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Discounted Cash Flow Valuation

Chapter 4

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Appreciate the significance of compound vs. simple interest

Describe and compute the future value and/or present value of a single cash flow or series of cash flows

Recognize and compute the impact of compounding periods on the true return of stated interest rates

Develop facility with a financial calculator and/or spreadsheet to solve time value problems

Comprehend and calculate time value metrics for perpetuities and annuities

Familiarization with loan types and amortization

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4.1 Valuation: The One-Period Case4.2 The Multi-period Case4.3 Compounding Periods4.4 Simplifications4.5 Loan Types and Loan Amortization4.6 What Is a Firm Worth?

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A dollar today is more valuable than a dollar to be received in the future

Why?◦ A dollar today is more valuable because: It can be invested to create more than a dollar in the

future It can be immediately consumed There is no doubt about its receipt

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If you know your required rate of return and the length of time before cash is harvested, you can calculate some critical metrics:

◦ The value today of a payment to be received in the future This measure is called a “Present Value”◦ The value in the future of a sum invested today This measure is called a “Future Value”

◦ Present and Future Values can be calculated over single and multiple periods

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If you were to invest $10,000 at 5-percent interest for 1 year, your investment would grow to $10,500.

$500 would be interest: $10,000 × 0.05$10,000 is the initial amount: $10,000 × 1$10,500 is the amount in the investment

after one year (i.e., at t=1), calculated as:$10,500 = $10,000×(1.05)

The total amount due at the end of the in-vestment is called the Future Value (FV).

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In the one-period case, the formula for FVcan be written as:

FV = C0×(1 + r)

Where C0 is cash flow today (time zero), and r is the appropriate interest rate.

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In the one-period case, the formula for FVcan be written as:

FV = C0×(1 + r)

Where C0 is cash flow today (time zero), and r is the appropriate interest rate.

NOTE: This C0 is oftenalso referred to as thePresent Value (PV)

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Present Value is today’s value of a sum to be received in the future given a specific rate of interest and time horizon.

Suppose you were promised $10,000 due in one year when interest rates are 5-percent. Your investment be worth $9,523.81 in today’s dollars.

The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is the Present Value (PV).

Note that $10,000 = $9,523.81×(1.05).

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In the one-period case, the formula for PVcan be written as:

Where C1 is cash flow at date 1, and r is the the appropriate required rate of return, also called the discount rate or (more generic-

ally) the interest rate.

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The Net Present Value (NPV) of an invest-ment is the present value of the expected cash flows, less the cost of the investment.

Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate (more pre-cisely called required return) is 5%.

Should you buy?

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The PV of the expected future cash inflow isGreater than the cost. In other words, the NetPresent Value is positive, so the investment

should be purchased.

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In the one-period case, the formula for NPV can be written as:

NPV = –Cost + PV of Future CFs

If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FV would be less than the $10,000 the investment promised, and we would be worse off in FV terms:

$9,500 × 1.05 = $9,975, or less than $10,000

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The previous examples considered only one period of time.

It is possible to compute PV & FV for multiple periods of time.

Doing so requires discrimination between simple and compound interest

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Suppose an investor has $1000 to invest at an interest rate of 9% (per year). After 1 year, he/she will have earned $90: $1000 x 0.09

If he/she takes the $90 earned out of the capital market and reinvests the $1000 capital at 9%, they will again earn $90 in a year’s time.

This process could continue ad infinitum with the annual interest earned never contributing further to the investor’s wealth

With simple interest, therefore, valuing future or present values never has to extend beyond the

single-period case

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Suppose an investor has $1000 to invest at an interest rate of 9% (per year). In 1 year, he/she will have earned $90: $1K x 0.09

Now suppose that the investor reinvests both the origi-nal $1K capital and $90 earnings for another year at 9%

In this 2nd year, he/she will earn $98.10, an amount that’s $8.10 greater than the previous year.

In this example, the interest in the 2nd year is higher than the first because it is paid on the initial capital andon prior earnings.

In other words, the interest also earns interest, or the interest is compounded.

Compounding may not seem very compelling the early years of an investment. But, we will see that it is a very

powerful long-term force.

