Multiple cash flow valuation

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Multiple cash flow valuation
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Now add cash flows

• Suppose you receive cash 1 period and 2 periods
  from today,
• PV C1/(1r1) C2/(1r2)2
• And if another is received each period
  thereafter,
• PV C1/(1r1) C2/(1r2)2 C3/(1r3)3 .

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General valuation formula

• So in general, we can say that,
• Note that this can incorporate any negative cash
  flows from investment, including those at time
  t0.
• Therefore, this is the net present value (NPV) of
  any cash flow stream (see eqn 4.5, on page 101).

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Aside on term structure

• The interest rate term structure (of some type of
  instrument) gives market determined discount
  rates over varying periods of time (terms).
• In many cases, assuming a fixed rate through time
  is appropriate
• many contracts (loans, mortgages) are written in
  this manner
• In other cases, we would want to discount cash
  flows with the appropriate discount rate for each
  particular term.
• We can also always solve for the constant rate
  the gives the same PV as the varying appropriate
  rates

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Pictures of the US Treasury Yield Curve
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Hypothetical example
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What about a long series of cash flows?

• Typical series of cash flows are longer than 5,
  as in the previous example!
• For example, a 30-year monthly mortgage has 360
  cash flows
• No one likes spreadsheets with 360 rows
• If the cash flows follow certain patterns, there
  are nice shortcuts!

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Perpetuities

• (1) What if r is constant, and equal cash flows
  are received forever?
• This is a perpetuity See page 107
• (1b) What if perpetuity starts t1 periods from
  today?
• .

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Annuities

• What if equal periodic payments have a finite
  life, ending at period t?
• Note that would be the difference between a
  perpetuity starting at period 1, and a perpetuity
  that starts at t1.

See eqn 4.15, page 111
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Growing perpetuity

• Now, suppose our perpetuity did not have a
  constant C, but C grows at some rate g?
• Then
• See eqn 4.12, page 108

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Growing annuity

• Suppose our annuity is not constant, but grows at
  some rate g?
• Similarly, this can be solved as the difference
  in two growing perpetuities
• see eqn 4.17, page 115

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Example Coupon bonds

• Suppose a 1,000 par value government bond pays
  an 8 APR coupon semi-annually with next payment
  due in 6 months, and it has 18 years remaining
  until maturity? Rates on similar investments
  yield 8 (your required yield to maturity is 8)
• PV PV coupon annuity PV single prin pymt
• PV401/.04 - 1/(.04(1.04)36)1000/(1.04)36
• PV756.33 243.67 1,000
• See discussion of level coupon bonds, pgs
  129-136

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Bond example

• Note that bond price par
• What if coupon rate gt yield to maturity?
• This is a premium bond
• What if coupon rate lt yield to maturity?
• This is a discount bond
• In each case the NPV of the investment zero
• in an efficient market, the NPV of an investment
  in any financial security is zero!

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Solving for the annuity payment

• Suppose you want to borrow 30,000, at 6 with
  annual payments over 25 years. What will be your
  annual payment?
• PVA1/r - 1/(r(1r)t)
• PV Ax
• A PV/x
• A 30,000/1/.06 - 1/(.06(1.06)25) 2,346.80

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Annuity components

• Note that the amount of principal paid in an
  annuity varies with each payment

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Annuity components

• This gives rise to the fact that when you borrow,
  your equity position doesnt improve at a linear
  rate

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Annuity example 2

• Suppose you borrow 250,000 for a house, with a
  5.99 APR compounded monthly, for 30 years. What
  is your payment?
• And components are

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How does different rates change relative size of
components?

• What if r12, then payment 2,571
• What if r18, then payment

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How does different rates change components?

• What if you borrow from Goodfellas Bank at a
  rate 100?

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Time to payoff an annuity

• Suppose you borrow 300,000 for 30 years at a
  fixed rate of 6 APR compounded monthly. What is
  your monthly mortgage payment?
• Now, suppose you decide to pay 2,000 per month
  instead. How long will it take to payoff your
  loan?

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Finding remaining payoff

• Suppose you borrow 300,000 for 30 years at a
  fixed rate of 6 APR compounded monthly. As
  shown above, your mortgage payment is 1798.65.
• Now, suppose you have 10 years remaining on the
  original mortgage, and you want to payoff the
  remainder of the loan. How much do you owe?

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Lets do some practice exercises .Break into
your tutorial teams and try the following