FINC3131 Business Finance

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FINC3131Business Finance

• Chapter 5 Time Value of Money Advanced
  Topics

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Learning Objectives

• Use a financial calculator to solve TVM problems
  involving multiple periods and multiple cash
  flows.
• Solve TVM problems when the period of compounding
  is less than a year.
• Tell the difference between an ordinary annuity
  and an annuity due.
• Solve TVM problems involving an annuity due.
• Prepare an amortization schedule

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Preparing BAII Plus for use

• Press 2nd and Format. The screen will
  display the number of decimal places that the
  calculator will display. If it is not eight,
  press 8 and then press Enter.
• Press 2nd and then press P/Y. If the display
  does not show one, press 1 and then Enter.
• Press 2nd and BGN. If the display is not
  END, that is, if it says BGN, press 2nd and
  then SET, the display will read END.

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The Formula for Future Value
Future Value
Number of periods
Rate of return or discount rate or interest rate
or growth per period
Present Value
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The Formula for Present Value
From before, we know that
Solving for PV, we get
Unless otherwise stated, r stated on an annual
basis.
Again, now we deal with PV problems where n gt 2
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Special keys used for TVM problems

• N Number of periods (e.g., years)
• I/Y Interest rate/ discounting/
• compounding rate per period
• PV Present value
• PMT The periodic fixed cash flow in an annuity
• FV Future value
• CPT Compute

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What is the future value (FV) of an initial 100
after 3 years, if I/YR 10?

• Finding the FV of a cash flow or series of cash
  flows is called compounding.
• FV can be solved by using the step-by-step,
  financial calculator, and spreadsheet methods.

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The step-by-step and formula methods

• After 1 year
• FV1 PV (1 I) 100 (1.10) 110.00
• After 2 years
• FV2 PV (1 I)2 100 (1.10)2 121.00
• After 3 years
• FV3 PV (1 I)3 100 (1.10)3 133.10
• After N years (general case)
• FVN PV (1 I)N

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The calculator method

• Solves the general FV equation.
• Requires 4 inputs into calculator, and will solve
  for the fifth.

3
10
0
-100
INPUTS
N
I/YR
PMT
PV
FV
OUTPUT
133.10
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Multi-period, Find PV

• Find the present value of 6,000 that occurs at t
  6. The discount rate is 14 percent.
• Use PV FV/(1r)6
• FV6000, N 6, I/Y 14, PMT 0.
• Press CPT and then PV

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Multi-period, find FV

• Suppose you deposit 150 in an account today and
  the interest rate is 6 percent p.a.. How much
  will you have in the account at the end of 33
  years?
• Use FV PV x (1r)33
• Press PV-150, N33,I/Y6, PMT0
• Press CPT then FV

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Multi-period, find r

• You deposited 15,000 in an account 22 years ago
  and now the account has 50,000 in it. What was
  the annual rate of return that you received on
  this investment?
• Use r (FV/PV)1/n 1.
• PV - 15000, N 22, PMT 0, FV 50000,
• I/Y ?

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Multi-period, find n

• You currently have 38,000 in an account that has
  been paying 5.75 percent p.a.. You remember that
  you had opened this account quite some years ago
  with an initial deposit of 19,000. You forget
  when the initial deposit was made. How many
  years (in fractions) ago did you make the initial
  deposit?
• PV - 19000, PMT 0, FV 38000, I/Y 5.75,
• N ?

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Perpetuity 1

• Perpetuity a stream of equal cash flows ( C )
  that occur at the end of each period and go on
  forever.
• PV of perpetuity
• C is the cash flow at the end of each period
• r is the discount rate

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Perpetuity 2

• So what?
• We use the idea of a perpetuity to determine the
  value of
• A preferred stock
• A perpetual debt

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Perpetuity questions

• Suppose the value of a perpetuity is 38,900 and
  the discount rate is 12 percent p.a.. What must
  be the annual cash flow from this perpetuity?
• Use C PV x r. Verify that C 4,668.
• An asset that generates 890 per year forever is
  priced at 6,000. What is the required rate of
  return?
• Use r C/PV. Verify that r 14.833 percent

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Annuity

• An annuity a cash flow stream where a fixed
  amount is received every period for a fixed
  number of periods.
• Example You rent out a property for 12,000 per
  year for ten years.
• In many TVM problems, the cash flow stream is
• An annuity combined with a single cash flow
  (often at the beginning or the end)
• A combination of two or more annuities.

