Financial Management

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Financial Management

• Lecture 1 The Discounted Cash Flow Analysis
• Ning Gong

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Organisation

• Principal and Interest
• Future Value and Compounding Present Value and
  Discounting
• Future and Present Values of Multiple Cash Flows
• Valuing Level Cash Flows Annuities and
  Perpetuities
• Arithmetic vs. Geometric Average.
• Nominal Rates vs. Real Rates
• Loan Types and Loan Amortisation

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1.1 Terminology Rate of Return, Principal and
Interest

• If you invest C0 at time 0, and receive C1 at
  time 1, then the rate of return of the investment
  is
• Principal and Interest
• The rate of return on a bank deposit is called
  the interest rate.
• The initial amount of your bank account is
  called the principal

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1.2 Terminology PV and FV

• Consider the time line below
• PV is the Present Value, that is, the value
  today.
• FV is the Future Value, or the value at a future
  date.
• The number of time periods between the Present
  Value and the Future Value is represented by t.
• The rate of interest per period is denoted by
  r.
• All time value questions involve the above four
  values PV, FV, r, and t. Given three of them, it
  is always possible to calculate the fourth.

0
1
2
3
t
. . .
PV
FV
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1.3 Future Value for a Lump Sum

• Notice that at r 10, an initial deposit of
  100 becomes
• At t 1 110 100 ? (1 .10)
• This means, after depositing your money with the
  bank for one year, you will get your 100
  principal back, plus you will earn 10100 10
  interest. Your total payoff (FV) is 110 at t
  1.
• At t 2 121 110 ? (1 .10) 100 ? 1.10 ?
  1.10
• 100 ? 1.102
• At t 3 133.10 121 ? (1 .10) 100 ?
  1.102 ?1.10 100 ? (1.10)3

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1.4 General Formula for Future Value

• In general, the future value, FVt, of 1 invested
  today at r for t periods is
• FVt 1 (1 r)t
• The expression (1 r)t is the future value
  interest factor.

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1.5 Example 1

• Q. Deposit 5,000 today in an account paying 10.
  How much will you have in 6 years? How much is
  simple interest? How much is compound interest?
• Answer FV 5000 ? (1 r )t 5000 ? (1.10)6
  8857.81
• At 10, the simple interest is .10 ? 5000
  500 per year. After 6 years, this is 6 ? 500
  3000 the difference between compound and
  simple interest is thus 3857.81 - 3000
  857.81
• Simple interest means that only the principal
  earns interest Compound interest means that the
  interest earned brings interest in the future.

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1.6 Future Value of 100 at 10 Percent

• Beginning Simple Compound
  Total Ending
• Year Amount Interest Interest
  Interest Earned Amount
• 1 100.00 10.00 0.00
  10.00 110.00
• 2 110.00 10.00
  1.00 11.00 121.00
• 3 121.00 10.00
  2.10 12.10 133.10
• 4 133.10 10.00
  3.31 13.31 146.41
• 5 146.41 10.00
  4.64 14.64 161.05
• Totals 50.00
  11.05 61.05

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1.7 Present Value for a Lump Sum Example 2

• Q. Suppose you need 20,000 in three years to pay
  for your round-the-world dream trip. If you can
  earn 8 on your money, how much do you need
  today?
• A. Here we know that the future value is 20,000,
  the rate is 8, and the number of periods is 3.
  What is the unknown present amount (i.e., the
  present value)?
• From the FV formula
• FVt PV x (1 r )t ? 20,000 PV x
  (1.08)3
• Rearranging the terms PV 20,000/(1.08)3
  15,876.64
• The PV of a 1 to be received in t periods when
  the rate is
• r is PV 1/(1 r )t --gt i.e., the discount
  Factor.

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1.8 Example 3 Finding the Rate r

• Q. You are offered an investment that costs you
  1,000 and will double your money in 8 years,
  what rate of return does this investment earn?
• A. The future value is 2,000 and the present
  value is 1,000 . There are 8 years involved, so
  we need to solve for r in the following
• 1000 2000/(1 r )8
• (1 r )8 2, ? 1 r 21/8 , ? r
  1.0905 1.
• Solving for r, r 9.05.

