Valuation Under Certainty

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Valuation Under Certainty

• Investors must be concerned with
• - Time
• - Uncertainty
• First, examine the effects of time for one-period
  assets.
• Money has time value. 100 in one year is not as
  attractive as 100 today.
• Rule 1 A dollar today is worth more than a
  dollar tomorrow, because it can be reinvested to
  earn more by tomorrow.

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Session 1

• Topics to be covered
• Time value of money
• Present value, Future value
• Interest rates
• compounding intervals
• Bonds
• Arbitrage

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Present Value

• The value today of money received in the future
  is called the Present Value
• The present value represents the amount of money
  we would be prepared to pay today for something
  in the future.
• The interest rate, i is the price of credit in
  financial markets.
• Interest rates are also known as discount rates.

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Present Value

• The Present Value Factor or Discount Factor is
  the number we multiply by a future cash flow to
  calculate its present value.
• Present Value (PV)Discount FactorFuture
  Value(FV)
• - Discount factor 1/(1i)
• Example (i10)
• - Discount factor 1/(1.10)
• - The present value of 200 received in 1 year
  is

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Future Value

• Alternatively, we may use the interest rate, i,
  to convert dollars today to their value in the
  future.
• Suppose we borrow 50 today, and must repay this
  plus 5 interest in one year.
• Future Value (FV) Present Value (PV)(1i)
• FV PV(1i) 50(1.05) 52.50

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Bonds

• A bond is a promise from the issuer to pay the
  holder
• - the principal, or face value, at maturity.
• - Interest, or coupon payments, at intervals up
  to maturity.
• A 100 face value bond with a coupon rate of 7
  pays 7.00 each year in interest, and 100 after
  a pre-specified length of time, called maturity.

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Zero Coupon Bonds

• A zero coupon bond has no coupon payments.
• The holder only receives the face value of the
  bond at maturity.
• Suppose the interest rate is 10. A zero coupon
  bond promises to pay the holder 1 in one year.
  Its price today is therefore
• The discount factor is just the price of a zero
  coupon bond with a face value of 1.

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Net Present Value

• The Net Present Value is the present value of the
  payoffs minus the present value of the costs.
• Suppose Treasury Bills yield 10.
• The present value of 110 in one year is
• Suppose we could guarantee this payoff by
  investing in a project that only costs 98 today.
• The NPV of this project is

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NPV

• The formula for calculating the NPV (one-period
  case) is
• Note that C0 is usually negative, a cost or cash
  outflow.
• In the above example, C0 -98 and C1 110.

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Rate of Return

• The rate of return is the interest rate expected
  to be earned by an investment.
• The rate of return for this project is
• We only want to invest in projects that return
  more than the opportunity cost of capital.
• The cost of capital in this case is 10.

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Decision Rules

• We know
• 1.) This project only costs 98 to guarantee
  110 in one year. In the market, it costs 100
  to buy 110 in one year.
• 2.) This project returns 12.2. In the market,
  our return is only 10.
• This project looks good.

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Decision Criteria

• We have equivalent decision rules for capital
  investment (with a ONE-PERIOD investment
  horizon)
• - Net Present Value Rule accept investments
  that have a positive NPV.
• - Rate of Return Rule accept investments that
  offer a return in excess of their opportunity
  cost of capital.
• These rules are equivalent for one period
  investments.
• These rules are NOT equivalent in more
  complicated settings.

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Example Market Value

• Continue to suppose you can borrow or lend money
  at 10.
• Assume the price of a one-year zero-coupon bond
  with a FV of 110 is 98.
• The price of this bond is less than its present
  value.
• We may use this example to illustrate the concept
  of arbitrage.

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No Arbitrage

• Arbitrage is a free lunch, a way to make money
  for sure, with no risk and no net cost.
• For example
• - Buy something now for a low price and
  immediately sell it for a higher price.
• - Buy something now and sell something else
  such that you have no net cash flows today, but
  will earn positive net cash flows in the future.
• Assets must be priced in financial markets to
  rule out arbitrage.

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Example (cont.)

