1 msbc 5060 chapter 4 discounted cash flow valuation and tvm

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MSBC 5060Chapter 4

Discounted Cash Flow Valuation and TVM

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Chapter Outline:1 One-Period Case

2 Multi-Period Case

3 Compounding Periods

4 Simplifications

5 Loan Amortization

6 What Is a Firm Worth?

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Key Concepts and Skills:• Compute the future value and/or present

value of a single cash flow or series of cash flows

• Compute the return on an investment• Use a spreadsheet to solve time value

problems• Understand Annuities and Perpetuities • Learn the “Rule of 72’s”

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Here is the Idea:• Get $100 today or Get $100 in one year.

– Which is better?

• Obviously getting the $100 today is better.

Why?• If you want to buy something today, you can.• If you want to buy something in 1 year instead:

– You can lend the $100 today for one year– And have more than $100 in one year

• So if I don’t get the money for one year,

I need to get more than $100– How much more? Talk about that in a soon!

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One Cash Flow

One- and Multi-Period Cases:

FV = C0 x (1+r)t

PV = Ct /(1+r)t

FV = Future Value

PV = Present Value

Ct = Cash Flow at time t

r = The interest rate

• We will solve for each of these variables – If we have the other 3.

• And talk about what the variables mean

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Future Value and Compounding• Future Value: What will a payment made today be worth

later?• Save $100 for 1 year at 10% interest. What will we have

in 1 year?

t = 1 r = 10% PV = $100 FV = ?

In 1 year the FV = PV( 1 + r)1 = $100(1.1)1 = $110• Save $100 for 2 years at 10% interest

Leave $110 in bank for a second year:

$110(1.1) = $121

$100(1.1)(1.1) = $100(1.1)2

• General Notation

$100(1 + r)t (1 + r)t is sometimes called the

Future Value Interest Factor

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Table A.3 (page 966 in the book)• Actually shows at TABLE of FVIFs• Use the table to look up FVIF for 10% in 5 years: 1.6105• So $100 in 5 years is worth $100(1.6105) = $161.05• But nobody uses tables anymore!• Use your spreadsheet

• This is equal to $100(1.10)5 = $100(1.6105) = $161.05

Using Excel:

=FV(rate, nper, pmt, [pv], [type])

=FV(.10,5,0,100) = -161.05

Why is the Excel answer negative?

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This is the formula that is in Excel:0 = PV + FV/(1+r)t

Solve for FV:

FV = -PV(1 + r)t

= -100(1.1)5 = -161.05• This formulation allows for “signing” cash flows:

– Positive is an inflow– Negative is an outflow

• The assumption is – If PV is positive (get money now)– Then FV must be negative (pay money later)

• But by convention, we report positive values

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Simple Interest vs. Compound Interest

$100 in Five Years at 10% per year:• Interest Factor: (1 + r)t = 1.15 = 1.61051• Future Value: $100(1.61051) = $161.05• 10% Interest on the $100 in each year is $10• Over five years it is $50• The extra $61.05 - $50 = $11.05 is interest on interest• Also called COMPOUND INTEREST

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Present Value• Present Value: What will a payment made later be worth

today?• Receive $5,000 in 12 years discounted by 6% interest. • Calculate the Present Value:• PV = FV/( 1 + r)t = $5,000/(1.06)12 = $2,484.85

Using Excel:

=PV(rate, nper, pmt, [fv], [type])

=PV(.06,12,0,5000) = -2484.85

• $2,484.85 < $5,000• $2,484.85 is the PV of $5,000 (at 6% over 12 years)• $5,000 is the FV of $2,484.85 (at 6% over 12 years)

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Review Question:• What is the PV of $10,000 if you receive the money in 5 years

and it is discounted at 12% per year?• What is the FV in 8 years of $30,000 paid today if it earns 9%

per year?

Note: Even though Excel will show negative numbers as FV and PV outputs, we still report positive values.

