Lecture 3Lecture 3

Managerial Finance

FINA 6335

Ronald F. Singer

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Present Value of Bonds Present Value of Bonds & Stocks& Stocks At this point, we apply the concept of

present value developed earlier to price bonds and stocks.

Price of Bond = Present Value of Coupon Annuity

Present Value of Principal

+

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ExampleExample

Consider a 20 year bond with 6% coupon rate paid annually. The market interest rate is 8%. The face value of the bond is $100,000. PV of coupon annuity = 20 6000 = 58.908 t=1 (1 + 0.08)

PV of principal = 100,000 = 21,455

(1 + 0.08)20

Present Value of Total = 80,363 OR  

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By Calculator

20 N

8 I%YR

6000 PMT

100000 FV

PV 80,363.71

By Bond tables in calculator = 80.364% of Par OR Price of bond = PVAF(20, 8%) x 6,000 + PVF(20, 8%) x 100,000

= 9.818 x 6,000 + 0.215 x 100,000

= 58,908 + 21,500 = 80,408

ExampleExample

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Yield to MaturityYield to Maturity

YTM: The Annual Yield you would have to earn to exactly achieve the cash flow promised by the bond

It is the internal rate of return of the bond

It is that interest rate which makes the price of the bond equal the present value of the promised payments.

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Consider a bond with principal of $100,000 and a coupon, paid semiannually, of 9%, selling for 99.375 (This is percent of the face value, so that the actual price is 100,000 x .99375 = $99,375.

Maturity date is August 30, 2002.The semiannual coupon payments are: 4.5% of 100,000 or 4,500.(As of August 30. 2001) 4,500 4,500 4,500 104,500

0 1 2 3 4

2/01 8/02 99,375 99,375 = 4500 + 4500 + 4500 + 104,500

(1+YTM) (1+YTM)2 (1+YTM)3 (1+YTM)4 2 2 2 2

The Yield to Maturity is 9.35%.

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By Calculator4 N 4.000

-99375 PV -99375.000

4500 PMT 4500.000

100000 FV 100000.000

CPT I/Y 4.6749

x 2 = 9.35

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US Exchange BondsUS Exchange Bonds

From the Wall Street Journal (January 26, 2001)U.S. Exchange Bonds

BONDS CUR VOL. CLOSE NET YLD CHG

DukeEn67/823… 7.3 10 941/8 _(coupon is 6.875) 1034.375   0 6/01 12/01 6/02 6/23

 

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Note: We assume that the bond matures at the end of June, and pays interest at the end of June and December. This is an arbitrary assumption. Also, U.S. Bonds are quoted "cum" coupon, so that the next coupon is always included in the calculation:  

NB: See Moody's Bond Guide or S & P for actual dates. – Principal: $1,000 (most US Corporate bonds have $1,000

principal).– Coupon (Annual): $68.75– Maturity : 2023– Current yield: 7.3%  

Current = Coupon = 6.875 = 7.3% Yield Price 94.125

  Price = 0.94125 x 1000 = $941.25  

What is the bond's Yield to Maturity?  

YTM = Accrual =  

Note in this case: YTM > Current Yield > Coupon: Why?

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Calculation of YTMCalculation of YTM

Suppose we know the appropriate Yield to Maturity ("Discount Rate")

For Example: 10% (NB: Bond Quotes are in simple interest) The Bond Value is

P0= 34.375 + 1000 t=1 (1.05)t (1.05)19

  1034.375 34.375..... └────┴───┴───┴──────────────────────────┘

0 1 2 3 45  

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Treasury Bonds, Notes and BillsTreasury Bonds, Notes and Bills

Maturity AskRate Mo/Yr Bid Asked chg. Yld.10 5/8 Aug 15 149:08 149:14 +16 5.61  Bid Price: what government security dealers were willing to

buy the bonds for at the end of the day  Asked Price: what government security dealers were willing

to sell the bonds for at he end of the dayThe amount after the : is the quote in 32nd of a percentage point

and the quote is in percent of par. so the quotes are, 149 + 8/32% of par bid

149.25%and 149 + 14/32% of par asked

149.4375%

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Treasury Bonds, Notes and BillsTreasury Bonds, Notes and Bills

Rate: the coupon rate as a percent of par.

the coupon is 10.625% of $100,000 or

$10,625 per year in two equal installments of $5312.50 each.

 Chg.: the change in the asked price from the prior day's close (in 32nds).