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The general formula for the future value of an investment over many periods can be written as:

FV = C0 × ( 1 + r )TWhere

C0 is cash flow at time 0,r is the appropriate interest rate (per period), andT is the number of periods over which the time-zero cash is invested.

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Suppose you invest $1100 today, with a gigantic 40%/year rate of return for the next five years.

What will the investment be woth in 5 years?

FV = C0 × ( 1 + r )T

= $1100 × 1.405 = $5916.06

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Notice that the value at t = 5, $5916.06, is considerably higher than the sum of the original investment plus five simple-interest increases of 40% on the original $1100:

$5916.06 > $1100 + 5×$1100×0.40 =$3300

This difference is due to compounding.

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The general formula for the present value of a future cash flow at a future point in time ( T ) can be written as:

PV = CT / ( 1 + r )TWhere

CT is cash flow at date T, r is the appropri-ate discount rate (or interest rate), and T is the number of periods over which the cash is invested.

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How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

0 1 2 3 4 5

$20,000PV

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Examples thus far have offered the time and interest rate and solved for PV or FV

Keep in mind that there are four variables:◦ PV◦ FV◦ T◦ R

If you have any three you can solve for the fourth The math can become cumbersome◦ Financial Calculators and Spreadsheets are very helpful

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If we deposit $5,000 today in an account pay-ing 10% (per year), how long does it take to grow to $10,000?

$10K/$5K = 1.10T

ln($10K/$5K) = T x ln(1.10)ln($10K/$5K) / ln(1.10) = T

T = 7.27 years

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Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your invest-ment to cover the cost of the education?

… or 21.15%/year

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Texas Instruments BA-II Plus◦ FV = future value◦ PV = present value◦ I/Y = periodic interest rate

P/Y must equal 1 for the I/Y to be the periodic rate Interest is entered as a percent, not a decimal

◦ N = number of periods◦ Remember to clear the registers (CLR TVM) after

each problem◦ Other calculators are similar in format

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Consider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the required rate or return (or, more generically, the inter-est rate) is 12%, what is the present value of this stream of cash flows?

If the issuer offers this investment for $1,500, should you purchase it?

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0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Present Value < Cost → Do Not Purchase

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First, set your calculator to 1 payment per year.Then, use the cash flow menu:

CF2

CF1

F2

F1

CF0

1

200

1

1,432.93

0

400

I

NPV

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CF4

CF3

F4

F3 1

600

1

800

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All examples thus far have assumed annual com-pounding

Instances of other compounding schedules abound:◦ Banks compound interest quarterly, monthly, etc.◦ Mortgage companies compound interest monthly

Yet, almost all interest rates are expressed annually If a rate is expressed annually, but compounded

more frequently, then the effective rate is higher than the stated rate

This concept is called the effective annual rate or EAR

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Compounding an investment m times a year for T years provides for the future value of wealth:

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FV = C0 × ( 1 + r )T

First, simply transform the exponent to reflect the appropriate numberof compounding periods: T = number of years (the old T) times number

of compounding periods in one year (m)

Next, similarly transform the rate to reflect the appropriate rate per com-pounding period: r = annual rate (the old r) divided by number of

compounding periods in one year (m)

Then just use the same basic compounding equation from earlier in thechapter, being careful to ensure consistency across the r and the T

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For example, if you invest $50 for 3 years at 10% compounded semi-annually, your investment will grow to

or, using Durham’s preferred approach1. recognize that the period of interest is a half-year2. transform the investment horizon (T) into half-years: 2 (number of com-

pounding periods in one year) x 3 (old T, in years) = 63. transform the rate (old r, per year) into a rate per half-year: 0.10 / 2 = 0.05

4. then use the same compounding equation from the prior slide:

FV = $50 x 1.056 = $67.00

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A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?”

The Effective Annual Rate (EAR) of interest is the annual rate that would give us the sameend-of-investment wealth after 3 years:

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So, investing at 10.00%/year with semi-annualcompounding is effectively the same as

investing at 10.25%/year with annualcompounding.

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Find the Effective Annual Rate (EAR) of an 18%APR loan that is compounded monthly.

What we have is a loan with a monthlyinterest rate rate of 1½%.

This is equivalent to a loan with an annualinterest rate of 19.56%.

