Lecture 9

Discounting and Valuation

The investment guru

Financial Markets and Corporate Strategy, David Hillier

Average rate of return on

investments is more than 20 percent

The purchase of Coca-Cola in 1988 netted him an 800 percent return over

twelve years

Warren Buffett only invests in firms whose

intrinsic value is greater than their cost

to purchase

Real investments

Financial Markets and Corporate Strategy, David Hillier

•Present value (PV)•Cash flows•Discount rate

Definitions:

Present Value: the market price of a portfolio of traded securities that tracks the future cash flows of the proposed project

Discount rate: the rate of return that discounts future cash flows to the present.

Unlevered Cash Flows

Financial Markets and Corporate Strategy, David Hillier

•Unlevered cash flows•Financing cash flows•EBIT - earnings before interest and taxes•Deriving unlevered cash flows from the accounting cash flow statement

Definitions:

Unlevered cash flows: cash flows generated directly from the real assets of the project or firm

Financing cash flows: associated with (1) issuance or retirement of debt and equity, (2) interest or dividend payments, and (3) any interest-based tax deductions that stem from debt financing.

Unlevered Cash Flows - Result 9.1

Financial Markets and Corporate Strategy, David Hillier

Deriving Unlevered Cash Flow from the Income Statement

Financial Markets and Corporate Strategy, David Hillier

Result 9.2  Unlevered cash flow = Profit before Interest and Taxes+ depreciation and amortization change in working capital capital expenditures+ sales of capital assets realized capital gains+ realized capital losses Profit before Interest and Taxes tax rate

Exhibit 9.1 Earnings and Cash Flows for Bayer Healthcare AG

Financial Markets and Corporate Strategy, David Hillier

Year

PBIT(a)

Depreciation

(b) PBITDA

Increase inWorkingCapital

(d)

PretaxCash Flow

(e) = (c) - (d)

Taxes(at 40%)

(f) = (a) (0.4)

UnleveredCash Flow(g) = (e) -

(f)1 €10,000 €100,000 €110,000 €10,000 €100,000 €4,000 €96,000

2 10,000 100,000 110,000 10,000 100,000 4,000 96,000

3 10,000 100,000 110,000 10,000 100,000 4,000 96,000

4 10,000 100,000 110,000 10,000 100,000 4,000 96,000

5 10,000 100,000 110,000 10,000 100,000 4,000 96,000

6 110,000 0 110,000 10,000 100,000 44,000 56,000

7 110,000 0 110,000 10,000 100,000 44,000 56,000

8 110,000 0 110,000 10,000 100,000 44,000 56,000

Example 9.3: Incremental Cash Flows: Cash Flow Differences in Two Scenarios

Financial Markets and Corporate Strategy, David Hillier

Flyaway Air is thinking of acquiring a fleet of new fuel-saving jets. The airline will have the following cash flows if it does not acquire the jets:

If it does acquire the jets, its cash flows will be

What are the incremental cash flows of the project?Answer: The incremental cash flows of the project are given by the difference between the two sets of cash flows:

Cash Flows (in € millions) at Date0 1 2 3

100 140 120 100

Cash Flows (in € millions) at Date0 1 2 3

80 180 110 130

Cash Flows (in € millions) at Date0 1 2 3

20 40 10 30

Exhibit 9.4 The Value of an Investment over Multiple Periods When Interest (Profit) Is Reinvested

Financial Markets and Corporate Strategy, David Hillier

0

1

2

.

.

.

t - 1

Beginning-of-PeriodDate

Initial Principal Balance

Interest (profit)Earned over Period End-of-Period Value

End-of-PeriodDate

1

2

3

.

.

.

t

Po

Po (1 = r)

Po (1 + r)2

.

.

.

Po (1 + r)t-1

Por

Po (1 = r)r

Po (1 + r)2r

.

.

.

Po (1 + r)t-1r

Po + Por = Po (1 + r)

Po (1 + r) + Po (1 + r) r = Po (1 + r)2

Po (1 + r)2 + Po (1 + r)2r = Po (1 + r)3

.

.

.

Po (1 + r)t-1 + Po (1 + r)t-1r = Po (1 + r)t

0

1

2

.

.

.

t - 1

Beginning-of-PeriodDate

Initial Principal Balance

Interest (profit)Earned over Period End-of-Period Value

End-of-PeriodDate

1

2

3

.

.

.

t

Po

Po (1 = r)

Po (1 + r)2

.

.

.

Po (1 + r)t-1

Por

Po (1 = r)r

Po (1 + r)2r

.

.

.

Po (1 + r)t-1r

Po + Por = Po (1 + r)

Po (1 + r) + Po (1 + r) r = Po (1 + r)2

Po (1 + r)2 + Po (1 + r)2r = Po (1 + r)3

.

.

.

Po (1 + r)t-1 + Po (1 + r)t-1r = Po (1 + r)t

ttt

t r

PPrPP

)1()1( 00

The Future Value (Equation 9.2) and Present value (Equation 9.3):

Example 9.4 and 9.5

Financial Markets and Corporate Strategy, David Hillier

Example 9.4: Computing the Time to Double Your MoneyHow many periods will it take your money to double if the rate of return per period is 4 percent?Answer: Using equation (9.2), find the t that solves

which is solved by t = ln (2)/ln(1.04) = 17.673 (periods).

