do
    not include the effects of compounding. This rate is
    called the nominal rate of interest.
   The rate which includes the effects of compounding is
    called the effective rate.
   As we saw in an earlier example, 12% per year nominal
    rate of interest compounded monthly actually yields
    12.68% return because of compounding effects.
   Nominal rates are given for simplicity and are almost
    always stated as an annual rate for purposes of
    comparing different alternative rates.
                  4. Effect of compounding periods

   What difference does the compounding period make if
    the nominal rate is the same?
   Consider the following loans all with a 12% nominal or
    annualized rate of interest:
                               number of periodic rate    effective
        Compound Period         periods   of interest        rate
     compounded annually                1           12        12.0000
     compounded quarterly               4             3       12.5509
     compounded monthly                12             1       12.6825
     compounded weekly                 52       0.2307        12.7341
     compounded daily                 365       0.0329        12.7475
     compounded every minute     525600    0.0000228          12.7497
     compounded continuously      ---          ---            12.7497

   Yield increases as the compounding period is shortened
                      5. Annuities

    An annuity is a series of equal payments, one
     per period equally spaced through time.
    Examples include monthly rental payments,
     semiannual corporate bond coupon
     payments and mortgage payments.
    Mathematically, an annuity can be solved as
     the sum of individual compound interest
    If time periods are not equally spaced or if
     the amounts vary, then the series of
     payments is not an annuity.
                       5. Annuities (cont)
    Annuity concepts are important in the accounting for
     bonds and leases.
    The present value of an annuity is its present day cash
     value -- conceptually you can sell or buy an annuity for
     this value.
    The future value of an annuity is the amount to which
     payments will grow if invested an left to compound.
    The non-discounted value of an annuity is the sum of
     the payments which is the number of payments times
     the payment amount.
    Annuities are of two types:
       Ordinary annuities, or
       Annuities due.
                 5.a. Ordinary annuities

     Ordinary annuities payments are due at the
      end of each period.
     Consider an ordinary annuity of $100 per
      period for five periods:

     The payments are made at the end of each
     Coupon payments on a bond are ordinary
      annuities; payment is made after the period.
              5.a. Ordinary annuities -- example

    Consider the same ordinary annuity of
     $100 per period for five periods:

    What is the present value if the appropriate
     rate of interest is 7%?
    This can be solved by several methods:
       Present value tables
       Computers or calculator
       Formula
                5.a. Ordinary annuities -- example
    One good way to understand annuities is to work
     the problem as five separate present value problems
     and then add the results:
     PVannuity = PV1 + PV2 + PV3 + PV4 + PV5
 = 100(1.07)-1 +100(1.07)-2 +100(1.07)-3 +100(1.07)-4 +100(1.07)-5
 = 100/(1.07) +100/(1.145) +100/(1.225) +100/(1.311) +100/(1.403)
 = 93.46 + 87.34 + 81.63 + 76.29 + 72.30 = $410.02
    Thus, the non-discounted value of the annuity is the
     sum of the payments ($500), but the value
     discounted at 7% is $410.02.
                  5.b. Annuities due

    Annuities due payments are due at the
     beginning of each period.
    Consider an annuity due of $100 per period
     for five periods:

    The payments are made at the beginning of
     each period.
    A monthly rent payment is an annuity due;
     you pay in advance of usage.
                    5.b. Annuities due -- example
    This problem is similar to the ordinary annuity
     except that all payments are moved forward by one
     period. The first payment is received immediately
     so it is not discounted. Note that using zero to
     designate the present makes the formula work:
    PVannuity = PV0 + PV1 + PV2 + PV3 + PV4
 = 100(1.07)-0 +100(1.07)-1 +100(1.07)-2 +100(1.07)-3 +100(1.07)-4
 = 100/(1) +100/(1.07) +100/(1.145) +100/(1.225) +100/(1.311)
 = 100 + 93.46 + 87.34 + 81.63 + 76.29 = $437.72
    Notice that the present value of the annuity due is
     exactly 1.07 times the present value of the ordinary
              5.c. Mathematical reconciliation

    An annuity due is the same as an ordinary
     annuity with each payment shifted forward one
    Since the annuity due is received earlier and
     money has time value, the annuity due is more
    Since each payment is shifted by one period, you
     can adjust from an annuity due to an ordinary
     annuity (or back) by the following formula:
        annuity due = (1+r)*ordinary annuity
                     6. Perpetuities

    Perpetuities are annuities that last forever.
    There are few real perpetuities, but they give
     good insight into annuities.
    The present value of a perpetuity is:
    Examples of perpetuities include some
     Canadian and some British government
                  7. Implicit interest rates

    The present value of a lump sum problem
     has four components:
      P, the present value
      F, the future value
      r, the rate of interest and
      n, the number of periods
    Which are related by the formula:
        Fn = P(1+r)n
    Any three of the components determines the
     fourth, or you can solve for any component if
     you know the other three.
               7. Implicit interest rates (cont)

    In implicit interest rate problems, we solve for
     the interest rate.
    That is, given a P, V and the number of
     periods, the r which makes the equation
     balance is know as the implicit interest rate,
     a.k.a. the internal rate of return.
    There is often no direct solution to these
     types of problems, instead, the solution is
     reached through iterative mathematical
                   Appendix Summary

     This appendix introduces compound interest
      problems and the related problems of present
      value, discounted cash flows and time value of
     The applications of present value and future
      value of both an annuity and a lump sum are
     Perpetuities and implicit interest rates are
     These methods are very valuable to the
      accountant in valuing liabilities.

To top