PV of $10,000 paid in 5 years at 12%:

=PV(rate, nper, pmt, [fv], [type])

=PV(.12,5,0,10000) = -5,674

FV in 8 years of $30,000 paid today at 9%:

=FV(rate, nper, pmt, [pv], [type])

=FV(.09,8,0,30000) = -59,777

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Interpreting PV and FV Example:• Example: Your company can pay $800 for an asset it believes it can

sell for $1,200 in 5 yrs. Similar investments pay 10%– What does similar mean?– Another investment with the same risk (a stock or bond issued

by another similar company) pays 10%• So is paying $800 for something that can be sold in 5 yrs for $1,200

a good idea?PV = FV/(1+r)t = $1,200/(1.1)5 = $745.11

In Excel: =PV(.10,5,0,1200) = -745.11

FV = PV(1+r)t = $800(1.1)5 = $1,288.41

In Excel: =FV(.10,5,0,800) = -1288.41

• It is a Bad Idea! You need to earn 10% so either:• You should pay less than $800 (pay $745.11) to get $1,200• You should pay $800 to get more than $1,200 (get $1,288.41)

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Review Question:• An investment costs $20,000 and you expect to hold it for 10 years. • Investments with similar risk earn 12% per year.• If the projected sale price is $75,000, is this a good investment idea?

If the projected sale price exceeds the calculated FV, then it is a good idea:

FV = PV(1+r)t = $20,000(1.12)10 = $20,000(3.10585) = $62,117

Or using Excel:

=FV(rate, nper, pmt, [pv], [type])

=FV(.12,10,0,20000) = -62,117

$75,000 > $62,117 so invest.

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Here’s the idea: • Over 10 years, the invest, which costs $20k and

pays $75k, has a return greater than 12% • If it paid only $62,117, the return would be 12%• Since it pays more ($75k) it must have a higher

return than the required return

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Determine the Discount Rate: Solve for r

PV(1 + r)t = FV

(1 + r)t = FV/PV

1 + r = (FV/PV)(1/t)

r = (FV/PV)(1/t) – 1

• What rate is need to increase $200 to $400 in 10 years?

r = ($400/$200)(1/10) – 1 = 0.071 = 7.18%

• What rate is need to increase $200 to $400 in 8 years?

r = ($400/$200)(1/8) – 1 = 0.0905 = 9.05%

In Excel:

=RATE(nper, pmt, pv, [fv], [type])

=RATE(8,0,200,400) = #NUM

=RATE(8,0,-200,400) = 9.05%

=RATE(8,0,200,-400) = 9.05%

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Review Question:• What rate is needed to increase $20,000 to $80,000 in 10 years?

r = (FV/PV)(1/t) – 1

= (80/20)(1/10) – 1

= (4)(1/10) - 1 = 0.1487 = 14.87%

Or

=RATE(nper, pmt, pv, [fv], [type])

=RATE(10,0,-20,80) = 0.1487 = 14.87%

Note:

=RATE(10,0,-1,4) = 0.1487 = 14.87%

=RATE(10,0,1,-4) = 0.1487 = 14.87%

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Determine the Number of Periods: Solve for t (or N)

PV(1 + r)t = FV

(1 + r)t = FV/PV

ln(1 + r)t = ln(FV/PV)

t[ln(1 + r)] = ln(FV/PV)

t = [ln(FV/PV)]/[ln(1 + r)]

But we’ll just use the machine:• How many years are needed to increase $200 to $400 at 7.18%

=NPER(rate, pmt, pv, [fv], [type])

=NPER(.0718,0,200,400) = #NUM!

=NPER(.0718,0,-200,400) = 10.00

=NPER(.0718,0,200,-400) = 10.00

(–PV and +FV or +PV and –FV both work)

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Review Question:• How many years are needed to get $1,000,000 if you

invest $22,095 and earn 10% per year?

t = [ln(FV/PV)]/[ln(1 + r)]

= ln(1,000,000/22,095)/ln(1.1) = 40

Or

=NPER(rate, pmt, pv, [fv], [type])

=NPER(.10,0,-22095,1000000) = 40

=NPER(.10,0,22095,-1000000) = 40

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Rule of 72’sIf FV/PV = 2 (your money doubles) then (r)(t) ≈ 72

•Example: Start with $200 and get $400 then FV/PV = 2

$200 to $400 in 10 years at 7.18%(10)(7.18) = 71.8 ≈ 72

$200 to $400 in 8 years at 9.05%(8)(9.05) = 72.4 ≈ 72

If the value of your stock doubles in 5 years, what is the approximate annualized compounded return?

(r)(t) ≈ 72 (r)(5) ≈ 72 (r) ≈ 72/5 = 14.4%

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Review Question:• You own a house that you believe has doubled in value over

the last 20 years. • Using the Rule of 72’s, estimate the approximate annual

return on the house.

