The Time Value of Money Lecture 3 and 4

Corporate Finance Ronald F. Singer

Fall, 2010

Basic Concepts of Time Value of Money

• What is the time value of money?• If I offered you either $6,000 or $6,500 which one would you choose?• If I offered you $6,000 today or $6,500 in two years, which one would you

choose?• The first problem is easy: It involves two different amounts received at the

same time.• The second problem is more difficult, as it involves different amounts at

different periods of time.

The interesting part of finance is that it involves cash flows that are received at different points in time. We must devise a way of "comparing" these two different amounts to be able to make a choice between them, (or to add them up).

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The Timeline

• A timeline is a linear representation of the timing of potential cash flows.

• Drawing a timeline of the cash flows will help you visualize the financial problem.

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The Timeline (cont’d)

• Assume that you lend $20,000 to a friend. You will be repaid in two payments, one at the end of each year over the next two years.

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The Timeline (cont’d)• Assume that you are lending $10,000 today and that the

loan will be repaid in two annual $6,000 payments.

The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow.

• Timelines can represent cash flows that take place at the end of any time period.

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Three Rules of Time Travel

• Financial decisions often require combining cash flows or comparing values. Three rules govern these processes.

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Present Value of a Lump Sum and the Discounting Process

There are three ways of computing the Present Value:

• 1. Use the Formula: Where R is the discount rate T is the number of periods

to wait for the Cash Flow• 2. Use a spreadsheet

• 3. Use a Financial Calculator ),,,(Pr FVPMTNRatePV esentValue ),,,(Pr FVPMTNRatePV esentValue

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)R+(1

Flow Cash=Value Present T)R+(1

Flow Cash=Value Present T

FVPMTPVYIN ,,,/,

Future Value of a Lump Sum and the Compounding Process

There are three ways of computing the Future Value: • 1. Use the Formula: Where: R is the discount rate T is the number of periods to wait for the

Cash Flow• 2. Use a Spreadsheet

• 3. Use a Financial Calculator Note: when using Spreadsheet, the rate say 12% is entered

as .12 or 12%. When using a typical calculator, enter 0nly 12Example

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)R+(1 x Flow Cash=Value Future T)R+(1 x Flow Cash=Value Future T

),,,( PVPMTNRateFV Value Future ),,,( PVPMTNRateFV Value Future

FVPMTPVYRIN ,,,/%, FVPMTPVYRIN ,,,/%,

The Relationship Between Present and Future Value

• PV = FVT = FVT x (1/(1+R)T )

(1+R)T

• FV =PV(1+R)T =

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Example Using Calculator

•Financial calculators recognize the formulas and relationship above so that they calculate present and future values by balancing the above equations.•Typical layout: N I/YR PV PMT FV

Now the idea here is that given N (number of periods) and I/YR the interest rate per period, then the equation PV = FVN * (1/(1+I%/YR)T must hold.

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Financial Calculator•Suppose you want to know what the (present) vaIue of receiving $2,000 in ten years. Perform the following operations: Enter 10N 8 2000 PV Compute --926.39

•What if you want the future value of 1000 after 5 years at 8%? 5 2-11

Examples

What is the Present Value of $1 received five years from today if the interest rate is 12%?

•  Using the formula:•  • Using the Spreadsheet function:

PV = PV(Rate, N, PMT, FV)

= PV(0.12, 5, 0,1)• Using the Calculator:

$0.567 = ).12 + (1 x $1 -5

Examples

What is the Future Value of $1 invested for five years if the interest rate is 12% ?

• Using the formula: • Using the spreadsheet formula

FV(RATE, NPER, PMT,PV) = FV(.12,5,0,1) • Using the calculator: 1 PV FV -1.7623

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$1.762 = ).12 + (1 x $1 5

Future Value of a Lump Sum and the Compounding Process

What is the future value of $100 in 3 years if the interest rate is 12% ?• Approach 1: Keep track of dollar amount being compounded:Period1 $100.00 + $100.00(0.12) OR $100(1.12) = $112.002 $112.00 + $112.00(0.12) OR $100(1.12)2 = $125.443 $125.44 + $125.44(0.12) OR $100(1.12)3 = $140.49 • Approach 2: Keep track of number of times interest is earned:Period1 $100(1.12) $100(1.12) = $112.002 $100(1.12)(1.12) $100(1.12)2 = $125.443 $100(1.12)(1.12)(1.12) $100(1.12)3 = $140.49 Notice that the process earns interest on interest. This is called compounding. The

further out in the future you go the more important is the effect of compounding

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Finding the Present Value of an Annuity

• An Annuity is a fixed payment received periodically over some time period.

