# appendix Appendix 1 Appendix – Compound Interest by fanzhongqing

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Appendix – Compound Interest:
Concepts and Applications

FINANCIAL ACCOUNTING
AN INTRODUCTION TO CONCEPTS,
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
analysis.
4. Learn how to find the interest rate to satisfy a stated set of
conditions.
5. Begin to learn how to construct a problem from a
Appendix-3
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
Appendix-4
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
decline
 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.
Appendix-5
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.
Appendix-6
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.
Appendix-7
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
future.
Appendix-8
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
Appendix-9
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.
Appendix-10
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.
Appendix-11
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
value:

Fn            where P is the principle
P
(1  r ) n       r is the rate of interest and
n is the number of periods
Appendix-12
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
annually.
 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?
Appendix-13
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.
Appendix-14
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
Appendix-15
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
problems.
   If time periods are not equally spaced or if
the amounts vary, then the series of
payments is not an annuity.
Appendix-16
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.
Appendix-17
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
period.
   Coupon payments on a bond are ordinary
annuities; payment is made after the period.
Appendix-18
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
Appendix-19
5.a. Ordinary annuities -- example
   One good way to understand annuities is to work
the problem as five separate present value problems
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.
Appendix-20
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.
Appendix-21
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
annuity.
Appendix-22
5.c. Mathematical reconciliation

   An annuity due is the same as an ordinary
annuity with each payment shifted forward one
period.
   Since the annuity due is received earlier and
money has time value, the annuity due is more
valuable.
   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
Appendix-23
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:
Pperpetuity=A*(1+1/r)
   Examples of perpetuities include some
bonds.
Appendix-24
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.
Appendix-25
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
methods.
Appendix-26
Appendix Summary

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

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