2504 Essentials of Business Finance

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Chapter 6

• 2504 Essentials of Business Finance

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Basic Idea for the Future Value of Multiple Cash
Flows

• FV (in two years) of todays deposit of A and
  another deposit of B in one year, given the
  interest rate of r is

0
1
2
Time (year)
A
A(1r) B A(1r)B
(A(1r)B)1r)
Time (year)
A
A(1r)2 B(1r) A(1r)2B(1r)
B

• Two ways (1) compound the accumulated balance
  forward one period at a time (2) calculate the
  FV of each cash flow and adding them up

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Basic Idea for the Present Value of Multiple Cash
Flows

• PV of an asset that will generate A in one year
  and additional B in two years, given the
  discount rate of r, is

0
1
2
Time (year)
B
B/(1r) A B/(1r)A
(B/(1r)A)/1r)
Time (year)
B/(1r)2 A(1r) B/(1r)2A(1r)
B
A

• Two ways (1) discount the last cash flow back
  one period and add it to the next-to-the-last
  cash flow (2) calculate the PV of each cash flow
  and adding them up.

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Basic Idea for the FV/PV of Multiple Cash Flows
Caution!

• Always draw the time line
• Assume all cash lows occur at the end of each
  period, and we are at the end of current period,
  time 0 (except an annuity due). An investment
  has the first-year (period) cash flow of 100,
  the second-year (period) cash flow of 200, and
  the third-year (period) cash flow of 300 means

0
1
2
3
100
200
300

• Note that
• PV of FV of multiple cash flows PV of the
  multiple cash flows
• FV of PV of multiple cash flows FV of the
  multiple cash flows.

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Ordinary Annuity

• A series of constant (level) cash flows that
  occur at the end of each period for some number
  of periods is called an ordinary annuity or,
  more correctly, the cash flows are in ordinary
  annuity form.

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Present Value Interest Factor for Annuity

• The present value of an annuity of C dollars per
  period for t periods when the rate of return is r
  is

Present Value (Interest) Factor
Present Value Interest Factor for Annuities.
PVIFA (r, t)

• You can find Annuity Present Value, C, or t, if
  you know the remaining three.
• However, to find r, you must use the Present
  Value of Annuity table (text book p. 804)

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Future Value Interest Factor for Annuity

• The future value of an annuity of C dollars per
  period for t periods when the rate of return is r
  is

Future Value (Interest) Factor
Future Value Interest Factor for Annuities. FVIFA
(r, t)

• You can find Annuity Future Value, C, or t, if
  you know the remaining three.
• However, to find r, you must use the Future
  Value of Annuity table (text book p. 806)

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Annuities Due

• With ordinary annuities, the cash flows occur at
  the end of each period.
• An annuity due is an annuity for which the cash
  flows occur at the beginning of each period.
• The relationship between the value of an annuity
  due and an ordinary annuity is (for both PV and
  FV)
• Annuity due value ordinary annuity value(1
  r).
• Calculate the present/future value of an annuity
  due as follows (1) calculate the present/future
  value as though it were an ordinary annuity and
  (2) multiply the answer by (1 r).

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Perpetuities

• An asset that generates a fixed amount of cash
  flows forever is called perpetuity. Perpetuity
  is an annuity in which the cash flows continues
  forever.
• The present value of a perpetuity, which generate
  an annuity of C, given the appropriate discount
  rate r is

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Growing perpetuities

• An asset that infinitely generates cash flows
  that will increase at a constant rate is called
  growing perpetuity.
• The present value of a growing perpetuity, which
  infinitely generates a stream of cash flows that
  is expected to increase at the rate of g forever,
  given the appropriate discount rate r is

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Growing Annuities

• Growing annuity is a FINITE number of growing
  cash flows.
• An asset generates cash flows will increase at a
  constant rate, g, for a fixed (or limited) number
  of periods, T. The cash flow it generated in the
  next period is C, and the appropriate discount
  rate is r. Then the PV of such an asset (or this
  growing annuity) is given by

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Effective Annual Rate

• Stated (quoted) interest rate, or annual
  percentage rate the interest rate charged per
  period of compounding multiplied by the number of
  compounding per year.
• Effective interest rate (EAR) the interest rate
  expressed as if it were compounded once per year.
  