1 + EAR = 1.1956thus, EAR = 0.1956

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keys: description:

[2nd] [ICONV] Opens interest rate conversion menu

[↓] [EFF=] [CPT] 19.56

Texas Instruments BAII Plus

[↓][NOM=] 18 [ENTER] Sets 18 APR.[↑] [C/Y=] 12 [ENTER] Sets 12 payments per year

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Perpetuity◦ A level stream of cash flows that lasts forever

Growing perpetuity◦ A stream of cash flows that grows at a constant

(same) rate forever Annuity◦ A level stream of cash flows that lasts for a fixed

number of periods Growing annuity◦ A stream of cash flows that grows at a constant

(same) rate for a fixed number of periods

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A constant stream of cash flows that lasts forever

0

…1

C

2

C

3

C

Mathematicallyequates to:

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What is the value of a British consol that promises to pay £15 every year for ever? The interest rate is 10% (per year).

0…

1

£15

2

£15

3

£15

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A growing stream of cash flows that lasts forever

0

…1

C

2

C×(1+g)

3

C ×(1+g)2

Mathematicallyequates to:

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The expected dividend next year is $1.30, and dividends are expected to grow at 4%/year forever.If the discount rate is 12%/year, what is the value of this promised dividend stream?

0

…1

$1.30

2

$1.30×1.04

3

$1.30 ×1.042

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A constant stream of cash flows with a fixed maturity

0 1

C

2

C

3

C

T

C

Mathematicallyequates to:

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The equation for PV of an annuity always tells you the value of the annuity one periodbefore the first cash flow (CF) in the annuity.Typically, the annuity’s first CF is at t=1, in which case the calculated PV is straightfor-wardly a value at t=0 (today)However, if the annuity’s first CF is at, say, t=4, the PV equation gives a value at t=3.

Then, if the goal is to calculate a valueat t=0, you must discount the

time-3 value from t=3 to t=0

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If you can afford a $400 monthly car pay-ment, how much car can you afford if inter-est rates are 7.2%/yr. on 36-month loans?

0 1

$400

2

$400

3

$400

36

$400

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What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

0 1 2 3 4 5

$100 $100 $100 $100$323.97$297.22

Step 1

Step 2

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A growing stream of cash flows with a fixed maturity

0 1

C

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

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A defined-benefit retirement plan offers a 40-year payout with $20000 to be paid in 1 year (i.e., at t=1) and with annual payments increasing by 3% each year. What is the present value at retirement if the discount rate is 10%/year?

0 1

$20,000

2

$20,000×(1.03)

40

$20,000×(1.03)39

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A Pure-Discount Loan is the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time.

An Interest-Only Loan requires an interest payment each period, with full principal due at maturity. [Bonds are this type of ‘loan’.]

Amortized Loans require repayment of principal over time, in addition to required interest.

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Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.

If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?◦ PV = 10,000 / 1.07 = 9,345.79

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Consider a 5-year, interest-only loan with a 7% interest rate (per year). The principal amount is $10,000. Interest is paid annually.◦ What would the stream of cash flows be?

Years 1 – 4: Interest payments of .07(10,000) = 700 Year 5: Interest + principal = 10,700

This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later.

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Consider a $50,000, 10-year loan at 8%/year interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.

Click on the Excel icon to see the amortization table

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Each payment covers the interest expense plus reduces principal

Consider a 4-yr., $5000 loan with annual payments. The interest rate is 8%/year.◦ What is the annual payment?

4 N 8 I/Y 5,000 PV CPT PMT = -1,509.60

Click on the Excel icon to see the amortiza-tion table

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Conceptually, a firm should be worth the present value of the firm’s cash flows.

The tricky part is determining the size, timing, and risk of those cash flows.

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You can solve time value problems in any of four ways:◦ Math (Formulae given above)◦ Tables (See Appendix A)◦ Financial Calculator◦ Spreadsheet Software

Financial calculators and spreadsheet soft-ware are the most common methods now.

I, personally, prefer the math equations.

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How is the future value of a single cash flow computed?

How is the present value of a series of cash flows computed.

What is the Net Present Value of an investment?

What is an EAR, and how is it computed? What is a perpetuity? An annuity? Contrast interest-only loans to amortized

loans.