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Annuity, find PV

• You are considering buying a rental property.
  The yearly rent from this property is 18,000.
  You expect that the property will yield (i.e.,
  generate) this rent for the next twenty years
  after which you will be able to sell it for
  250,000. If your required rate of return is 12
  percent p.a., what is the maximum amount that you
  would pay for this property?
• PMT18000, FV250,000, I/Y12, N20, PV?

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Annuity, find FV

• You open an account today with 20,000 and at the
  end of each of the next 15 years, you deposit
  2,500 in it. At the end of 15 years, what will
  be the balance in the account if the interest
  rate is 7 percent p.a.?
• PV-20000, PMT-2500, N15, I/Y7, FV?

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Annuity, find I/Y

• You lend your friend 100,000. He will pay you
  12,000 per year for the ten years and a balloon
  payment at t 10 of 50,000. What is the
  interest rate that you are charging your friend?
• PV-100,000, FV50,000, PMT12,000, N 10,
  I/Y?

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Annuity, find PMT

• Next year, you will start to make 35 deposits of
  3,000 per year in your Individual Retirement
• Account (so you will contribute from t1 to
  t35).
• With the money accumulated at t35, you will
• then buy a retirement annuity of 20 years with
  equal yearly payments from a life insurance
  company (payments from t36 to t55).
• If the annual rate of return over the entire
  period is 8, what will be the annual payment of
  the annuity?

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Uneven Cash Flows
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Uneven cash flows 1

• Your account pays interest at a rate of 5 percent
  p.a. You deposit 8,000 in it today. You must
  have exactly 3,000 in the account at the end of
  two years. How much should you withdraw at the
  end of the first year to ensure this?

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What is the PV of this uneven cash flow stream?
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What is the PV of this uneven cash flow stream?
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Solving for PVUneven cash flow stream

• Input cash flows in the calculators CF
  register Press CF key
• CF0 0, ENTER,
• C01 100, ENTER, F011, ENTER,
• C02 300, ENTER, F022, ENTER,
• C03 50, /- key, ENTER,
• Press NPV key
• I 10, ENTER, press CPT key to get NPV
  530.087. (Here NPV PV.)

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Uneven cash flows 2

• An asset promises to produce the following series
  of cash flows. At the end of each of the first
  three years, 5,000. At the end of each of the
  following four years, 7,000. And, at the end of
  each of following five years, 9,000. If your
  required rate of return is 10 percent, how much
  is this asset worth to you?
• Find PV of this series of cash flows.
• PV 46,612.68

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Uneven cash flows 3

• You will need to pay for your sons private
  school tuition (first grade through 12th grade) a
  sum of 8,000 per year for Years 1 through 5,
  10,000 per year for Years 6 through 8, and
  12,500 per year for Years 9 through 12. Assume
  that all payments are made at the beginning of
  the year, that is, tuition for Year 1 is paid now
  (i.e., at t 0), tuition for Year 2 is paid one
  year from now, and so on. In addition to the
  tuition payments you expect to incur graduation
  expenses of 2,500 at the end of Year 12. If a
  bank account can provide a certain 10 percent
  p.a. rate of return, how much money do you need
  to deposit today to be able to pay for the above
  expenses?

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Special topics

• Compounding period is less than 1 year
• Continuous compounding
• Annuity due
• Loan amortization

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Compounding period is less than 1 year

• Saying that compounding period is less than 1
  year is equivalent to saying that
• frequency of compounding is more than once per
  year

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Common examples
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Example (1)

• Suppose that your bank states that the interest
  on your account is eight percent p.a.. However,
  interest is paid semi-annually, that is every six
  months or twice a year.
• The 8 is called the stated interest rate.
• (also called the nominal interest rate)
• But, the bank will pay you 4 interest every 6
  months.

