• Financial Management

Sponsored Links

  •   
  • FileName: Timeval.pdf [preview-online]
    • Abstract: Financial ManagementTime Value of Money1Time Value of Money (TVM)Present value and future value– how much is $1 now worth in the future?– how much is $1 in the future worth now?Business planning– compare current to future cash flows

Download the ebook

Financial Management
Time Value of Money
1
Time Value of Money (TVM)
Present value and future value
– how much is $1 now worth in the future?
– how much is $1 in the future worth now?
Business planning
– compare current to future cash flows
• is investment in a particular project worthwhile?
• is one project a better investment than another?
Personal financial planning
– should you consume today or invest and consume later?
– how will your money grow over time?
– how much do you need to save today to retire later? 2
Professor Gordon Phillips 1
$1 Today versus $1 Later
Interest rates:
– the supply and demand of money must be equal
– the value of being patient; the cost of impatience
– depends on horizon (maturity)
– reflects expected inflation and a “real return”
What about $ 1 for certain in five year’s time versus $1
on average in five year’s time?
– Discount rates reflect both the “riskless” time value of money
as well as the riskiness of the cash flows
3
What techniques will we review?
What is the value today of $CT at time T?
What is the value at time T of $C0 today?
Compounding
– Effective annual interest rate (EAIR) vs.
Stated annual interest rate (SAIR)
– Continuous vs. discrete compounding
Multiple cash flow streams
– perpetuity
– growing perpetuity
– annuity
4
Professor Gordon Phillips 2
Time Value of Money: Notation
|________________________________|
C0 CT
PV FV
$ invested today $ received in future
Define some terms
CT = Future value at time T = FV; or cash flow at T
C0 = Current value = PV; or cash flow at O
r = Discount or interest rate
T = Number of future time periods
5
Present Value and Discounting
Discounting: if we receive $1 in two years, what
is it worth to us today (r = 9%)?
Year 0 1 2
C $0.842 $1
2
Present value (C0) = $1 / 1.09 = $0.842
The interest rate (9%) is also called the discount rate
6
Professor Gordon Phillips 3
Present Value - another example
How much would an investor have to set aside
today in order to have $20,000 five years from
now if the current rate is 15%?
PV $20,000
0 1 2 3 4 5
$20,000
$9,943.53 =
(1.15) 5
7
Future Value
Compounding: how much will $1 invested today
at 9% be worth in two years?
The Time Line
Year 0 1 2
C $1 $1.1881
2
Future value (C2) = $1 x 1.09 = $1.1881
8
Professor Gordon Phillips 4
Compounding
Compounding: interest on interest
– Benjamin Franklin: “Money makes money, and the money that
money makes makes more money”
Which one would you prefer?
1. simple interest: no compounding
FVT = PV0(1 + r*T)
2. annual compound interest
FVT = PV0( 1 + r )T
3. Semi-annual compound interest (bonds and banks)
FVT = PV0( 1 + r/2 )2T, where r is annualized,T in years
4. continuous compound interest: compound “every second”
FVT = PV0*erT (to be shown later)
9
“Money Makes Money…”
INTEREST RATE EXAMPLE
re turn 10%
contribution 100
ye a rs
0 1 2 3 4 5 10 15 30
1. S imp le Inte re s t 100 110 120 130 140 150 200 250 400
2. Co mp ound Inte re s t 100 110 $121.00 $133.10 $146.41 $161.05 $259.37 $417.72 $1,744.94
3. Co ntinuo us Compo und Inte re s t 100 $110.52 $122.14 $134.99 $149.18 $164.87 $271.83 $448.17 $2,008.55
10
Professor Gordon Phillips 5
Compounding
2500
1. Simple Interest
FV of $100 2000
1500 2. Compound
Interest
1000
3. Continuous
500 Compound
Interest
0
0 1 2 3 4 5 10 15 30
Years
11
Effective Annual Rates (EAIR)
The formula for computing the return when there are
multiple compounding periods is:
(1+SAIR/m)m = (1+EAIR)
m is the number of times interest is paid annually
SAIR represents the stated annual interest rate
EAIR represents the effective annual interest rate
If the SAIR is 12%, how much will I receive at the end of the year if
interest is paid monthly vs. semi-annually vs. yearly?
12
Professor Gordon Phillips 6
EAIR: Example
If you invest $50 for 1 year at 12% (the SAIR) compounded
semi-annually - what is the EAIR on that investment?
.12 2
FV = $50 × (1 + ) = $50 × (1.06) 2 = $56.18