YLD.: the Yield to Maturity based on the closing asked price

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Valuation of Common StockValuation of Common Stock

The Annual Expected Return on a share of common stock is composed of two components:

Dividends and Capital Gains Expected Returns:

E(R0) = Dollar Return = E(Div1) + E (P1) - Po

Price P0

 Where P0 = The current per share price

E(Div1) = Expected dividend per share at time 1

E(P1) = Expected price per share at time 1

E(Ro) = Expected Return E(R0) = expected dividend yield + expected capital gain return

 

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New York Stock Exchange New York Stock Exchange Composite TransactionsComposite Transactions

52 Weeks Yld Vol Net

High Low Stock Sym Div % PE 100s Hi Lo Close Chg

223/8 151/8 OcciPete OXY 1.00 5.1 dd 4660 195/8 193/8 191/2 ...

   Current (Annual)Yield = Dividend = 0.051

Price

  

P-E Ratio = Closing Price = (dd)

Current Earnings

Current = Closing Price

Earnings P-E Ratio

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Note, we don't observe E(Ro) but we observe prices and promised payoffs. 

If we solve for Po, the current value of the stock 

Po = E(Div1) + E (P1) 1 + E(R)  = Expected Payoff x Discount

at Time 1  Factor This relation will hold through time, therefore,  

P1 = E (Div2) + E(P2) 1 + E(R) 

Substitute for P1 

Po = E(Div1) + E(Div2) + E(P2) 1 + E(R)  (1 + E(R))2 (1 + E(R))2

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In general, Po = T E (Divt) + E(PT) t=1 (1 + E(R))t (1 + E(R))T

 You can think of E(PT) as a liquidating dividend equal to the value of firm's assets at time T.

 As T ----> 00,

 Present Value of E(PT)----> 0 And the stock price is the present value of all future dividends paid to existing stockholders

Po = 00 E (Divt) t=1 (1 + E(r))t  What happened to capital gain? 

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Consider the value of the stock (or the per share Price of the stock)

The basic rule is: The value of the stock is the present value of the cash flows to the stockholder. This means that it will be the present value of total dividends (or dividends per share), paid to current stockholders over the indefinite future.

That is: V(o) = Ó E{ Dividend(t)} t=1 (1 + r)t

or: P(0) = Ó E{ DPS(t) }t=1 (1 + r)t

This equation represents: The Capitalized Value of Dividends

Capitalized Value of DividendsCapitalized Value of Dividends

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Capitalized Value of DividendsCapitalized Value of DividendsThe problem is how to make this OPERATIONAL.That is, how do we use the above result to get at actual

valuation? We can use two general concepts to get at this result: They all

involve the above equation under different forms. (1) P0 = EPS1 + PVGO

r (2) P0 = Ó (Free Cash Flow per Share)t t=1 (1 + r)t

EPS1 is the expected earnings per share over the next period.PVGO is the "present value of growth opportunities.r is the "appropriate discount rate

Free Cash Flow per Share is the cash flow available to stockholders after the bondholders are paid off and after investment plans are met.

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Capitalized Dividend ModelCapitalized Dividend Model

Simple versions of the Capitalized Dividend ModelDIV(1) = DIV(2) = . . . = DIV(t) = ...The firm's dividends are not expected to grow. essentially, the firm is planning no additional investments to propel growth. thus:

with investment zero:DIV(t) = EPS(t) = Free Cash Flow(t)PVGO = 0therefore the firm (or stock) value is simply:

P0 = DIV= EPS r r

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Constant Growth ModelConstant Growth Model

Next suppose that the firm plans to reinvest b of its earnings at a rate of return of i throughout the indefinite future. Then growth will be a constant level of: g = b x i,note that:DIV(t) = (1 - b)EPS(t) = Free Cash Flow(t)and we can write the valuation formula as:

P0 = DIV(1) = (1-b)EPS(1) = Free Cash Flow(1) r - g r - g r - g

= EPS(1) + PVGO r

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Example: ABC corporation has established a policy of simply maintaining its real assets and paying all earnings net of real depreciation out as a dividend. suppose that:

r = 10% Net Investment = ?

Current Net Earning per Share is 10. then: EPS(1) = EPS(2) . .=. . EPS(t). = 10

Year 1 2 3 ....

growth 0 0 0dividends 10 10 10free cash flow 10 10 10

and: Po = 10 = 100 0.10

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Now let this firm change its policy: Let it take the first dividend (the dividend that would

have been paid at time 1) and reinvest it at 10%. then continue the policy of paying all earnings out as a dividend.

We want to write the value of the firm as the present value of the dividend stream, the present value of free cash flow and the present value of Constant Earnings Per Share plus PVGO.time 1 2 3 . . . . . earnings 10 dividends 0 free cash flow investment

Present Value of DividendsPresent Value of Free Cash Flow

Present Value of current Earnings plus Present Value of growth opportunities. Suppose return on investment were 20%? Suppose it were 5% ?

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This value of the firm can be represented by

vo = EPS1 + PVGO:

r

where,

PVGO = NPV(t)

t=1 (1+r)t

Notice: if the NPV of future projects is positive then the value of the stock, and its price per share will be higher, given its current earnings and its capitalization rate