• If cash flows (i.e., an annuity or perpetuity)
  occur semi annually, you must use the rate per
  semi-annual (i.e., APR compounded semi-annually,
  divided by 2).

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Continuous Compounding

• Sometimes investments or loans are calculated
  based on continuous compounding
• EAR eq 1
• The e is a special function on the calculator
  normally denoted by ex
• Example What is the effective annual rate of 7
  compounded continuously?
• EAR e.07 1 .0725 or 7.25

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Check Points

• Basic idea underlying the PV/FV of multiple cash
  flows? Two ways to calculate it?
• Annuities? How to find the PV/FV of an annuity?
• Annuities due?
• Perpetuity?
• Growing perpetuity?
• Growing annuities?
• Effective annual rate? Relationship to stated
  (quoted) interest rate (annual percentage rate)?

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Check Points CAUTION!!!

• Draw time lines in order to avoid silly mistakes.
  
• Cash flows stated to occur in one year means
  at the end of one year (expect the cases of
  annuities due).
• C in formulas is the cash flows to occur in the
  first (or next) period, NOT current period (i.e.,
  time 0). If cash flows occur also at time 0, you
  must treat it outside of the formula.
• If you are confused about the timing of cash flow
  occurrences (in tests/exam), ASK ME.

see problem 10 in Practice Problems for May 24/25
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Check Points CAUTION!!!

• Three types of interest rates EAR, APR (or
  stated/quoted rate), and rate per period (APR
  divided by number of compounding).
• Frequency of cash flows and frequency of
  compounding of interest rate must match. For
  example, if cash flows occur quarterly, r in the
  formula must be rate per quarter (i.e., APR
  compounded quarterly divided by 4).
• t must be number of cash flow occurrences rather
  than number of year. For example, if cash flows
  occur daily and you would like to know the FV two
  years from now, t 3652.
• If is very easy to get confused and make silly
  mistakes try as many practice problems as
  possible.

see problem 12, 13, and 14 in Practice Problems
for May 24/25
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Mortgages

• In Canada, financial institutions are required by
  law to quote mortgage rates with semi-annual
  compounding
• Since most people pay their mortgage either
  monthly (12 payments per year), semi-monthly (24
  payments) or bi-weekly (26 payments), you need to
  remember to convert the interest rate before
  calculating the mortgage payment!
• Financial institutions offer mortgages with fixed
  interest rates for various period (generally up
  to 5 years).

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Mortgages Example 1

• Theodore D. Kat is applying to his friendly,
  neighbourhood bank for a mortgage of 200,000.
  The bank is quoting 6. He would like to have a
  25-year amortization period and wants to make
  payments monthly. What will Theodores payments
  be?

• First, calculate the EAR
• Second, calculate the effective monthly rate
• Then, calculate the monthly payment

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Pure Discount Loans

• Treasury bills are excellent examples of pure
  discount loans. The principal amount is repaid
  at some future date, without any periodic
  interest payments.
• If a T-bill promises to repay 10,000 in 12
  months and the market interest rate is 4 percent,
  how much will the bill sell for in the market?
• PV 10,000 / 1.04 9,615.38, OR,

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Interest Only Loan

• The borrower pays interest each period and repays
  the entire principal at some point in the future.
• A 3-year, 10, interest only loan of 1,000 the
  borrower would pay 1,0000.1 100 in interest
  at the end of first and second years, and 1,000
  100 1,100 at the end of the third year.
• This cash flow stream is similar to the cash
  flows on corporate bonds and we will talk about
  them in greater detail later.

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Amortized Loan with Fixed Principal Payment

• A 50,000, 10-year loan at 8 interest. The loan
  agreement requires the firm to pay 5,000 in
  principal each year plus interest for that year.

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Amortized Loan with Fixed Payment

• Each payment covers the interest expense plus
  reduces principal
• A 4-year loan with annual payments. The interest
  rate is 8 and the principal amount is 5000.
• What is the annual payment?

Note The ending balance of .01 is due to
rounding. The last payment would actually be
1,509.61.
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Any questions? End see you next week