• To arbitrage this opportunity, we
• 1.) buy the bond
• 2.) borrow 100 for one year.
• The cash flows from this strategy today and at
  the end of one year are
• Today One Year
• Buy the bond -98 110
• Borrow 100 (1 yr) 100 -110
• Net cash flow 2 0

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Short Selling

• Suppose the price of the zero-coupon bond were
  102.
• Our arbitrage strategy would be reversed.
• - Lend 100 for one year.
• - Short Sell the zero-coupon bond.
• The cash flows from this strategy would be
• Today One Year
• Sell the bond 102 -110
• Lend 100 (1 yr) -100 110
• Net cash flow 2 0

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Market Value

• As the above example illustrates, the only price
  for a bond which rules out arbitrage is 100.
• 100 is also the present value of the payoff of
  the bond.
• RULE 2 Assets must be priced in the market to
  rule out arbitrage (i.e., no arbitrage)
• Therefore, the present value of an asset is its
  market price.

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Compound Interest Vs. Simple Interest

• Next we consider assets that last more than one
  period.
• How is multi-period interest paid?
• Invest 100 in bonds earning 9 per year for two
  years
• - After one year 100(1.09) 109
• - Reinvest 109 for the second year 109(1.09)
  118.81
• We do NOT earn just 9 2 18 .
• We earn interest on our interest, or COMPOUND
• SIMPLE INTEREST interest paid only on the
  initial investment
• COMPOUND INTEREST interest paid on the initial
  investment and on prior interest.

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Example Simple Interest

• 100 invested at 10 with no compounding becomes

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Example Compound Interest

• 100 invested at 10 compounded annually becomes
  

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Compound Interest

• A present value PV invested for n years at an
  interest rate of i per year grows to a future
  value
• (1 in)n is the Compound Amount Factor.
• Above, the FV of 100 compounded annually at 10
  for 3 years is
• In principle, the interest rate in may vary with
  the length of the investment horizon, n. More
  later . . .

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Present Value

• We may use the above relation to calculate the
  present value of an n-period investment, with
  compound interest
• where is the discount factor, or
  present value factor.
• For example, the present value of 100 in 6 years
  at 10 per year with annual compounding

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Semi-Annual Compounding

• So far, we assume cash flows occur at annual
  intervals.
• - In Europe, most bonds pay interest annually.
• - In the U.S., most bonds pay interest
  semiannually.
• A 100 bond pays interest of 10 per year, but
  payments are semi-annual.
• - Half of the interest (5 or 50) is paid after
  6 months.
• - Reinvest this 50 for the second 6 months.
• By the end of the year, we would have

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Example

• This return is as if we earned
• if we had only received our payment at the end
  of the year.
• 10 compounded semiannually is equal to 10.25
  compounded annually.
• - 10 is called the nominal interest rate.
• - 10.25 is called the effective interest rate.

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Example

• Suppose you buy 100 of a 7-year Treasury note
  that pays interest at a nominal rate of 10 per
  year, compounded semiannually.
• Define one period as 6 months
• The interest rate per period is 5.
• There are 14 (6-month) periods until the 7-year
  maturity.
• So, we can use our general formula for future
  values to compute the value at maturity

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Extending the PV Formula

• RECALL For a project with one cash flow, C1, in
  one year,
• If a project produces one cash flow, C2 after TWO
  years, then the present value is
• If a project produces one cash flow, C1 after one
  year, and a second cash flow C2 after two years ,
  then

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General Present Value

• By extension, the present value of an extended
  stream of cash flows is
• This is called the Discounted Cash Flow or
  Present Value formula
• Similarly, the Net Present Value is given by

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Example

• Suppose a project will produce 50,000 after 1
  year, 10,000 after 2 years, and 210,000 after 4
  years.
• It costs 200,000 to invest.
• We may earn 9 per year (compounded annually) on
  1, 2, or 4 year zero-coupon bonds.
• The present value of this project is
• The NPV of this project is

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Net Present Value Rule

• In the last example, the PV of payoffs exceeded
  the PV of the costs, so the project is a good
  one.
• Investment Criterion (The NPV Rule) Accept a
  project if the NPV is greater than 0.
• This criterion is a good general rule for all
  types of projects.
• The NPV Rule can also be used to rank projects
  a project with a larger (positive) NPV is better
  than one with a smaller NPV.