FNCE 100 Corporate Finance
Discounting

1

First Basic Principle of Finance



A dollar today is worth more than a dollar tomorrow
» But how much more?..

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Topic Overview





Compounding & Future Value
Discounting & Present Value
Multiple Cash Flows
“Special” Streams of Cash Flows
» Perpetuities
» Annuities



Interest Rates
» APR versus EAR

3

Lottery Example


You just won a lottery which gives you two options:
(1) Receive $100 today
(2) Receive $120 in one year
$100

$120

0



Money
Time

1

Which option should one take?

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Lottery Example: Future Value


If you take money now, you can put them in the bank at the current interest rate “r” of 5%, and have the following amount in one year:
V0=$100
0



Money
Time

V1=$105
1

The amount in one year – future value (FV) – is calculated as
FV = Principal + r × Principal = $100 + 0.05 × $100 = $105



Which option should we take now?
5

Lottery Example: Present Value


Alternatively, we can compare the value at time 0:

V0=$114.3
0


Money
Time

V1=$120
1

The amount in a given year – Present Value (PV) – is:
$120 = PV+ r × PV = PV+ 0.05 × PV → PV = $114.3



Which option should we take?

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Basic Terminology



Timeline: a linear representation of the timing of potential cash flows.
Two types of cash flows:
1. Inflows (i.e., money we get) are represented by positive numbers
2. Outflows (i.e., money we give) are represented by negative numbers



Example:
» Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments.

7

Money’s Time Units


Think of money as having a “time unit” denoting when it is received (or paid) 

We can only compare money in the same time units:

» Just like currency
» It doesn’t make sense to add $50 US to ₤50; and
» It doesn’t make sense to add $50 received today with $50 received next year.


Discounting and Compounding are the tools to manipulate money’s time units » Discounting converts money’s time units back in time
» Compounding converts money’s time units forward in time

→ ? value
→ ? value

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Terminology






Present Value (PV) – converts back in time
Future Value (FV) – converts forward in time
Principal – the amount of money
Interest rate – “exchange rate” between money today and tomorrow: »
»
»
»

Discount Rate
Hurdle Rate
Opportunity cost of capital
Required Return

9

Compounding


The Future Value (FV) of a cash flow T-years from today is:

FV  C 1  r 

T

» C = Cash Flow (or “CF”)
» r = discount rate


Example:
» Would you rather receive $1,000 today or $1,210 in two years if you can earn
10% per year on the $1,000?

Timeline and Future Value = ?

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Discounting


The Present Value (PV) of a cash flow T-years from today is:
PV 
» C = Cash Flow (or “CF”)
» r = discount rate



C

1  r 

T

 C 1  r 

T

Example:
» What is the price of a savings bond that will pay $15,000 in ten years if the annual interest rate is 6%?

Timeline = ?
Present Value = ?

11

The Power of Compounding


In 1626, Peter Minuit purchased Manhattan Island for $24. Given today’s real estate values in New York, this appears to be a great deal for Minuit.



But consider the value of the $24 in 2010 if it had been invested at an interest rate of 8% since 1626, for 384 years (2009 -1626 = 384):
Vn = V0 × (1+r)T
V384 = $ 24 × 1.08384 ~ $164 trillion
More than the value of the entire US stock market!

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Example: Investing in Real Estate


Consider buying a house that is currently worth $1.5 million. You know for certain that the house will be worth $1.59 million next year. Assume that your local bank is offering you a special interest rate of 3% per annum on loans. Should you buy the house (100% financing)?

13

Example: Present Value


You have decided to pursue an MBA 5 years from today. The first year tuition will be $35,000. If the annual interest rate is 4%, how much money do you have to save today?

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Example: Solving for the Interest Rate


You are looking at an investment that will pay $1,200 in 5 years from now if you invest $1,000 today. What is the implied rate of interest?

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Example: Solving for the Horizon


You want to purchase a new car and you are willing to pay $21,000. If you can invest at 10% per year, and you are currently have $15,000, how long will it be before you have enough money to pay cash for the car?

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Present Values and Discount Factor




The Multiplier

PV 

Present Value:

C

1  r 

T

 C 1  r 

T

1 is called the discount factor (DF)
(1  r )T
• Distant cash flows are discounted at a ... rate

1.00
0.90
0.80

Discount Factor

0.70

• A dollar two years from now is worth ... than a dollar one year from now

r =6%

0.60
0.50
0.40
0.30

• What happens introduce risk?