(r)(t) ≈ 72

(r)(20) ≈ 72 (r)(20) ≈ 72/20 = 3.6

Check this answers using the calculator’s TVM function:

N= 20 PV = -1 FV = 2 I/YR = 3.53 ≈ 3.6

Bonus Question: • Assume you will earn 12%? How long to quadruple in price?• Quadruple is double twice: (r)(t) ≈ 72 72/r ≈ t 72/12 = 6 years• So double in 6, double twice in approximately 12 years• Check this: r = 12 PV = -1 FV = 4 N = 12.23 ≈ 12

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Recap:FV = PV(1+r)t

•Solve for any of the four variables

1.FV2.PV3.r (also called RATE)4.t (also called N or NPER)

• Use the Excel functions to solve for the one variable not given

• Be sure to understand the economic meaning of the values

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The FV of Multiple CFs• For any one CF: FV = PV(1+r)t

• For multiple CFs, the FV is the sum of each FV

Example: • Receive $100 at t = 0 and t = 1. • Calc the FV at t = 2 if the rate is 8%

– The first $100 increases twice. – The second $100 increases once.

FV = $100(1.08)2 + $100(1.08) = $224.64

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Another Example:• You currently have $7,000 in an account (at t = 0)• You will deposit $4,000 at the end of each of the next 3

years (at t = 1, t = 2 and t = 3)• How much will you have at time 3 at 8%?

$7,000 at t = 0 with 3 years of interest $7,000(1.08)3 = $8,818$4,000 at t = 1 with 2 years of interest $4,000(1.08)2 = $4,666$4,000 at t = 2 with 1 year of interest $4,000(1.08)1 = $4,320$4,000 at t = 3 $4,000 = $4,000

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Example continued• Same Example, but now…• How much will you have at time 4 at 8%?

$7,000 at t = 0 with 4 years of interest $7,000(1.08)4 = $9,523$4,000 at t = 1 with 3 years of interest $4,000(1.08)3 = $5,039$4,000 at t = 2 with 2 years of interest $4,000(1.08)2 = $4,666$4,000 at t = 3 with 1 year of interest $4,000(1.08)1 = $4,320

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Calculations:

FV at t = 3:$7,000(1.08)3 = $8,818

$4,000(1.08)2 = $4,666

$4,000(1.08)1 = $4,320

$4,000(1.08)0 = $4,000

$21,804

FV at t = 4:$7,000(1.08)4 = $9,523

$4,000(1.08)3 = $5,039

$4,000(1.08)2 = $4,666

$4,000(1.08)1 = $4,320

$23,548

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Now Calculate the PV of Multiple CFs:You need $1,000 at t = 1 and $2,000 at t = 2

How much do you need to invest today if you earn 9%?

Or what is the PV of these cash flows at 9%?

$1,000/(1.09) + $2,000/(1.09)2 = $2,600.79

0

$1,683.36$2,600.79

$917.43

2

$1,000

1

$2,000

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Think about the PV this way:• Invest $2,601 at 9%.• Show that you can

withdraw $1,000 at t = 1 and

withdraw $2,000 at t = 2:$2,600.79(1.09) = $2,834.86 (at t = 1)

$2,834.86 - $1,000 = $1,834.86 (withdraw $1,000 at t = 1)

$1,834.86(1.09)2 = $2,000 (available at t = 2)

• So if you invest $2,601 at 9%, you can withdraw $1,000 at time 1 and $2,000 at time 2

• The PV of $1,000 at time 1 and $2,000 at time 2 is $2,601

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Review Question:• If your investment earns 10%, how much do you need to

invest now to be able to withdraw $500 in one year and $800 in two years?

• In order to withdraw $500 in one year and then $800 in two years, you must invest the sum of the PVs of these withdrawals.

PV of $500 in one year = $500/(1.1) = $455

PV of $800 in two years = $800/(1.1)2 = $661

Sum = $455 + $661 = $1,116

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Multiple CFs: Some Terms:1. Annuity

– A stream of constant cash flows that lasts for a fixed number of periods

2. Growing Annuity– A stream of cash flows that grows at a constant rate

for a fixed number of periods

3. Perpetuity– A constant stream of cash flows that lasts forever

4. Growing Perpetuity– A stream of cash flows that grows at a constant rate

forever

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The Word “Annuity” has two definitions

Economic Definition:1. All CFs are the same2. CFs occur at regular intervals (Annually, Semi-annually, Quarterly,