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1.9 Find the Number of Periods t

• Suppose we were interested in purchasing a house
  that requires 50,000 down payment. We currently
  have 25,000 in savings. If we can earn 12 on
  this 25,000, how long until we have the required
  down payment for the house?
• Apply the PV formula, ? 25,000 50,000/(112)t
• Solving for t,

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1.10 Basic Valuation Model for Multiple Cash
Flows

• The DCF model
• Suppose you will receive Ct at period t
  until time period T. T can be infinity. Then the
  PV (at t 0) of this income stream is
  (applying the principle of value additivity)

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1.11 Net Present Value

• If we invest C0 now (at t 0), and we expect to
  receive C1, C2, CT from this project, then
• The Net Present Value (NPV) is
• Intuitively, we will invest in a project if and
  only if the NPV is positive.

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1.12 Example 4 Net Present Values
Assume that the cash flows from the construction
and sale of an office building is as follows.
Given a 5 required rate of return, create a
present value worksheet and show the net present
value.
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1. 13 Example 4 -- continued
Assume that the cash flows from the construction
and sale of an office building is as follows.
Given a 5 required rate of return, create a
present value worksheet and show the net present
value.
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1.14 Annuities and Perpetuities Summary of
Basic Formulas

• Perpetuity Present Value
• PV C/r
• Annuity Future Value
• FVt C ? (1 r )t - 1/r
• Annuity Present Value
• PV C ? 1 - 1/(1 r )t/r
• The formulas above are the basis of many of the
  calculations in finance. It will be worthwhile to
  keep them in mind!

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1.15 Example 5 Annuity Present Value

• Suppose you need 20,000 each year for the next
  three years to make your tuition payments.
• Assume you need the first 20,000 in exactly
  one year. Suppose you can place your money in a
  savings account yielding 8 compounded annually.
  How much do you need to have in the account
  today?
• (Note Ignore taxes, and keep in mind that you
  dont want any funds to be left in the account
  after the third withdrawal, nor do you want to
  run short of money.)

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1.16 Example 5 Annuity Present Value
(continued)

• Here we know the periodic cash flows are 20,000
  each. Using the most basic approach
• PV 20,000/1.08 20,000/1.082
  20,000/1.083
• 18,518.52 17,146.77 15,876.65
• 51,541.94
• Heres a shortcut method for solving the problem
  using the annuity present value factor
• PV 20,000 ? 1 - 1/(1.08)3/.08
• 20,000 ? 2.577097 51,541.94

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1.17 Example 6 Annuity Future Value

• Suppose a 30 years old investor, Bill, would
  invest a fixed amount annually to accumulate the
  one million he needs for his retirement. If the
  first deposit is made in one year, and deposits
  will continue through age 65, how large must they
  be? Assume r 10.
• Set this up as a FV problem
• 1,000,000 C ? (1.10)35 - 1/.10
• C 1,000,000/271.0244 3,689.70
• Becoming a millionaire is not too difficult, if
  the govt does not tax your super!

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1.18 Example 6 Annuity Future Value (continued)

• Unfortunately, most people dont start saving
  for retirement that early in life. (Many dont
  start at all.)
• Suppose Bill just turned 45 and has decided its
  time to get serious about saving. Assuming that
  he wishes to accumulate 1 million by age 65, and
  will begin making equal annual deposits in one
  year and make the last one at age 65, how much
  must each deposit be?
• Setup 1 million C ? (1.10)20 - 1/.10
• Solve for C C 1 million/57.2750
  17,459.62
• By waiting, Bill has to set aside close to five
  time as much money each year!

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1.19 Example 6 Annuity Future Value (concluded)

• We can also solve this problem in two steps
• Step 1 Find the PV of 1 million.
• PV 1,000,000/(1.10)20 148,643.63
• Step 2 Find C by using the annuity present
  value formula
• 148,643.63 C/ 1 1/(1.10)20 /.10
• ? C 17,459.62

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1.20 Example 7 Perpetuity Calculations

• Suppose we expect to receive 1000 at the end of
  each of the next 5 years. Our opportunity rate is
  6. What is the value today of this set of cash
  flows?
• PV 1000 ? 1 - 1/(1.06)5/.06
• 1000 ? 1 - .74726/.06
• 1000 ? 4.212364
• 4212.36
• Now suppose the cash flow is 1000 per year
  forever. This is called a perpetuity. And the PV
  is easy to calculate
• PV C/r 1000/.06 16,666.66
• So, payments in years 6 thru ? have a total PV
  of 12,454.30!