0.20
0.10
0.00
0

10

20

30

40

if

we

50

Period

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Multiple Cash Flows


Present Value (PV) and Future Value (FV) are linear operators
» PV(C1 , C2) = PV(C1)+PV(C2)
» FV(C1 , C2) = FV(C1)+FV(C2)



Example: If we can earn a 10% annual interest rate and save $1000 today, and $1000 at the end of each of the next two years how much will we have in 3 years?
Timeline = ?
FV = ?

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General Stream of Cash Flows


Present Value

PV 

N

 PV (C ) n n0




N





n0

Cn
(1  r )n

The PV of a stream of cash flows is just the sum of the PVs.
Future Value (same idea):
FV 

N

 FV (C ) n n0



N



n0

Cn 1  r   PV 1  r  n N

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Perpetuities


A perpetuity is a stream of cash flows with no end:

Cash Flows 0

C

C

C

C

C

C

…

Periods 0

1

2

3

4

5

6

…



Examples:
» Cencus Agreements issued in 12th century in Italy, France, and Spain to circumvent usury laws of Catholic Church (no principal = no loan)
» Hoogheemraadschap Lekdijk Bovendams
– 17th century Dutch Water Board to upkeep local dikes (they still pay interest!)

» British consol bonds
» Panama Canal perpetuities


How do we compute present value (PV) of perpetuity?

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Valuing Perpetuities


Step 1: Write out the PV of the perpetuity
PV C 1r





1



1

 1 r

C  r
1

  ...

2

3



2

 
1r



3

... 


Step 3: Multiply both sides by 1/(1+r)
1
PV 1 r  C 1 r






Step 2: Pull out the cash flow, C
PV C 1 r




C 1r



2

 
1r



3

 
1r



4


...


Step 4: Subtract (3) from (2)
1
1
PV 1  1  r    C 1  r 





Step 5: Do some algebra
PV 

C r 21

Example: Perpetuity



What does the timeline look like



The stream of cash flows is a ?

?

with a PV = ?

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Growing Perpetuities


A growing perpetuity is a stream of cash flows that grow at a constant periodic rate, g, with no end.

Cash Flows 0
Periods 0


C

C(1+g)

1

C(1+g)4

2

3

C(1+g)5

…

5

C(1+g)2 C(1+g)3

6

…

4

Infeasible to calculate by brute force - is there a shortcut?
PV 

C rg » You should be able to derive this

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Annuities


An annuity is a level stream of regular payments that last for a fixed number of periods

Cash Flows 0

C

C

C

C

C

0

Periods 0

1

2

3

N-1

N

N+1



0…
…

Examples:
» Mortgages
» Lottery prizes (sometimes…)
» Retirement savings plans



How do we compute PV?
» We replicate an annuity with some securities we know how to price!

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Valuing Annuities: Replicating Securities


An annuity is just the difference in two perpetuities starting at different times! » Perpetuity #1 starts today:

Cash Flows 0

C

C

C

C

Periods 0

1

2

…

N-1

C
N

C

…

N+1

…

– It has present value at time 0 equal to C/r.

» Perpetuity #2 starts in period N:
Cash Flows 0

0

0

0

0

0

C

…

Periods 0

1

2

…

N-1

N

N+1

…

– It has present value at time N equal to C/r and at time 0 equal to (C/r)(1+r)-N

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Valuing Annuities


Subtracting the cash flow streams of the two perpetuities gives us the cash flow stream for our annuity

Cash Flows 0

C

C

C

C

Periods 0

1

2

…

N-1



C
N

0

…

N+1

…

Therefore, difference in present values for the two perpetuities must equal the present value of our annuity

C
N 
N
C C
PV  PV  Perpetuity #1  PV  Perpetuity #2       1  r    1  1  r  

r r
 r


What’s the future value of an annuity

C
N
N 
N
N
C
FV  PV 1  r    1  1  r    1  r   1  r   1

 r r

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Example: Annuity



PV of Option A:
» What is the timeline ?

» What is the present value of all the cash flows



PV of Option B =

?

?
27

Example: Loan



The “usual” consumer loan has fixed regular payments – it is an annuity
You just borrowed $10,000 for three years at 6%. Three equal payments are due at the end of each year. What is the size of regular payments?

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Valuing Growing Annuities


A growing annuity is a constant growing stream of regular payments that last for a fixed number of periods

Cash Flows 0
Periods 0


C

(1+g)C

1

2

(1+g)2C
3

…

(1+g)N-2C

(1+g)N-1C

N-1

N

0

…

N+1 …

The present value of this stream is
PV 

T
C   1 g  
1  

r  g   1 r  



» You should be able to derive this

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Compounding Periods


So far everything was on an annual basis
» Cash flows were every year
» Interest was on an annual basis (i.e., compounded once a year)



Often, compounding occurs more frequently than annually
» E.g, semi-annual, quanrterly, monthly, continuous, etc.