Monthly…)3. All CFs are discounted at the same rate

The Financial Product:4. Pay an insurance company (or a bank) a lump sum today5. Receive CFs at regular intervals for a fixed period or until you die6. Sometimes you pay now (or make regular payments starting now)

and then receive payments when you retire at 65

Same pattern of Cash Flow rules for:• Loans (you pay) • A Purchased Annuity (you are paid)

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Formula for PV of an Annuity (PVA)

PVA = (C/r)[1 - 1/(1 + r)t] (Same thing but typed)

Text Book’s Notation:1/(1 + r)t = Present Value Factor (PVF)

PVA = C{[1 - PVF]/r}

Other Notation:{[1 - 1/(1 + r)t]/r} = Present Value Annuity Factor (PVAF)

PVA = C{PVAF}

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PV of $1,000 per for 5 years @ 6%:

We’ll use Excel’s PV function:=PV(rate, nper, pmt, [fv], [type])

=PV(.06,5,1000) = -4,212.36

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Future Value of an Annuity• You will receive $50 per year for next 10 years• When you get the money, you will deposit it in a bank

and earn 7%• Calculate the FV of a 10 yr, $50, 7% annuity?

Formula: FVA = C{[(1 + r)t -1]/r}

= C{FVAF}

Excel: =FV(rate, nper, pmt, [pv], [type])

=FV(0.07,10,50) = -690.82

Note: 10 x $50 = $500 < $690.82

$690.82 - $500 = $190.82 is interest and interest-on-interest

PV of a Growing AnnuityA growing stream of cash flows with a fixed

maturity:

0 1

C

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

Growing Annuity Example• A defined-benefit retirement plan pays a growing

amount each year for 40 years• The 1st payment is $20,000• Payments increase by 3% each year. • Calculate the PV at retirement if the discount rate is

10%?

0 1

$20

57.121,265$10.1

03.11

03.10.

000,20$40

PV

2

$20×(1.03)

40

$20×(1.03)39

Growing Annuity Example• What if you will not retire for five more years?• Calculate today’s value of the retirement plan

Recall:• A defined-benefit retirement plan pays $20,000 per year for 40

years. Payments increase by 3% each year and the discount rate is 10%

At retirement the plan is worth: PV = C/(r-g){1 – [(1 + g)/(1 + r)]T} = 20,000/(.1 – .03){1 – [1.03/1.1)]40} = $265,121.57

=PV((1+r)/(1+g)-1, nper, pmt, [fv], [type])/(1+g)

=PV((1.1)/(1.03)-1,40,20000)/(1.03) = -265,121.57

Five years earlier it is worth:265,121.57/(1.1)5 = 164,619.64

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PV Annuity vs. Growing Annuity

=PV((1+r)/(1+g)-1, nper, pmt, [fv], [type])/(1+g)

𝑃𝑉Growing Annuity= 𝐶𝑟 −𝑔 [1−( 1+𝑔1+𝑟 )

𝑡 ]

𝑃𝑉 Annuity=𝐶𝑟 [1−( 1

1+𝑟 )𝑡]

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Annuities Due• An Annuity Due means the payments are made at the

beginning of each period, not at the end of each period:• So the First payment is made immediately, not at the end

of the first period.• The figure below shows the payment timing of a four

year $100 Annuity and an Annuity Due:

• In Excel, use the “Type” indicator• =PV(rate, nper, pmt, [fv], [type])• Type = 1 for payments at the beginning of the period

Example

Annuity Annuity Due $100

$100$100 $100 $100

40 1 2

$100 $100

3

$100

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Compare an Annuity Due to an Annuity• Is the PV of a 4 yr Annuity Due greater than or less than

the PV a regular 4 yr annuity?– Would you rather be paid the Annuity Due or the Annuity?– Assume a 10% discount rate

• The 4 yr Annuity Due is the same as a 3 yr (regular) Annuity plus an extra $100 now (at time zero):

• 3 yr Annuity: =PV(.10,3,100) = 249• PV 3 yr Annuity + $100 = 349

• In Excel =PV(rate, nper, pmt, [fv], [type]) = PV(.10,4,100,,1) = 349=PV(rate, nper, pmt, [fv], [type]) = PV(.10,4,100) = 317

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Review Question:• You will pay $1,000 per month to rent apartment for a year• The lease requires monthly payments at the beginning of each month. • Assume a 12% APR-Monthly discount rate. • Note: 12% APR means 1% per month• Calculate the NET BENEFIT (in present value terms) to the landlord of

receiving the rent payments at the beginning of each month as opposed to the end of each month.