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Example (2)

• Ok, so we know how much interest is paid every 6
  months. Over a year, what is the percentage
  interest I actually earn?
• In other words,
• I want to know the effective annual interest rate
  
• (or effective interest rate, or annual
  percentage yield)

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Example (3)

• Suppose you deposit 100 into the account today.
• Account balance at end of 6 months
• 100 x 1.04 104
• Account balance at end of 1 year
• 104 x 1.04 108.16
• Effective interest rate
• (108.16 100)/100 0.0816 or 8.16

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When frequency of compounding is more than once a
year
n number of years m frequency of
compounding per year r stated interest rate
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Can the effective rate ever be equal to the
nominal rate?

• Yes, but only if annual compounding is used,
  i.e., if m 1.
• If m gt 1, effective rate will always be greater
  than the nominal rate.

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Effective rate example

• You have decided to buy a car whose price is
  45,000. The dealer offers to finance the entire
  amount and requires 60 monthly payments of 950
  per month. What are the yearly stated and
  effective interest rates for this financing?
• Answer
• stated 9.723 p.a.
• effective 10.168 p.a.

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Car buying with down payment

• You are considering buying a new car. The sticker
  price is 15,000, and you have 3000 for down
  payment. You obtain a 5-year car loan at a
  nominal annual interest rate of 12. What is your
  monthly loan payment?
• Read question carefully when you work out the
  size of the loan. What is PV?

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Annuity with monthly compounding

• Compute the future value at the end of year 25 of
  a 100 deposited every month for 10 years (with
  the first deposit made one month from today) into
  an account that pays 9 percent p.a.

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Annuity with semiannual compounding

• You would like to accumulate 16,500 over the
  next 8 years. How much must you deposit every
  six months, starting six months from now, given a
  4 percent per annum rate with semiannual
  compounding?

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Effective rate

• Your banks stated interest rate on a three month
  certificate of deposit is 4.68 percent p.a. and
  the interest is paid quarterly. What is the
  effective interest rate?

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Find period

• The stated interest rate for a bank account is 7
  percent and interest is paid semi-annually. How
  many years will it take you to double your money
  in this account?

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More frequent compounding, more

• All else constant, for a given nominal interest
  rate, an increase in the number of compounding
  periods per year will cause the future value of
  some current sum of money to
• Increase
• Decrease
• Remain the same
• May increase, decrease or remain the same
  depending on the number of years until the money
  is to be received.
• Will increase if compounding occurs more often
  than 12 times per year and will decrease if
  compounding occurs less than 12 times per year.

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Annuity Due 1

• Up till now, we deal with ordinary annuities.
• For an ordinary annuity, payment occurs at the
  end of each period.
• For an annuity due, payment occurs at the
  beginning of each period.
• The difference becomes clear when we look at time
  lines.

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Consider an annuity that pays 300 per year for
three years.

• If ordinary annuity, time line is
• If annuity due, time line is

300
300
300
T 1
T 3
T 0
T 2
300
300
300
T 3
T 0
T 1
T 2
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Is there a relationship between ordinary annuity
and annuity due?

• Yes !
• PV of annuity due
• (PV of ordinary annuity) x (1 r)
• FV of annuity due
• (FV of ordinary annuity) x (1 r)
• ordinary annuity and regular annuity mean the
  same thing.

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Example

• You have a rental property that you want to rent
  for 10 years. Prospective tenant A promises to
  pay you a rent of 12,000 per year with the
  payments made at the end of each year.
  Prospective tenant B promises to pay 12,000 per
  year with payments made at the beginning of each
  year. Which is a better deal for you if the
  appropriate discount rate is 10 percent?
• Set PMT 12,000 N 10 I/Y 10
• To answer question, focus on dollar amount of
  each PV.