• Suppose we have the following cash flow stream, and that the "interest rate" is 10%:

Time Line: 200 200 200 200 200

0 1 2 3 4 5This is a 5 year annuity of 200 each year.

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Finding the Present Value of an Annuity

• Formula: Present Value of an Annuity = PMT (1 – 1/(1+rate)T) rate(Not recommended)

• Spreadsheet FormulaPVA(Rate,N,PMT,FV)

• Calculator PMT

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Finding the Present Value of an Uneven Cash Flow

• Suppose we have the following cash flow stream and the discount rate is 10%

• Time Cash Flow 1 800 2 300

3 to 5 200Find its present value

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Method 1: The Sum of the Present Values of each payment: 800 300 200 200

200$800 X (.909) = $727.20── $300 X (.826) = 247.80────────┘ $200 X (.751) = 150.20──────────────┘ $200 X (.683) = 136.60────────────────────┘ $200 X (.621) = 124.20─────────────────────────┘ Present Value $1,386.00

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• Method 2: Recognize that this is a combination of two lump sums and an annuity that begins two periods in the future and lasts for three periods.

That is: 0 1 2 3 4 5 └───── ───── ───── ───── ────┘ ┴ ┴ ┴ ┴ 800 300 200 200 200

is equivalent to

0 1 2 3 4 5 Plus 0 1 2 3 4 5 └─ ─ ── ── ─── ───┘ └─ ─ ── ── ─── ───┘┴ ┴ ┴ ┴ ┴ ┴ ┴ ┴ 800 300 0 0 0 0 0 200 200 200

Which in turn is equivalent to:

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Present Values: 0 1 2 3 4 5• └───── ───── ───── ───── ────┘┴ ┴ ┴ ┴PV(.10, 1, 0, 800) $727.20 800 300 0 0 0 PLUSPV(.10, 2, 0, 300) 247.80

PLUS PLUS

0 1 2 3 4 5 └───── ───── ───── ───── ────┘┴ ┴ ┴ ┴PVA(.10, 5, 200 ) 758.20 200 200 200 200 200

MINUS MINUS

PVA(.10,2,200) 347.20 1 2 3 4 5 └───── ───── ───── ───── ────┘┴ ┴ ┴ ┴ 200 200 0 0 0

TOTAL $1,380• Using Financial Calculator: $1,386.26 (Hewlett Packard)

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Method 3: Treat this as two lump sum payments plus an annuity that begins in period 2. 0 1 2 3 4 5

└───── ───── ───── ───── ────┘┴ ┴ ┴ ┴ 800 300 200 200 200

Is equivalent to 0 1 2 3 4 5

└───── ───── ───── ───── ────┘┴ ┴ ┴ ┴ 800 300 PVA(10%,3,200) 727.27

247.80 411.05 497.37

$1,386.12

Note the Annuity begins IN PERIOD 2 and must be discounted back 2 (not three) years.

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PerpetuitiesSome Securities last "forever," and generate the equivalent of a perpetual

cash flow. Clearly, we cannot evaluate these perpetual cash flows in the

conventional manner. We do however, have formulas which allows us to evaluate these cash

flows.• A Perpetuity is a series of equal payments that continues forever.

0 1 2 3 4 5 .......... 98 99 100......└──── ──── ──── ──── ──── ──┴ ┴ ┴ ┴ ┴ ..........── ──── ─┘┴ ┴ 15 15 15 15 15 .......... 15 15 15 .......• The Present Value of a Perpetuity is:

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Rate Discount

Period per Flow Cash = PV Rate Discount

Period per Flow Cash = PV

r

CF=PV r

CF=PV

Perpetuities

Example: A British Government Bond pays 100,000 pounds a year forever (Consul). The market rate of interest is 8%. How much would you pay for this bond?