How does change that affect the computations of the interest received or paid during the year?

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Compounding and Interest Rate


Earning 5% annually is not the same as earnings 2.5% twice a year (every six months) because of compounding:
$1



 1  0.025

$1.025

 1  0.025

$1.050625

With several compounding periods during the year, the effective annual interest rate (EAR) is higher than a simple annual percentage rate (APR) that ignores discounting
» The APR is 5%
» The EAR is 5.0625%

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Annual Percentage Rate (APR)


The Annual Percentage Rate (APR), indicates the amount of simple interest earned in one year.
» Simple interest is the amount of interest earned without the effect of compounding. » APR is what banks typically quote, e.g. credit card with 10% APR



The APR itself cannot be used as a discount rate.
» The APR with m compounding periods is a way of quoting the actual interest earned each compounding period:

Interest Rate per Compounding Period  i 

APR m periods / year

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Effective Annual Rate (EAR)


The Effective Annual Rate (EAR) indicates the total amount of interest that will be earned at the end of one year
» Incorporates the effect of compounding
» Also referred to as the effective annual yield (EAY) or annual percentage yield (APY)
» We can use this to discount cash flows, as long as cash flows occur on annual basis and we express time in annual units (i.e., years)
Interest Rate per Compounding period = (1+EAR)1/m-1

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Example: EAR and APR

?



Timeline:



Monthly rate =



Periodic Cash Flow =

?
?

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EAR vs APR


How do I convert an APR (not a discount rate) to an EAR (a discount m rate)?

APR 

1  EAR  1 

m


»
»


EAR
?
with the frequency of compounding
If compounding is once per year (m=1) then EAR= ?

We also defined an interest rate per compounding period: i = APR/m = interest rate per compounding period



Periodic interest rate can be used as a discount rate as long as we express time in corresponding compounding periods (quarters, months, etc.)

35

APR to EAR Conversions


Assume the APR is 6%. Then:
Compounding Interval
Annual (m=1)
Semiannual (m=2)
Monthly (m=12)
Daily (m=365)
Continuous (m=∞)



EAR
(1+0.06)1-1=6%
(1+0.06/2)2-1=6.09%
(1+0.06/12)12-1=6.1678%
(1+0.06/365)365-1=6.1831% e0.06-1 = 6.1837%

Continuous compounding m APR 

APR limm 1 
 =e m 
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Converting the APR to a Discount Rate
Example



Strategy: Compute the PV of the lease and compare it with the $150,000
Timeline: ?



This cash flow stream is an



?

with

?

periodicity

37

Converting the APR to a Discount Rate
Example (Cont.)


Computing the monthly discount rate:
» Method 1:
– We’re given an APR of 5% with semiannual compounding, which implies the EAR = ?
– Convert annual discount rate into monthly discount rate

?

» Method 2:
– Compute an effective periodic (6-month) interest rate from the APR,
– Convert six-month discount rate into monthly periodic rate:

?

?

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Converting the APR to a Discount Rate
Example (Cont.)


With the monthly discount rate in hand, the PV of the annuity is ?



The PV of the lease is greater than the upfront payment of
$150,000 so purchase the system outright

39

More Examples


What is the present value of $100 to be received 20 years from now assuming an interest rate of 9% and semi-annual compounding? 

If you have a twenty-year mortgage with a fixed interest rate
(APR) of 5% and monthly payments of $3500, what is the principal amount of the loan?

40

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What Formulas Should I Know?


Notation:
»
»
»
»
»
»



r = EAR
R = APR
T = # of years m = number of compounding periods per year
N = Total number of periods = T * m (years * periods/year) i = R/m = effective periodic interest rate

Converting from an APR to an EAR:

m

m

m
 APR 
 R
1  EAR  1 
  1  r  1    1  i  m 
 m

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What Formulas Should I Know? (Cont.)



Present and Future Value of a Single Cash Flow

PV  FV 1  i 


 PV  FV 1  r 

T

Present Value of a Growing Annuity
N

1  g  
1  


 (1  i)  


Implies: 1) Future Value of a Growing Annuity, 2) Future Value of an Annuity, and 3) Present Value of an Annuity

PV  C 

»


N

1
(i  g )

Present Value of a Growing Perpetuity

PV 
»

C
(i  g )

Implies: 1) Future Value of a Growing Perpetuity, 2) Future Value of a Perpetuity, and 3) Present
Value of a Perpetuity

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Summary


Money has a “time unit”
» Can only compare money in same units!
» Compound to get future values
» Discount to get present values



Future and Present Values are linear



Special streams of cash flows

» Use them on “streams” of cash flows
» Perpetuity
» Annuity


Interest Rates
» APR vs. EAR

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