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Another example

• What is the present value of an annuity of 1200
  per year for 10 years (with the first payment to
  be made today and the last payment to be made 9
  years from today) given an interest rate of 5.5
  percent p.a.?

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Loan Amortization

• Amortization is the process of separating a
  payment into two parts
• The interest payment
• The repayment of principal
• Note
• Interest payment decreases over time
• Principal repayment increases over time

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Example of loan amortization 1

• You have borrowed 8,000 from a bank and have
  promised to repay the loan in five equal yearly
  payments. The first payment is at the end of the
  first year. The interest rate is 10 percent.
  Draw up the amortization schedule for this loan.
• Amortization schedule is just a table that shows
  how each payment is split into principal
  repayment and interest payment.

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Example of loan amortization 2

• 1) Compute periodic payment.
• PV8000, N5, I/Y10, FV0, PMT?
• Verify that PMT 2,110.38
• Amortization for first year
• Interest payment 8000 x 0.1 800
• Principal repayment
• 2,110.38 800 1310.38
• Immediately after first payment, the principal
  balance is 8000 1310.38 6,689.62

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Example of loan amortization 3

• Amortization for second year
• Interest payment 6689.62 x 0.1 668.96
• (using the new balance!)
• Principal repayment
• 2,110.38 668.96 1441.42
• Immediately after second payment, the principal
  balance is 6,689.62 1441.42 5,248.20
• Verify the entire schedule (on following slide)

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Verify the amortization schedule
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Using financial calculator to generate
amortization schedule 1

• Very often, amortization problems involve long
  periods of time, e.g., 30 year mortgage with
  monthly payments gt 360 periods.
• To generate amortization schedule in such
  problems, its more efficient to use the
  financial calculator.
• Lets reuse the last problem (Problem 7.25).
  First, find the monthly payment. Key in
• PV8000, N5, I/Y10, FV0, PMT?
• We already worked out that PMT 2,110.38.

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Using financial calculator to generate
amortization schedule 2

• Suppose we want to work out the remaining balance
  immediately after the 2nd payment.
• Press 2ND, AMORT to activate the Amortization
  worksheet in BA II Plus.
• Press P12, ENTER, ?,
• Press P22, ENTER, ?,
• You will see BAL5,248.20

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Using financial calculator to generate
amortization schedule 2

• Press ? again and you see the portion of the year
  2 payment going towards repaying principal, i.e.,
  PRN -1,441.42
• Press ? again and you see the portion of year 2
  payment going towards interest, i.e., INT
  -668.96
• To get out of the Amortization schedule, press
  2ND, Quit.

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All together now (1)

• Which of the following statements is most
  correct?
• A 5-year 100 annuity due will have a higher
  future value than a 5-year 100 ordinary annuity.
• A 15-year mortgage will have smaller monthly
  payments than a 30-year mortgage of the same
  amount and same interest rate.
• All else being constant, for a given nominal
  interest rate, an increase in the number of
  compounding periods per year will cause the
  future value of some current sum of money to
  increase.
• Statements A and C are correct.
• All of the statements above are correct.

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All together now (2)

• Which of the following statements is most
  correct?
• An investment that compounds interest
  semiannually, and has a nominal rate of 15
  percent, will have an effective rate less than 15
  percent.
• The present value of a three-year 1000 annuity
  due is less than the present value of a
  three-year 1000 ordinary annuity.
• The portion of the payment of a fully amortized
  loan that goes toward interest declines over
  time.
• Statements A and C are correct.
• None of the answers above is correct.

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Summary

• TVM problems with multiple periods and multiple
  cash flows
• Solving TVM problems using financial calculator
  and time lines
• Special topics
• Compounding period lt one year
• Continuous compounding
• Annuity due
• Loan amortization

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Assignment

• Chapter2
• Self-test ST-3 ST-4
• Questions 5-2 5-3 5-4
• Problems 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9
  5-10 5-12 5-13 5-14 5-15 5-16 5-17 5-18 5-19 5-21
  5-22 5-23 5-24 5-25 5-27 5-33 5-34