FIN 319: Intermediate Financial Management Spring 2005

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FIN 319 Intermediate Financial
ManagementSpring 2005

• SLIDE SET 2 TIME VALUE OF MONEY
• (BASED ON RWJ CHAPTER 4)

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Money has time value

• Why?
• Because interest rates (and expected returns) are
  positive.
• Implies that
• A dollar today is worth more than a dollar
  tomorrow.
• Cash flows at different points in time cannot be
  added together.

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To illustrate the basicsBank account
accumulations.

• You place PV 1000 in an account that earns r
  8 interest, paid at the end of each year. Let
  FVn denote the balance after n periods.
• n 1. FV1 1000 1000(.08) 1000(1.08)
  1080.00
• n 2. FV2 1080 1080(.08) 1080(1.08)
  1000(1.08)2 1166.40
• n 3. FV3 1166.4 1166.4(.08)
  1166.4(1.08) 1000(1.08)3 1259.71
• In general, the balance after n periods can be
  determined from the initial balance and the
  interest rate as
• FVn PV(1r)n

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Present Values (Discounting)

• The same basic formula can be used to determine
  the current amount that is equivalent to (i.e.
  can grow to) a stated future amount, since
• PV FVn/(1r)n
• Example You need 20 million five years from now
  to fund a capital investment. If r 6, what
  amount can be set aside now to fund the
  investment?
• PV 20.0/(1.06)5 20.0/1.33823 14.945
  million.
• Terminology Here, R is referred to as the
  discount rate, and PV is referred to as the
  present value of FVn .
• What if you cant really earn r 6?

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PRESENT VALUES
Present value of 1
PRESENT VALUE Year 5 10 15
1 .952 .909 .870 2 .907 .826 .756
5 .784 .621 .497 10 .614 .386 .247
20 .377 .149 .061
Years
r 5
r 10
r 15
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Present Value Of A Series of Future Cash Flows

• The present value of a stream of cash flows can
  be found using the following general valuation
  formula

• In words (1) discount each cash flow back to
  the present using the appropriate discount rate,
  and (2) sum the present values. The latter step
  is OK, because each future cash flow has been
  restated as its equivalent amount at a common
  point in time.

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Example -- solve for the interest rate

• A Risk-Free Zero Coupon Bond Is Currently Selling
  for 680.58. The Bond will make a single
  payment of 1000 at the end of year 5. What is
  the interest rate (yield) on the bond?
• Timeline 0 1 2 3 4 5
• PV 680.58 FV 1000
• 680.581000/(1r)5 On HP10BII Amber
  C All
• 0.68058 1/(1r)5 - 680.58 PV
• 1.469335 (1r)5 1000 FV
• 1.08 1r. r 8. 5 N
• 0 PMT
• push the I/YR key to get 8.000
• Note Must be sure that P/YR is set 1
  !!!
• Here, we were able to solve for r directly. For
  instruments that make more than one future
  payment, the only way to solve for r is by trial
  and error.

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Alternate Compounding Periods

• Interest is sometimes Compounded Over Periods
  Other Than Annually. In terms of bank account
  examples, this means the interest is credited to
  the account more frequently.
• Caveat All of the time value of money formulas
  use the implicit assumption that the compounding
  interval is the same as the payment interval.
  Eg.
• Mortgage loans call for monthly payments.
• Bonds make coupon payments semiannually.
• Thus, a Warning In all present value
  calculations you must use the effective rate per
  period, not the nominal rate. A period is the
  length of time between cash flows, typically 1,
  3, 6, or 12 months

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Alternate Compounding Periods (Cont.)

• Let m denote the number of compounding intervals
  per year, n the number of years, and r the stated
  (or simple) annual rate.
• The relation between present and future values is
  restated as
• FVn PV(1 r/m)nm
• Eg., if PV 1000, r .12 and m 1, then FV2
  is
• FV2 1000(1 .12)2 1254.40,
• while if m 4 (quarterly compounding), then
• FV2 1000(1 .12/4)24
• FV2 1000(1 .03)8 1266.77

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Example

• Find the PV of 500 to be received in 5 years,
  with
• 12 nominal rate, annual compounding,.

