Discounting
Finance 100
Prof. Michael R. Roberts
Copyright © Michael R. Roberts
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Copyright © Michael R. Roberts
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Topic Overview
z
z
z
z
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The Timeline
Compounding & Future Value
Discounting & Present Value
Multiple Cash Flows
“Special” Streams of Cash Flows
» Perpetuities
» Annuities
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Interest Rates
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The Timeline
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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
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Example:
» Assume that you are lending $10,000 today and that the loan will be
repaid in two annual $6,000 payments.
Copyright © Michael R. Roberts
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Money’s Time Units
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Think of money as having a “time unit” denoting when it is
received (or paid)
» Just like currency
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We can only compare money in the same time units:
» 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.
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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
Copyright © Michael R. Roberts
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Compounding
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The Future Value (FVn) of a cash flow T-years from today is:
FV = C (1 + r )
T
» C = Cash Flow (or “CF”)
» r = discount rate
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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 = ?
Copyright © Michael R. Roberts
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Discounting
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The Present Value (PV) of a cash flow T-years from today is:
PV =
z
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 = ?
Copyright © Michael R. Roberts
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Multiple Cash Flows
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Present Value (PV) and Future Value (FV) are linear operators
» PV(C1 + C2) = PV(C1)+PV(C2)
» FV(C1 + C2) = FV(C1)+FV(C2)
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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 = ?
Copyright © Michael R. Roberts
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General Stream of Cash Flows
z
Present Value
PV =
N
∑ PV (C )
n = 0
z
z
n
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 = 0
n
=
N
∑
n = 0
Cn (1 + r ) = PV (1 + r )
n
N
Copyright © Michael R. Roberts
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Perpetuities
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A perpetuity is a stream of cash flows with no end:
Cash Flows 0
Periods 0
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C1
C2
C3
C4
C5
C6
…
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
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How do we compute PV?
Copyright © Michael R. Roberts
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Valuing Perpetuities
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Step 1: Write out the PV of the perpetuity
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Step 2: Pull out the cash flow, C
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Step 3: Multiply both sides by 1/(1+r)
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Step 4: Subtract (3) from (2)
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Step 5: Do some algebra
Copyright © Michael R. Roberts 10
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Perpetuity
Example
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What does the timeline look like ?
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The stream of cash flows is a ?
with a PV = ?
Copyright © Michael R. Roberts 11
Growing Perpetuities
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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
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C
1
C(1+g)
C(1+g)2 C(1+g)3
2
3
4
C(1+g)4
C(1+g)5
…
5
6
…
Again, infeasible to calculate by brute force so is
there a shortcut?
PV =
C
r−g
» You should be able to derive this
Copyright © Michael R. Roberts 12
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Annuities
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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
…
Periods 0
1
2
3
N-1
N
…
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Examples:
» Mortgages
» Lottery prizes (sometimes…)
» Retirement savings plans
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How do we compute PV?
Copyright © Michael R. Roberts 13
Valuing Annuities – Part I
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An annuity is just the difference in two perpetuities starting at
different times!
» Perpetuity #1 starts today:
Cash Flows 0
Periods 0
CF
CF
CF
CF
1
2
…
N-1
CF
CF
…
N
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
CF
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
…
…
Copyright © Michael R. Roberts 14
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Valuing Annuities – Part II
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Subtracting the cash flow streams of the two perpetuities gives
us the cash flow stream for our annuity
Cash Flows 0
Periods 0
z
CF
CF
CF
CF
1
2
…
N-1
CF
0
…
N
N+1
…
Therefore, difference in present values for the two perpetuities
must equal the present value of our annuity
PV = PV ( Perpetuity #1) − PV ( Perpetuity #2 ) =?
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What’s the future value of an annuity ?
Copyright © Michael R. Roberts 15
Annuity
Example
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PV of Option A:
» What is the timeline ?
» What is the present value of all the cash flows ?
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PV of Option B =?
Copyright © Michael R. Roberts 16
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Valuing Growing Annuities
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A growing annuity is a constant growing stream of regular
payments that last for a fixed number of periods
Cash Flows 0
CF
Periods 0
1
z
(1+g)CF (1+g)2CF
2
3
…
(1+g)N-2CF
(1+g)N-1CF
N-1
N
0
…
N+1 …
The present value of this stream is
PV =
C ⎡ ⎛ 1+ g ⎞
⎢1 − ⎜
⎟
r ⎢⎣ ⎝ 1 + r ⎠
T
⎤
⎥
⎥⎦
» You should be able to derive this
Copyright © Michael R. Roberts 17
Internal Rate of Return (IRR)
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The Internal Rate of Return (IRR) is the one interest rate that sets the net
present value of the cash flows equal to zero
N
∑
z
z
Example 1:
n = 0
Cn
− Initial Cost = 0
(1 + IRR) n
» The IRR of a security (e.g., bond, stock, CD, etc.) is just the one interest rate
that sets the present value of all the cash flows equal to the price (a.k.a. PV) of
the security:
N
Cn
− Price = 0
∑
n
n = 0 (1 + IRR)
Example 2:
» The IRR of an investment project (e.g., acquisition, merger, capital
expenditure, etc.) is just the one interest rate that sets the present value of all
the cash flows equal to the initial outlay (a.k.a. PV) of the investment:
N
∑
n = 0
Cn
− Initial Outlay = 0
(1 + IRR) n
Copyright © Michael R. Roberts 18
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Computing the Internal Rate of Return
Example
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The Timeline = ?
z
Present Value = ?
z
IRR = ?