• PV of a 12 month, $1,000 (Regular) Annuity discounted at 12% APR. • 12% APR means 1% per month• =PV(.01,12,1000) = 11,255

• PV of a 12 month, $1,000 Annuity Due discounted at 12% APR:• =PV(.01,12,1000,,1) = 11,368

Net Benefit = $11,368 – 11,255 = $113

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Review Question:• You could have done this:• Since all payments, expect the first and last, are the

same, compare just those two:

PV of one $1,000 payment at time 12 discounted at 1% per period:

=PV(.01,12,0,1000) = 887

Note: In Excel, a single payment is an FV, not a PMT

PV of one $1,000 payment at time 0 is $1,000

• Net Benefit = $1,000 - $887 = $113

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PV of a Perpetuity• A perpetuity is a level stream of CFs that lasts forever• The PV of a Perpetuity equals:

• The denominator in each successive term in the brackets has a larger exponent, so the value approaches zero

• This is known as a “convergent sequence”• The value to which it converges is:

𝑃𝑉=𝐶

(1+𝑟 )+

𝐶(1+𝑟 )2

+𝐶

(1+𝑟 )3+⋯

𝑃𝑉=𝐶 [ 1(1+𝑟 )

+1

(1+𝑟 )2+

1

(1+𝑟 )3+⋯  ]

𝑃𝑉=𝐶 [ 1𝑟 ]=𝐶𝑟

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Perpetuity Example• A company’s preferred stock will pay $5

annual dividend forever• Preferred stock of similar risk has a 6%

return• Calculate the price of the preferred stock• PV = C/r = $5/.06 = $83.33

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PV of a Growing Perpetuity• CFs grow at a constant rate (g) forever• The PV of a Growing Perpetuity equals:

If g < r (and it pretty much has to be), then

𝑃𝑉=𝐶0 (1+𝑔 )1

(1+𝑟 )1+𝐶0 (1+𝑔)2

(1+𝑟 )2+𝐶0 (1+𝑔)3

(1+𝑟 )3+⋯

𝑃𝑉=𝐶0 [ (1+𝑔)1

(1+𝑟 )1+

(1+𝑔 )2

(1+𝑟 )2+

(1+𝑔)3

(1+𝑟 )3+⋯   ]

𝑃𝑉=𝐶0 [( 1+𝑔1+𝑟 )1

+(1+𝑔1+𝑟 )2

+( 1+𝑔1+𝑟 )3

+⋯  ]( 1+𝑔1+𝑟 )<1

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PV of a Growing Perpetuity

• If g < r, then each fraction is less than one• So this is also a convergent sequence• If converges to:

𝑃𝑉=𝐶0 [( 1+𝑔1+𝑟 )1

+(1+𝑔1+𝑟 )2

+( 1+𝑔1+𝑟 )3

+⋯  ]

𝑃𝑉=𝐶0 [ 1+𝑔𝑟 −𝑔 ]=𝐶0 (1+𝑔)𝑟 −𝑔

=𝐶1

𝑟 −𝑔

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Growing Perpetuity Example• A company has a fixed finance and operating policy such

that its common stock dividend will grow at a fixed rate of 4% forever.

• It is expected to pay a dividend of $2 in one year• Similar stocks have a required return of 14%• Calculate the price of the common stock• PV = C1/(r – g) = $2/(0.14 – 0.04) = $20

• Now assume increased efficiency will allow the company increase its growth rate to 6% forever

• But the change will not affect the next dividend• Calculate the new price of the common stock• PV = C1/(r – g) = $2/(0.14 – 0.06) = $25

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Loan AmortizationHow will loan’s principal will be repaid?• Pure Discount Loan

– Two CFs: Borrower receives money at beginning– Borrows makes a single payment at end– The single payment covers both interest and principal– Zero-coupon Bonds, CP, T-Bills and CDss

• Interest-Only Loans– Borrower makes periodic payments which are just interest– Principal (and the final interest payment) paid at the end– Most bonds (Gov, Corp, Muni)

• Amortizing (or Self-Amortizing) Loans– Periodic payments include principal and interest– Payments are calculated using Excel PMT function– Loan amount is PV of the “annuity” payments– Consumer loans

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Self-Amortization LoansExample: • Consider a 5 year, $3,000 self-amortizing at 10%

annual interest.• Calculate the annual payments:• Use Excel PMT function:

=PMT(rate, nper, pv, [fv], [type])

=PMT(.10,5,3000) = -791.39

• The Amortization Schedule shows the portion of each fixed payment that is interest the portion that repays the loan