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1.21 Annual Percentage Rate (APR) vs. Effective
Annual Rate (EAR)

• Banks traditionally quote simple interest rate
  (annual percentage rate, or APR) for loans, but
  calculate monthly interest payments by
  compounding.
• If you borrow at a 10 annual rate (APR) and make
  monthly payments, the monthly interest rate is
  calculated as .10/12 0.00833.
• 1 invested for a year at a monthly rate of
  0.00833 will grow in one year to 1.0083312
  1.10467 dollars. Therefore the effective annual
  rate (EAR) is 10.467.

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1.22 EAR and APR (continued)

• Future value with compounding is
• Thus, if a loan compounds m times a year the
  effective annual rate is

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1.23 Continuous Compounding

• If the time of compounding with a year m
  increases infinitely, then we have the case of
  continuous compounding. This is a very useful
  concept, especially in your further study of
  option pricing.
• The effective annual rate is
• The future value is

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1.24 Compounding Periods, EARs, and APRs

• Compounding Number of times Effective
• period compounded annual rate
• Year 1 10.00000
• Quarter 4 10.38129
• Month 12 10.47131
• Week 52 10.50648
• Day 365 10.51558
• Hour 8,760 10.51703
• Minute 525,600 10.51709
• Continuous ?

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1.25 Geometric Rates of Return

• If you earn an interest rate of 5 for the first
  year and 10 for the second year, what is the
  average rate of return on your investment?
• The arithmetic average is 7.5.
• To think about this question carefully, let us
  start with 100 investment today. You will get
  1001.051.10115.5 in two years.
• The effective rate of return would be
• Thus, the effective rate of return is 7.47.
• The rate is called the geometric average rate of
  return.

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1.26 Nominal vs. Real Interest Rates

• If the bank says the interest rate on a car loan
  is 8, then the nominal interest rate is 8
  (nominal actual). In almost all corporate
  finance problems, you should use the actual
  interest rate.
• The real interest rate is the interest rate after
  taking out the effects of inflation. The nominal
  interest rate is the interest rate before taking
  out the effects of inflation.

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1.27 Calculation of Real Rate

• Suppose the actual interest rate is 8 and there
  is 3 inflation. You have 100 and are
  considering buying something that costs 100.
• If you put your money in the bank for a year, you
  will have 108 and the item will cost 103.
• So you can buy 108/103 1.0485 units.
• The real interest rate is 4.85.
• In general, the Fisher equation tells us that
• (1 real rate) (1 nominal rate)/(1
  inflation rate)

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1.28 Principal-and-Interest Loan Contracts
Mortgages

• Using the annuity formula to calculate the
  monthly mortgage payments.
• Interest payment equals the monthly rate
  multiplied by the mortgage loan still owed to the
  bank.
• The principal payment is equal to the monthly
  payment minus the interest payment.
• Decomposing the monthly mortgage payments into
  two parts principal and interest payments is
  important.
• For Mortgage-back securities
• For taxation purposes
• For refinancing decision.

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1.29 Monthly Mortgage Payments

• If you borrow 200,000 for 30 years with monthly
  payments and the APR is 9, then
• Monthly interest rate 0.09/12, and the number
  of payments 12 x 30 360,
• so the payment is (using Excel function)
  PMT(0.09/12, 360,
  200000) 1,609.25. Or, you can get C by
  applying the annuity formula.
• The first months interest payment is
  200,0009/12 1,500.
• The first months principal payment is 1,609.25
  - 1,500 109.25.
• The second months interest payment is (200,000
  109.25) 9/12 1,499.18.
• The first months principal payment is 1,609.25
  - 1,499.18 110.07

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1.30 Amortization Table An Example

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1.31 Formulas for Amortization Table
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1.32 Example 8 Calculation of Loan Balance

• We can use the Annuity formula to get the
  remaining principal quickly without using the
  Excel spreadsheet.
• Suppose Mike borrowed 300,000 for a 25-year loan
  at a fixed APR 6. After paying three years, what
  is the amount of money still owed to the bank?
• After 10 years?