Copyright © Michael R. Roberts 19
Effective Annual Rate (EAR)
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The Effective Annual Rate (EAR) indicates the total amount
of interest that will be earned at the end of one year
» Considers 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 we express time in
annual units (i.e., years)
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So far everything was on an annual basis
» Cash flows were every year
» Interest was on an annual bases (i.e., compounded once a year)
» Therefore, distinction was irrelevant: EAR = r
Copyright © Michael R. Roberts 20
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Adjusting the Discount Rate to Different
Time Periods
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Earning 5% annually is not the same as earnings 2.5% every six months
because of compounding
× (1 + 0.025 )
$1
z
$1.050625
So, if the EAR is 5% but we have semi-annual discounting the Equivalent
Periodic Rate (EPR) is
(1 + EPR )
z
× (1 + 0.025)
$1.025
2
− 1 = 5% ⇒ EPR = (1 + 0.05 )
1/2
− 1 = 0.0247 = 2.47% < 2.5%
More generally,
EPR = (1 + EAR )1/ m − 1
» where m = # of compounding periods per year (e.g., semi-annual Æ m = 2,
quarterly Æ m = 4, monthly Æ m = 12, …)
» EPR is just an n-period discount rate
Copyright © Michael R. Roberts 21
EAR and EPR
Examples
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If the EAR is 10% and we have quarterly
compounding, what is the EPR ?
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If the EPR is 0.6% and we have monthly
compounding, what is the EAR?
Copyright © Michael R. Roberts 22
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Valuing Monthly Cash Flows
Example
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Timeline: ?
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Monthly EPR = ?
Periodic Cash Flow = ?
z
Copyright © Michael R. Roberts 23
Annual Percentage Rate (APR)
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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.
» The APR is typically less than the effective annual rate (EAR) which
incorporates the effect of compounding
– Counterexample?
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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
Copyright © Michael R. Roberts 24
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EAR vs APR
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How do I convert an APR (not a discount rate) to an EAR (a
discount rate)?
m
APR ⎞
⎛
1 + EAR = ⎜ 1 +
⎟
m ⎠
⎝
» EAR increases with the frequency of ?
» If compounding is once per year (m=1) then EAR= ?
» Continuous Compounding:
– In limit as m Æ ∞, (1+APR/m)m Æ exp(APR)
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Some notation
» R = APR (not a discount rate!)
» i = APR/m = interest rate per compounding period
Copyright © Michael R. Roberts 25
Valuing Monthly Cash Flows Revisited
Example
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Recall the problem on slide 22:
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What is the APR (R) on this account ?
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How much interest is earned each period ?
» Monthly interest with an EAR of 6%
» Same as before so…
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How much do you have to save at the end of each month to accumulate $100,000 in
10 years ?
» Same as before!
Copyright © Michael R. Roberts 26
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Converting the APR to a Discount Rate
Example
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Strategy: Compute the PV of the lease and compare it with the
$150,000
Timeline: ?
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This cash flow stream is an ?
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with ?
periodicity
Copyright © Michael R. Roberts 27
Converting the APR to a Discount Rate
Example (Cont.)
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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 interest rate from the APR, ?
– Convert six-month discount rate into monthly periodic rate: ?
Copyright © Michael R. Roberts 28
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Converting the APR to a Discount Rate
Example (Cont.)
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With the monthly discount rate in hand, the PV of the
annuity is ?
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The PV of the lease is greater than the upfront
payment of $150,000 so purchase the system outright
Copyright © Michael R. Roberts 29
Nominal Versus Real Interest Rates
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Nominal Interest Rate: The rates quoted by
financial institutions and used for discounting or
compounding cash flows, r
z
Real Interest Rate: The rate of growth of your
purchasing power, after adjusting for inflation, rr
Growth in Purchasing Power = 1 + rr =
rr =
1 + r
1 + π
=
Growth of Money
Growth of Prices
r − π
≈ r − π
1 + π
Copyright © Michael R. Roberts 30
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US Interest Rates and Inflation
Copyright © Michael R. Roberts 31
What Formulas Should I Know?
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Notation:
»
»
»
»
»
»
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r = discount rate
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
Given a discount rate r for one period, convert to an n-period
discount rate:
n − period discount rate = (1 + r ) − 1
n
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Converting from an APR to an EAR:
⎛ APR ⎞
1 + EAR = ⎜1 +
⎟
m ⎠
⎝
m
Copyright © Michael R. Roberts 32
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What Formulas Should I Know? (Cont.)
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Real Interest Rates
rr =
z
Present and Future Value of a Single Cash Flow
PV = FV (1 + i )
z
⇔ PV = FV (1 + r )
−N
−T
Present Value of a Growing Annuity
PV = C ×
»
z
1+ R
−1
1+ π
1
(i − g )
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
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
Copyright © Michael R. Roberts 33
Summary
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Money has a “time unit”
» Can only compare money in same units!
» Compound to get future values
» Discount to get present values
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Future and Present Values are linear
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Special streams of cash flows
» Use them on “streams” of cash flows
» Perpetuity
» Annuity
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Interest Rates
» APR vs. EAR
» Real vs. Nominal
Copyright © Michael R. Roberts 34
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