• 12 nominal rate, semiannual compounding,

• 12 nominal rate, quarterly compounding,

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Stated And Effective Annual Rates

• Notice that the use of more frequent compounding
  effectively increases the interest rate.
• The Effective Annual Rate (EAR) is the annual
  interest rate that would produce the same answer
  with annual compounding as is actually obtained
  with more frequent compounding. It can be
  obtained as
• EAR (1 r/m)m - 1
• Eg., if r .12 and m 4, then EAR (1.03)4 -
  1 .1255.

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Example

• A bank quotes a mortgage rate of 8, but will
  compute monthly loan payments using standard time
  value formulas, implying monthly compounding.
  What is the effective annual interest rate on the
  loan?

So the loan effectively costs 8.30 per year.
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Valuing Streams of Structured Future Cash Flows

• Perpetuity A stream of equal payments, made at
  equal time intervals, forever.
• The present value of a perpetuity that pays the
  amount C at the end of each period, when the
  discount rate is r, is

• Note that this gives the value as of time 0, one
  period before the first payment arrives.

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Example

• You wish to endow a Permanent Chair in finance at
  Portland State University. The chair needs
  150,000/yr forever (in perpetuity).
• The trustees of U of Utah can invest your money
  at 6/yr through guaranteed investment contracts
  with the Neverbankrupt Insurance Co.
• How much money do you need (what is the PV of the
  perpetuity) ?
• PV C / keff 150,000 / 0.06 2,500,000

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Valuing Streams of Cash Flows

• Growing Perpetuity
• A growing perpetuity is a stream of periodic
  payments that grow at a constant rate and
  continue forever.
• The present value of a perpetuity that pays the
  amount C1 next period, growing at the rate g
  indefinitely when the discount rate is r is

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Examples
Suppose that the Finance professor to be hired
for the Chair wants 5 raises per year
guaranteed Growing perpetuity 150,000
received at time t 1, growing at
5 per period forever and discounted at
6 per period C
C(1 g) C(1 g) 2 0 1 2 3
PV C/(r g ) 150,000/(0.06 0.05)
15,000,000
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Sidebar Valuation by Earnings or Cash Flow
Multiples

• Analysts and investors sometimes take a shortcut
  approach to valuation
• multiply earnings or cash flow for the current
  year by an appropriate multiple.
• The best known of these is the price/earnings
  ratio.
• Valuation by multiple in the case of
  infinite-lived, constant growth, cash flows

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Multiples Appropriate for Valuing Infinite-Lived
Cash Flows with Constant Growth
Larger multiples are justified by higher cash
flow growth, or by lower interest (discount) rates
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Annuities

• An annuity is a series of equal payments made at
  fixed intervals for a specified number of periods
• e.g., 100 at the end of each of the next three
  years
• If payments occur at the end of each period it is
  an Ordinary Annuity.
• If payments occur at the beginning of each period
  it is an Annuity Due.

Ordinary Annuity
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Valuing Annuities

• Notice that an annuity that pays for T periods is
  equivalent to a perpetuity starting now less a
  perpetuity starting at date T1.
• The present value of a T period annuity paying a
  periodic cash flow of C when the discount rate is
  r is

• The textbook uses the shorthand notation Atr to
  denote the factor in brackets.
• If we have an annuity due instead, the net effect
  is that every payment occurs one period sooner,
  so the value of each payment is higher by the
  factor (1r).
• To obtain the PV with beginning-of-period
  payments, simply multiply the answer obtained
  based on end-of-period payments by the factor
  (1r).