2
The EAIR is the annual rate that would give us the
same end-of-investment wealth after 1 year:
$50 × (1 + EAIR ) = $56.18
13
EAIR: Solution
FV = $50 × (1 + EAIR) = $56.18
$56.18
(1 + EAIR) =
$50
EAIR = .1236
So, investing at12.36% compounded annually (i.e. the
EAIR) is the same as investing at the SAIR of 12%
compounded semiannually.
14
Professor Gordon Phillips 7
Continuous Compounding
The general formula for the future value of an investment
compounded continuously over many periods can be
written as:
FV = C0×erT
Where
C0 is cash flow at date 0,
r is the stated annual interest rate,
T is the number of periods over which the cash is
invested, and
e is a special number (= 2.718...). ex is a key on your
calculator, and the inverse function is the natural
logarithm (ln). 15
Present Values with
Continuous Compounding
Two types of compounding - be careful!
– Discrete Compounding FV
PV =
• annual or semi-annual most common: (1 + r )T
– Continuous Compounding: FV
PV =
e r *T
– very close for small r but not for large r: watch out for those “small”
discrepancies that make your bankers rich!
While no bank offers continuously compounded rates of return,
continuous compounding is often used in finance because it
facilitates calculations.
16
Professor Gordon Phillips 8
How long does it take to double your money?
If we deposit $5,000 today in an account paying 10%
annually, how long does it take to grow to $10,000?
FV = C0 × (1 + r )T $10,000 = $5,000 × (1.10)T
$10,000
(1.10)T = =2
$5,000
ln( 1 . 10 ) T = ln 2 ⇒ T ln( 1 . 10 ) = ln 2
ln 2 0.6931
T= = = 7.27 years
ln(1.10) 0.0953 17
Double your money ... using continuous compounding
Continuous compounding: FV/PV = erT
Double your money FV/PV = 2
erT = 2 ⇒ rT = ln (2) (if y=ex, then x = ln(y))
rT = .69
– if r=.10, T = 6.9 years (compare to 7.3 on last slide)
– if r=.20, T = 3.5 years
The rate or time required to double your money can be quickly figured out using the
“Rule of 72” (an approximate rule based on annual compounding).
18
Professor Gordon Phillips 9
What Rate Is Enough?
Assume the total cost of a college education will be $50,000
when your child enters college in 12 years. You have
$5,000 to invest today. What annual rate of interest must
you earn on your investment to cover the cost of your child’s
education? About 21.15%.
FV = C0 × (1 + r )T $50,000 = $5,000 × (1 + r )12
$50,000
(1 + r )12 = = 10 (1 + r ) = 101 12
$5,000
r = 101 12 − 1 = 1.2115 − 1 = .2115
19
Multiple Cash Flows
Now, a more realistic and complex setting
– Most investments generate multiple cash flows over time
– Special cases:
• Annuities: same cash flow repeated for a fixed number of periods
• Perpetuities: same cash flow repeated periodically, forever
• Growing annuities and perpetuities
Objective: understand Discounted Cash Flow (DCF) and
Net Present Value (NPV) so as to:
– value and price securities and companies
– evaluate projects: “capital budgeting”
20
Professor Gordon Phillips 10
Present and Future Value:
Multiple Cash Flows
Net Present Value and Investment Projects
– Net Present Value is a measure of the difference between the
market value of an investment and its cost. If the NPV is positive,
this indicates a profitable investment (if the NPV < 0, the
investment should not be made.
Estimating Net Present Value
Discounted cash flow (DCF) valuation consists of finding the
market value of assets or their benefits by taking the present
value of future cash flows, i.e., by estimating what the future
cash flows would trade for in today's dollars.
21
Net Present Value (NPV)
Some rules for computing NPV:
– use a simple time line to lay out all the cash flows
– only do simple addition or subtraction of cash flows if
they are from the same time period
– use an appropriate discount rate: later in the course
The general formula for calculating NPV:
NPV = -C0 + C1/(1+r) + C2/(1+r)2 + .. + CT/(1+r)T
C1 C2 C3
CT
...
-C0 22
Professor Gordon Phillips 11
Simple NPV Example
40 50
35
Assume r = 10%
-100
NPV = -100 + 40/(1+.10) + 35/(1+.10)2 + 50/(1+.10)3 = 2.85
23
Annuities
Series of equal period payments
0 1 2 3 4 .... T
|____|_____|______|______|____________|
C1 = C2 = C3 = C4 = CT
– For simplicity, we will assume a single rate of interest
