Lecture 5 & 6 Security Valuation

Corporate FinanceFINA 4330

Ronald F. SingerFall, 2009

Present Value of Bonds & 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

+

Example

Consider a 20 year bond with 6% coupon rate paid annually. The market interest rate is 8%. The face (par) value of the bond is $100,000.• PV of coupon annuity = 20 6000 = 58,908 t=1(1 + 0.08)t

• PV of principal= 100,000 = 21,455 (1 + 0.08)20 • Present Value of Total = 80,363 • OR  

By Calculator

• 20 N

• 8 I%YR

• 6000 PMT

• 100000 FV

• PV 80,363.71

Yield 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.

• 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 31, 2008.The semiannual coupon payments are: 4.5% of 100,000 or 4,500.(As of August 25, 2006) 4,500 4,500 4,500 104,500

0 1 2 3 4

2/10 8/11 • 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%.

• By Calculator

4 N 4.000

-99375 PV -99375.000

4500 PMT 4500.000

100000 FV 100000.000

ComPuTe I/Y 4.6749

x 2 = 9.35

Corporate BondsYahoo.com

• Bond Center Bond Center > Bond Screener > Bond Screener Results• Type Issue Price Coupon(%) Maturity YTM(%) Current• Yield(%) Fitch Ratings Callable

Sprint 8.375s ’12 (5 payments)

Sprint 8.375s ‘12 “Name of Bond”– Principal or Par: $1,000 (most US Corporate bonds

have $1,000 principal).– Coupon (Annual): $83.75 or 8.375% of par – Maturity : March 15, 2012– Coupon Payment Dates: March 15, and

September 15 through March 15, 2012 (every 6 months)

– Current yield: 8.292%  Current = Coupon(% of par) = 8.375 =

8.292% Yield Price (% of par) 101.00Actual Price = 101 times 1000 = $1010.00

The Bond's Yield to Maturity?   YTM = 7.961

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

Calculation 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= 5 41.875 + 1000 t=1 (1.05)t (1.05)5

41.875..41.875..41.875……………………… 1041.875 └────┴───┴───────┴────┴────────┴───9/09 3/10 9/10 3/11 9/11 3/12

  P0 = PVA(5,10%,41.875) + PV(5,10%,1000) + 41.875

Treasury Yield Curves

• www.bloomberg.com

Stocks

• www.finance.yahoo.com• SPRINT NXTEL CP (NYSE:S)

Sprint Nextel (S) (8/21/09) Range: 3.84-3.93

Last Trade: 3.90 52 wk Range: 1.35-9.35

Trade Time: Aug 21 Volume: 28,768,634

Change 0.07 (1.83%) Avg Vol (3m): 42,768,900

Prev Close: 3.83 Market Cap: 11.22B

Open: 3.90 P/E (ttm): NA

1y target Est. 5.35 EPS (ttm): -1.02

DIV & Yield N/A (N/A)

Sprint Nextel Corp

   Current (Annual)Yield = Dividend

Price  

P-E Ratio = Closing Price Current Earnings

Current = Closing Price EPS P-E Ratio

• Stock Valuation

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

Po = E(Div1) + E (P1) 1 + E(R)  = The Present Value of the Expected Payoffs to the

Stockholder.

This can be thought of as simply the Present Value of the Dividend plus the price per share that you expect to receive after a 1 year holding period.

Stock Valuation

This relation will hold through time, therefore,  

P1 = E (Div2) + E(P2) 1 + E(R) 

Substitute for E(P1 ) in previous equation:

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

 

The Value of Stock

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 

And:

Po = E(Div1) + E(Div2) + E(Div3) + E(P3) 1 + E(R)  ( 1 + E(R))2 ( 1 + E(R))3  ( 1 + E(R))3 

So in general, we can think of a stock as equal to the present value of a dividend stream over some time period plus what you can get for the stock if you sold it at the end of the time period. That is:

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

 Or, as the time period gets very large,

 Present Value of E(PT)----> 0

 And the stock price is the present value of all future dividends paid to existing stockholders

Po = E (Divt) t=1 (1 + E(R))t 

 

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.(Ignore Changes in Working Capital)

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 .10

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 . . . . . EPS 10 11 11 DIVIDEND 0 11 11 FCFE 0 11 11INVESTMENT 10 0 0

Present Value of Dividends 100 Present Value of Free Cash Flow to equity 100

Suppose return on investment were 20%? Suppose it were 5% ?

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) = E{ Dividend(t)} t=1 (1 + r)t

or: P(0) = E{ DPS(t) }t=1 (1 + r)t

This equation represents: The Capitalized Value of Dividends

Capitalized Value of Dividends

Capitalized 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 =EPS(1) + PVGO r

(2) P0 = (FCFE per Share)t t=1 (1 + r)t

EPS(1) is the expected Earnings per share over the next period.PVGO is the "present value of growth opportunities.r is the "appropriate discount rate

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

Capitalized Dividend Model

Simple versions of the Capitalized Dividend Model

DIV(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 = FCFE r r r

Constant Growth ModelNew let the firm plan to reinvest b of its earnings at a rate of return of i through

the indefinite future. Then annual growth in earnings, dividends, and the price per share will be a constant equal to: g = b x i,b is the “Plowback Ratio” or (1 – “Payout ratio”) or the Net Investment as a percentage of earnings. i is th Return on Equity (ROE)

note that: in the special case that Change in Working Capital is zero: EPS = Would be Free Cash Flow to equity if Net Investment were zero, so that: DIV(t) = (1 - b)EPS(t) = Free Cash Flow to Equity(t) and we can write the valuation formula as:

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

= EPS(1) + PVGO r

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’s of future projects are positive then the value of the stock, and its price per share will be higher, given its current earnings and its capitalization rate

General Equation for Firm Valuation

• Stock Value can be represented by the PV of a Dividend Annuity plus the predicted stock price at the end of the Annuity.

• DIV DIV + P(T)