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Amortization Schedule• 5 year, $3,000, 10% annual, self-amortizing loan • Annual Payments are $791.39

• Beg Bal = Previous Period’s End Bal = 2,508.61• Interest = Beg Balance x Rate = 2,508.61 x 0.1 = 250.86• Principal = Payment – Interest = 791.39 – 250.86 = 540.53• End Bal = Beg Bal – Prin = 2,508.61 – 540.53 = 1,968.08 • At the end of year 2, $1,968.08 not yet repaid

Period Beg Bal PMT Interest Principal End Bal

1 $3,000.00 $791.39 $300.00 $491.39 $2,508.61

2 $2,508.61 $791.39 $250.86 $540.53 $1,968.08

3 $1,968.08 $791.39 $196.81 $594.58 $1,373.50

4 $1,373.50 $791.39 $137.35 $654.04 $719.46

5 $719.46 $791.39 $71.95 $719.44 $0.01

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Amortization Schedule• What is the ending balance at time 2?

• It is the amount that still has to be paid• It is also the PV of remaining payments• At time 2, there are 3 more $791.39 payments at 10%

=PV(rate, nper, pmt, [fv], [type])

=PV(.1,3,791.39) = 1,968.08• Also the value of a “balloon payment” due at time 2

Period Beg Bal PMT Interest Principal End Bal

1 $3,000.00 $791.39 $300.00 $491.39 $2,508.61

2 $2,508.61 $791.39 $250.86 $540.53 $1,968.08

3 $1,968.08 $791.39 $196.81 $594.58 $1,373.50

4 $1,373.50 $791.39 $137.35 $654.04 $719.46

5 $719.46 $791.39 $71.95 $719.44 $0.01

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Compounding Periods• Annual Percentage Rate (APR)• APR = Periodic Rate x # of Periods• Periodic Rate = APR / # of Periods• A credit charges 18% APR monthly• The rate is actually 18/12 = 1.5% per month• What annual rate is equivalent?

– Called the Effective Annual Rate (EAR)

EAR = (1.015)(1.015)(1.015)…(1.015) – 1 = (1 + 0.18/12)12 – 1 = 19.56%

• So indifferent between paying 18% APR Monthly and paying 19.56% per year

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Example:• An investment pays 12% APR Semi-Annual• Calculate the FV of $1,000 invested for 3 years

EAR = (1 + APR/m)m = (1 + 0.12/2)2 - 1 = 12.36 %FV = C(1 + EAR)T = $1,418.52 = $1,000(1.1236)3 = $1,418.52

Combine the calculations:FV = C(1 + APR/m)(m x T)

= $1,000(1 + .12/2)(2 x 3) = $1,418.52

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Another ExampleFormulas:• EAR = (1 + APR/m)m – 1• APR = m[1 + EAR)1/m – 1]

Example:• Payday Loans cost $75 for a $500 two-week loan• Calculate the APR and EAR

– Assume there are exactly 26 two-week periods per year.• Periodic rate = $75/$500 = 15%• APR = Periodic Rate x # of Periods = 0.15 x 26 = 390%• EAR = (1 + APR/m)m – 1 = (1.15)26 – 1 = 3686%

Yes, Really.

Three thousand, six hundred and eighty-six percent!

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Compounding Periods in ExcelEAR is called EFFECTAPR is called NOMINAL

Assume 8% mortgage (monthly rate). Calculate the EAR:• EFFECT(nominal_rate, npery) • EFFECT(.08,12) = 8.30%

A one-year CD pays 5%. What is the equivalent quarterly rate?• NOMINAL(effective_rate, npery) • NOMINAL(.05,4) = 4.91%• The periodic rate is 4.91%/4 = 1.23%

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Continuous Compounding• EAR = (1 + APR/m)m – 1• As m increases, the EAR for 10% increases

• As m approaches infinity EAR = er – 1• Called “Continuous Compounding”• 10% Continuously Compounded EAR = e.10 – 1 = 10.52%

m EAR1 10.00%2 10.25%4 10.38%

12 10.47%

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Where do we go from here? • Apply these TVM calculations to o corporate

decisions making – Cash Flows

• And Corporate securities– Stocks and Bonds

• A company is worth the present value of its cash flows• The problems are

– Determining the size of the cash flows– Determining the timing of the cash flows– Determining the correct discount rate for the cash flows

• But we will value corporate projects and stocks and bonds by discounting the cash flows paid to the owners