which allows us to easily use the interest tables.
24
Professor Gordon Phillips 12
PV of an Annuity
1
1−
PV of annuity: T
C ( 1 + r)T
PV = ∑ t
=C⋅
t =1 (1 + r ) r
– NOTE: first payment is received in one year’s time
– Table A.2 in RWJ Appendix shows annuity factor values
– generalization: RWJ give the formula for a growing annuity
(equation 4.15 on page 92)
How much are 5 equal annual payments of $8,190
(starting one year from today) worth today if the annual
interest rate is 10%?
– Answer = $31,046.54
25
Annuity Practice Problem
Installment purchases:
– you are considering purchasing a house that costs
$70,000
– you intend to put 20% down and finance the
remainder
– if the 30 year mortgage rate is 7%, what will be your
monthly mortgage payment?
Note: we implicitly assume monthly
compounding with end of period payments
26
Professor Gordon Phillips 13
Amortization Table
Initial Data
LOAN DATA TABLE DATA
House Price $700,000.00
Loan amount: $560,000.00 Table starts at date: 1/1/03
Annual interes t rate: 7.00% or at payment number: 1
Term in years: 30
Payments per year: 12 PVIFA 150.3075679
Firs t payment due: 1/1/03
PERIODIC PAYMENT
Entered payment: The table uses the calculated periodic payment amount,
Calculated payment: $3,725.69 unless you enter a value for "Entered payment."
After-Tax payment $2,680.07
CALCULATIONS
Use payment of: $3,725.69 Beginning balance at payment 1: $560,000.00
1st payment in table: 1 Cumulative interest prior to payment 1: $0.00
Table
Payment Beginning Ending Cumulative
No. Date Balance Interest Principal Balance Interest
1 1/1/03 560,000.00 3,266.67 459.03 559,540.97 3,266.67
2 2/1/03 559,540.97 3,263.99 461.70 559,079.27 6,530.66
3 3/1/03 559,079.27 3,261.30 464.40 558,614.87 9,791.95
4 4/1/03 558,614.87 3,258.59 467.11 558,147.76 13,050.54
5 5/1/03 558,147.76 3,255.86 469.83 557,677.93 16,306.40
6 6/1/03 557,677.93 3,253.12 472.57 557,205.36 19,559.52
7 7/1/03 557,205.36 3,250.36 475.33 556,730.03 22,809.89
8 8/1/03 556,730.03 3,247.59 478.10 556,251.93 26,057.48
9 9/1/03 556,251.93 3,244.80 480.89 555,771.03 29,302.28
10 10/1/03 555,771.03 3,242.00 483.70 555,287.34 32,544.28
11 11/1/03 555,287.34 3,239.18 486.52 554,800.82 35,783.45
12 12/1/03 554,800.82 3,236.34 489.36 554,311.46 39,019.79 27
FV of an Ordinary Annuity
( 1 + r)T −1
Future Value FV = C ⋅
r
– FV annuity factors in Table A4 of RWJ
– Note that you can just calculate the PV of the annuity, and then find
the FV by compounding the PV forwards (in other words, you don’t
need to learn yet another formula)
You wish to take a year off work in 5 years
– If you save $8,190 each year for 5 years, how much will you have saved
by the end of the time period if the interest rate is 10%? (assume you
begin saving at the end of the year).
– Hint: see page 30 (take FV of the PV calculated – you get $50K)
28
Professor Gordon Phillips 14
Perpetuities
A perpetuity is a stream of infinite cash flows with the
first cash flow beginning one period from now
0 1 2 3 4 ........…..
C C C etc
|______|______|______|______|___________|
C
PV of a perpetuity: PV =
r
The value of $10 every year forever if the interest
rate is 5% is $200 (= $10 / .05)… quite simple! 29
Valuing Growing Perpetuities
A cash flow stream growing at a steady rate
– cash flow each year; the first one (in one year’s time) is $60,
but cash flow grows at 2% a year
– if the discount rate is 7%, how much would you be willing to
pay? (Answer = $1200)
Present value of growing perpetuity:
C
PV = , for r > g
r−g
Useful when valuing companies (or equity).
30
Professor Gordon Phillips 15
Summary
Present values and Future values allow us to easily switch back
and forth between cash flows separated in time
– calculations are mechanical: look for shortcuts, use formulas / tables
Simplifications: if possible reduce PV problems to
– Perpetuity: an everlasting stream of constant cash flows
– Growing perpetuity: stream of cash flows that grows at a constant rate
forever
– Annuity: a stream of constant cash flows that lasts for a fixed number of
periods
– Growing annuity: a stream of cash flows that grows at a constant rate for a
fixed number of periods
Important question for the future:
– where do the cash flows come from?
– what are their risks?
31
Professor Gordon Phillips 16


Use: 0.6604