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Discounting

Capital Budgeting and Corporate Objectives

Professor Ron Kaniel

Simon School of Business

University of Rochester

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Topic Overview

The Timeline

Compounding & Future Value

Discounting & Present Value

Multiple Cash Flows

“Special” Streams of Cash Flows» Perpetuities

» Annuities

Interest Rates

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The Timeline

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.

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Money’s Time Units

Think of money as having a “time unit” denoting when it is received (or paid)» Just like currency

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.

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

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Compounding

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

» 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?

1T

FV C r

Timeline and Future Value = ?

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Discounting

The Present Value (PV) of a cash flow T-years from today is:

Example:» What is the price of a savings bond that will pay $15,000 in ten years if

the annual interest rate is 6%?

1

1

T

T

CPV C r

r

Timeline = ?

Present Value = ?

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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

The PV of a stream of cash flows is just the sum of the PVs.

Future Value (same idea):

0 0

( ) (1 )

N Nn

n nn n

CPV PV C

r

0 0

( ) 1 1N N

N n N

n nn n

FV FV C C r PV r

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Perpetuities

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

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 PV ?

0 1 2 3 4 5 6 …Periods

Cash Flows 0 C1 C2 C3 C4 C5 C6 …

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Valuing Perpetuities

Step 1: Write out the PV of the perpetuity

Step 2: Pull out the cash flow, C

Step 3: Multiply both sides by 1/(1+r)

Step 4: Subtract (3) from (2)

Step 5: Do some algebra

1 2 31 1 1 ...PV C r C r C r

1 2 31 1 1 ...PV C r r r

1 2 3 41 1 1 1 ...PV r C r r r

1 11 1 1PV r C r

CPV

r

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PerpetuityExample

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.

Again, infeasible to calculate by brute force so is there a shortcut?

0 1 2 3 4 5 6 …Periods

Cash Flows 0 C C(1+g) C(1+g)2 C(1+g)3 C(1+g)4 C(1+g)5 …

CPV

r g

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Annuities

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

Examples:» Mortgages

» Lottery prizes (sometimes…)

» Retirement savings plans

How do we compute PV?

0 1 2 3 N-1 N …Periods

Cash Flows 0 C C C C C …

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Valuing Annuities – Part I

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

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

» Perpetuity #2 starts in period N:

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

0 1 2 … N-1 N N+1 …Periods

Cash Flows 0 CF CF CF CF CF CF …

0 1 2 … N-1 N N+1 …Periods

Cash Flows 0 0 0 0 0 0 CF …

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Valuing Annuities – Part II

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

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

What’s the future value of an annuity

0 1 2 … N-1 N N+1 …Periods

Cash Flows 0 CF CF CF CF CF 0 …

Perpetuity #1 Perpetuity #2PV PV PV

?

?

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PV of Option A:» What is the timeline

» What is the present value of all the cash flows

PV of Option B =

Annuity Example

?

?

?

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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

The present value of this stream is

0 1 2 3 … N-1 N N+1 …Periods

Cash Flows 0 CF (1+g)CF (1+g)2CF (1+g)N-2CF (1+g)N-1CF 0 …

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1

TC g

PVr g r

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Internal Rate of Return (IRR)

The Internal Rate of Return (IRR) is the one interest rate that sets the net present value of the cash flows equal to zero

Example 1:» 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) ofthe security:

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:

0

Initial Cost = 0(1 IRR)

Nn

nn

C

0

Price 0(1 IRR)

Nn

nn

C

0

Initial Outlay 0(1 IRR)

Nn

nn

C

1010

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Computing the Internal Rate of Return Example

The Timeline =

Present Value =

IRR =

?

?

?

Jessica takes out a $1 million loan today. Her bank offers her the following repayment option: pay $100,000 at the end of the first year, afterwhich the repayment amount will increase by 4% per annum forever. What is the IRR?

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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» 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)

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

1111

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Adjusting the Discount Rate to Different Time Periods

Earning 5% annually is not the same as earnings 2.5% every six months because of compounding

So, if the EAR is 5% but we have semi-annual discounting the Equivalent Periodic Rate (EPR) is

More generally,

» where m = # of compounding periods per year (e.g., semi-annual m = 2, quarterly m = 4, monthly m = 12, …)

» EPR is just an n-period discount rate

2 1/21 1 .05 1 0.05 1 0.0247 0.025EPR EPR

1/(1 ) 1mEPR EAR

1 0.025 1 0.025 $1 $1.025 $1.050625

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EAR and EPRExamples

If the EAR is 10% and we have quarterly compounding, what is the EPR

If the EPR is 0.6% and we have monthly compounding, what is the EAR

?

?

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Valuing Monthly Cash FlowsExample

Monthly EPR =

Timeline: ?

?

Periodic Cash Flow = ?

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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.

» The APR is typically less than the effective annual rate (EAR) which incorporates the effect of compounding

– Counterexample?

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:

APRInterest Rate per Compounding Period

periods / yeari

m

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EAR vs APR

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

» EAR increases with the frequency of

» If compounding is once per year (m=1) then EAR=

» Continuous Compounding:– In limit as m ∞, (1+APR/m)m exp(APR)

Some notation» R = APR (not a discount rate!)

» i = APR/m = interest rate per compounding period

1 1 m

APREAR

m

?

?

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Valuing Monthly Cash Flows Revisited Example

Recall the problem on slide 22:» Monthly interest with an EAR of 6%

What is the APR (R) on this account

How much interest is earned each period

How much do you have to save at the end of each month to accumulate $100,000 in 10 years

?

?

?

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Converting the APR to a Discount RateExample

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

Timeline:

This cash flow stream is an with periodicity

?

? ?

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Converting the APR to a Discount RateExample (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 interest rate from the APR,

Convert six-month discount rate into monthly periodic rate:

??

?

?

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Converting the APR to a Discount RateExample (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

?

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Nominal Versus Real Interest Rates

Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows, r

Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation, rr

1 Growth of MoneyGrowth in Purchasing Power 1

1 Growth of Prices

rrr

1

rrr r

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US Interest Rates and Inflation

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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» Use them on “streams” of cash flows

Special streams of cash flows» Perpetuity» Annuity

Interest Rates» APR vs. EAR» Real vs. Nominal