PV = Cash Flow = 100,000 = 1,250,000of perpetuity r 0.08How much is the bond worth if the first coupon is payable

immediately? PV of Bond = PV Immediate Payment Plus Value of Perpetuity

= 1,250,000 + 100,000 = 1,350,000

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Growing Perpetuity

If the cash flow grows at a constant rate, then the perpetuity is called a growing perpetuity

where CF1= Cash flow next year r = Market rate interest g = Constant Growth rate What is the present value of a cash flow stream that pays

$105,000 at the end of this year, and grows at5% per year forever?

PV of growing = 105,000 = 3,500,000 perpetuity 0.08 -0.05

= 105,000 = 3,500,000 0.032-24

g-rCF=Perpetuity Growing of PV 1

g-rCF=Perpetuity Growing of PV 1

A Clarification on Different Compounding Periods

• We have assumed that we are dealing with compounding only once a year.

But what happens when the compounding is done more than annually?

• Given the periodic interest rate, you can use the tables to find the present value of a single payment, the present value of a periodic annuity, as well as the future values.

• Example: Suppose you will receive $1,000 per month for 12 months. at an annual (simple) interest rate of 18%, compounded monthly, what is the present value of this cash flow?

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Definition of Rates

• Periodic Interest Rate: the interest earned inside the compounding period.

Example: 18% compounded monthly has a periodic rate of 1.5%

• Nominal (Simple) Interest Rate: interest is not compounded. the amount you would earn, annually, if the interest were withdrawn as soon as it is received. (This is the APR (Annual Percentage Rate) you find on credit card and bank statements)

Example: Invest $1,000 today at 18%, APR paid monthly. you would have $1,180 at the end of one year.

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Definition of Rates•Effective Interest Rate: the annual amount you would have if the interest is allowed to compound. (This is the actual interest earn over the year allowing for compounding) Example: invest $1,000 today at 18%, compounded monthly. Then the periodic interest rate is 1.5% per month. The nominal rate is 18%. Then, allowing for compounding, the effective rate is: .............................. 0┴───1┴───2┴───3┴───4┴─────────────┴───┘  (1+.015)12 - 1 = 19.56% thus if you invested $1,000 at 18% compounded monthly you would have   $1,195.62 • To convert from simple rates to effective rates, use the formula: effective rate = (1 + r/m)m – 1. r is the simple rate m is the number of compounding periods per year.

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• Now the present value of monthly cash flow over one year is calculated as the present value of an annuity, received for 12 periods at a periodic rate of 1.5%.

Thus you want to use 12 as N and 1.5% as I%YR in the calculator.

• (Alternatively, if your calculator has an option to set the payments per year you could set it to 12 but this is not recommended. There is a tendency to forget to reset it to annual payments for the next problem, and what to use for N gets confusing)

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• What if, at the same compounding interval, you received only 2 cash flows of 6,000 each in month 6 and month 12?

6000 6000 │ │

│ ......... │ 0─1─2─3─4 ─5──6──…..──11──12

• Finally, what if the compounding of 18% occurs only twice per year?

effective rate present value

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• To pay an Annual Interest Rate r, compounded m times during the year means pay r/m for m-times in a year

Example, to pay 10% compounded quarterly means 2.5% is paid 4 times a year

The Effective Interest Rate is = (1 + 0.025)4 – 1 = 0.1038 or 10.38%

The Effective Interest Rate exceeds 10% since interest is paid on interest.

• When the compounding interval approaches zero, we have continuous compounding (1 + r/m)m - 1 = er - 1 = (2.7183)r - 1.

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• If 1 dollar is continuously compounded at rate r, at the end of the year 1 dollar will grow to er

where e = 2.7183 (e is the base of the natural log) after n years, 1 dollar will grow to = enr

Example: ABC bank offers 10.2% compounded quarterly. XYZ bank offers 10.1% interest continuously compounded. which is better for you?

in ABC bank 1 dollar deposit grows to, after 1 year,= (1 + 0.102/4)4 = (1.0255) 4 = 1.1060 in XYZ bank 1 dollar deposit after 1 year grows to= e.101 = 1.1063

therefore, even though XYZ only pays 10.1%, continuous compounding makes XYZ interest a better deal. notice that both of these offers are better than 10.5% simple

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