Introduction to Finance: Perpetuities and Annuities

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Introduction to FinancePerpetuities and
Annuities
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Valuing Shares of Stock

• Recall, a shareholder receives dividends.
• Dividends are a cash in-flow.
• Assume we hold the stock forever.
• Assume the dividends are paid forever (each
  year).
• Assume the discount rate stays the same forever.
• Sonys stock pays a dividend of 50 yen per year.
• What is the price of Sonys stock?

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Dividend Discount Model ???????
Foerever, Indefinitely
?50
?50
?50
1
2
3
t0
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DDM (Dividend Discount Model)

• How would you go about finding this present value
  which goes into infinity?
• Is there a simple way of obtaining this number?
• The perpetuity formula.

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Perpetuity (????)
A perpetuity pays a fixed income each year
(forever)
Notice r must be greater than g for the growing
perpetuity, why?
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Cash Flow ????????
C(1g)t, ggt0
C is the same, g0
Time ??
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Notation

• C cash flow, ????????
• r discount rate, ???
• g growth rate of cash flows, ???

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With Growth

• We could modify our perpetuity DDM model by
  assuming the dividends (cash flows) increase each
  year at a constant growth rate.
• This gives us a perpetuity with growth.

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Dividends for the first five years. Note
dividends are paid out to infinity. Initial
dividend is 50 yen. Dividends grow at 10 per
year (growth model).
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Inputs in Blue Shaded Area
Perpetuity Model
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Growth Rate (???)

• Analyst forecasts.
• payout ratio and ROE approach
• Take averages across firms with similar
  characteristics (same industry, risk class, etc.)
• Compare the g with growth rate of the entire
  economy-GNP or GDP (or industry). The firm
  cannot grow significantly faster than the economy
  forever!

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Examples
If the discount rate is 10 and you expect to
receive 100 yen each year forever. What is the
value of this stream of payments (today)? What if
the yearly payment grows at a rate of 1 per year.
PVC/r100 yen/0.11000 yen PVC/(r-g)100
yen/(0.1-0.01) 1111.11
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If a project has the following cash flows year 1
500 year 2 300 year 3 750 and the
discount rate is 12, what is the projects PV? If
cash flows were expected to be 500 forever ?
PV500/(1.12) 300/(1.12)2 750/(1.12)3 PV500/
0.12 (perpetuity case)
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Consol Bond (undated)

• A bond pays a coupon of 100 yen each year
  indefinitely.
• The discount rate is 20.
• Assume the Principal is not repaid.
• What is the price of the bond?

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Annuity
Annuity pays a fixed sum each year for a
specified number of years
Take the difference of two perpetuities to get
the annuity formula
perpetuity from year 1
perpetuity from year t1
annuity from year 1 to t1
year 1
year t1
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Annuity (??????)
Perpetuity from year t1 discounted back to year
one.
Perpetuity from year 1