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Annuity Example

• Compute the present value of a 3 year ordinary
  annuity with payments of 100 at r10

Or
If this had been an annuity due instead, the
value would be PV 248.68(1.1) 273.55
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Example A five year annuity paying 2000 per
year, with r 5.

• Valuing the payments individually we get

• Using the annuity formula we get
• Do on HP PMT2000, N5, I/YR5, FV0, push PV
  and get 8658.95 (note it displays as a negative
  number)

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• Alternatively, suppose you were given 8,658.95
  today instead of the annuity

• Notice that you can replicate the annuity cash
  flows by investing the present value to earn 5.
  
• This demonstrates that present value calculations
  provide a literal equality, in that the future
  cash flows can be converted into the present
  value and vice versa, if (and only if) the
  selected discount rate is representative of
  actual capital market conditions.

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Growing Annuities. A stream of regular payments
that grows at a constant rate, g.

• The present value of a T period growing annuity
  that pays C1 next period, with subsequent
  payments growing at rate g, when the discount
  rate is r is

or
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Example (growing annuity)

• A new venture is expected to generate 1 million
  in free cash flow during its first year.
  Subsequent cash flows will grow by 4 per year
  due to market growth and cost savings. Cash
  flows will continue for 20 years. The
  appropriate discount rate is 14.

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Growing Annuity Example (Continued)

• Impact of alternate cash flow growth rates

Higher future cash flow growth implies greater
value today!
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Annuities with Deferred Starts

• The standard formulas for level and growing
  annuities assume that the first payment occurs
  one period from now.
• If the first payment is deferred until later than
  one period from now, the standard formulas can
  still be used, but
• The answer denotes the value as of one period
  before the first payment, not the value now.

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A Valuation Problem
What is the value of a 10-year annuity that pays
300 a year at the end of each year, if the first
payment is deferred until 6 years from now, and
if the discount rate is 10? 0 1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 300 300
300
300 The value of the annuity payments as of
five years from now is Now discount this
equivalent payment back 5 years to time zero
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Application of Time Value Analysis Planning for
Retirement.

• You have determined that you will require 2.5
  million when you retire 25 years from now.
  Assuming an interest rate of r 7, how much
  should you have already saved? set aside each
  year from now until retirement?
• Determine the present equivalent of the targeted
  2.5 million.
• PV 2,500,000/(1.07)25
• PV 2,500,000/5.42743 460,623.
• Hmmm, you dont have that much saved??
• Determine the annuity payment that has an
  equivalent present value

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Retirement Planning Example, Continued.

• You expect your earnings to grow by 4 per year.
  Your retirement goal is unchanged but, instead of
  making equal payments, you wish to make payments
  that grow along with your earnings. How large
  should the first payment be? How large should
  the 25th payment be?

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A college planning example

• You have determined that you will need 60,000
  per year for four years to send your daughter to
  college. The first of the four payments will be
  made 18 years from now and the last will be made
  21 years from now. You wish to fund this
  obligation by making equal annual deposits over
  the 21 years. You expect to earn r 8 per
  year.
• Time Line

60
60
60
60
0
1
2
3
17
18
19
20
21
22
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College planning, cont.

• Step 1 Determine the time 17 value of the
  obligation

• Step 2 Determine the equivalent time zero
  amount

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College planning example, continued.

• Step 3 Determine the 21-year annuity that is
  equivalent to the stipulated present amount.

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Verification of Answer Obtained in
College-Planning Example
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Time Value Summary

• Discounted cash flow, or present value, analysis
  is the foundation for valuing assets or comparing
  opportunities.
• To use DCF you need to know three things
• Forecasted cash flows
• Timing of cash flows
• Discount rate (reflects current capital market
  conditions and risk).
• If there are no simple cash flow patterns, then
  each cash flow must be valued individually, and
  the present values summed. You can look for
  annuities or growing annuities to allow short
  cuts.
• Different streams of cash flows can be
  meaningfully compared (or equated) after they are
  each converted to their equivalent present
  values.