Title: FIN 319: Intermediate Financial Management Spring 2005
1FIN 319 Intermediate Financial
ManagementSpring 2005
- SLIDE SET 2 TIME VALUE OF MONEY
- (BASED ON RWJ CHAPTER 4)
2Money 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.
3To 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
4(No Transcript)
5Present 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?
6PRESENT 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
7Present 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.
8Example -- 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.
9Alternate 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
10Alternate 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
11Example
- 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,
12Stated 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.
13Example
- 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.
14Valuing 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.
15Example
- 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
16Valuing 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
17Examples
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
18Sidebar 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
19Multiples 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
20 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
21Valuing 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).
22 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
23Example 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)
24- 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.
25Growing 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
26Example (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.
27Growing Annuity Example (Continued)
- Impact of alternate cash flow growth rates
Higher future cash flow growth implies greater
value today!
28Annuities 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.
29A 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
30Application 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
31Retirement 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?
32A 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
33College planning, cont.
- Step 1 Determine the time 17 value of the
obligation
- Step 2 Determine the equivalent time zero
amount
34College planning example, continued.
- Step 3 Determine the 21-year annuity that is
equivalent to the stipulated present amount.
35Verification of Answer Obtained in
College-Planning Example
36Time 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.