Example 9.5: Determining the Yield on a Zero-Coupon BondCompute the per period yield of a zero-coupon bond with a face value of €100 at date 20 and a current price of €45.Answer: Using the formula presented in equation (9.4):  

or about 4.07 percent per period.

tPP )04.1(2 00

040733.145

100 20

1

r

Value Additivity and Present Values of Cash Flow Streams

Financial Markets and Corporate Strategy, David Hillier

Result 9.4 Let C1, C2, . . . , CT denote cash flows at dates 1, 2, . . . , T, respectively. The present value of this cash flow stream 

(9.5) if for all horizons the discount rate is r.

TT

r

C

r

C

r

CPV

)1(...

)1()1( 22

11

Example 9.6: Determining the Present Value of a Cash Flow StreamCompute the present value of the unlevered cash flows of the MRI of Bayer Healthcare AG, computed in Exhibit 9.1. Recall that these cash flows were €96,000 at the end of each of the first five years and €56,000 at the end of years six to eight. Assume that the discount rate is 10 percent per year.

Answer: The present value of the unlevered cash flows of the project is

€450,3871.1

€56,000

1.1

€56,000

1.1

€56,000

1.1

€96,000...

1.1

€96,000

1.1

€96,00087652

PV

Inflation

Financial Markets and Corporate Strategy, David Hillier

•Nominal discount rates

•Nominal cash flows

•Inflation-adjusted cash flows

•Real cash flows

•Real discount rates

11

1 min

i

rr alno

real

Result 9.5 Discounting nominal cash flows at nominal discount rates or inflation-adjusted cash flows at the appropriately computed real interest rates generates the same present value.

Annuities and Perpetuities

Financial Markets and Corporate Strategy, David Hillier

Perpetuities: r

CPV

Example 9.7: The Value of a Perpetuity

In 1752, the British government decided to consolidate all of its debt into one perpetuity that paid a 3 ½ percent coupon. The bond, which is known as a consol, still exists today except that its coupon has now changed to 2 ½ percent payable. Assuming that the discount rate on the bond is 5 percent per annum, what is the value of the bond today?

Answer: Using Result 9.6, the value is PV = £50 = £2.50/.05

Result 9.6 If r is the discount rate per period, the present value of a perpetuity with payments of C each period commencing at date 1 is C/r.

Annuities and Perpetuities - continue

Financial Markets and Corporate Strategy, David Hillier

Annuities :

Trr

CPV

)1(

11

Example 9.9: Computing Annuity Payments Flavio has just borrowed £100,000 from his rich professor to pay for his doctoral studies. He has promised to make payments each year for the next 30 years to his professor at an interest rate of 10 percent. What are his annual payments? Answer: The present value of the payments has to equal the amount of the loan, £100,000. If r is .10, equation (9.8) indicates that the present value of £1 paid annually for 30 years is £9.427. To obtain a present value equal to £100,000, Flavio must pay

at the end of each of the next 30 years.

Result 9.7 If r is the discount rate per period, the present value of an annuity with payments commencing at date 1 and ending at date T is

607,10£ 427.9

000,100C

Trr

CPV

)1(

11

Growing Perpetuities and Annuities

Financial Markets and Corporate Strategy, David Hillier

Growing Perpetuities: gr

CPV

Result 9.8 The value of a growing perpetuity with initial payment of C dollars one period from now is

gr

CPV

Growing Annuities :

T

T

r

g

gr

CPV

)1(

)1(1

Example 9.10: Valuing a Share of Equity

Financial Markets and Corporate Strategy, David Hillier

Agfa-Gevaert N.V. (Agfa) is an imaging technology firm that is listed on NYSE Euronext and is a component of the Bel-20 index, an index of the twenty largest Belgian firms. For the last four years, it paid a gross dividend of €0.50 per year. Assume that next year, Agfa will pay a dividend of €0.50 and that this will grow thereafter by 7 percent per annum forever. If the relevant discount rate is 10 percent, how much should you pay for Agfa equity?

Answer: Using equation (9.11), the per share value is 

The actual price of Agfa-Gevaert N.V. at the end of October 2007 was €15.40.

6.671€ 07.010.0

50.0

PV

Simple Interest: Time Horizons and Compounding Frequencies

Financial Markets and Corporate Strategy, David Hillier

•Annualized Rates•Equivalent Rates

Exhibit 9.5 Translating Annualized Interest Rates with Different Compounding Frequencies into Interest Earned per Period

Annualized Interest Rate Quotation Basis

Interest per Period Length of a Period

Annually compounded R 1 yearSemiannually compounded r /2 6 monthsQuarterly compounded r /4 3 monthsMonthly compounded r /12 1 monthWeekly compounded r /52 1 weekDaily compounded r /365 1Compounded m times a year r /m 1/m years

Example 9.11: Finding Equivalent Rates with Different Compounding Frequencies

Financial Markets and Corporate Strategy, David Hillier

An investment of £1.00 that grows to £1.10 at the end of one year is said to have a return of 10 percent, annually compounded. What are the equivalent semiannually and continuously compounded rates of growth for this investment?

Answer: Its semiannually compounded rate is approximately 9.76 percent and its continuously compounded rate is approximately 9.53 percent. These are found respectively by solving the following equations for r

and

2

2110.1

r

rte10.1

Thank You