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# Valuation and Rates of Return

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Valuation and Rates of Return
Chapter 10 McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter Outline Valuation of assets, based on the present value of future cash flows The required rate of return in valuing an asset based on the risk involved Bond valuation and determination of present value of interest and maturity payments Preferred stock valuation based on dividend paid Stock valuation and determination of present value of future benefits

Valuation of Financial Assets
Helps in evaluating financial commitment a firm needs to make to: Stockholders and bondholders Attract investment Cost of corporate financing (capital) is used in analyzing the feasibility of an investment on an ensuing project The relationship between time value of money, required return, cost of financing, and investment decisions is shown in the following diagram:

Valuation Concepts Valuation of a financial asset is based on determining the present value of future cash flows Required rate of return (the discount rate) Depends on the market’s perceived level of risk associated with the individual security It is also competitively determined among companies seeking financial capital Implying that investors are willing to accept low return for low risk and vice versa Efficient use of capital in the past results in a lower required rate of return for investors

Valuation of Bonds A bond provides an annuity stream of interest payments and a principal payment at maturity Cash flows are discounted at Y (yield to maturity) Value of Y is determined in the bond market The price of the bond is equal to: The present value of regular interest payments discounted at Y Added to the present value of the principal (also discounted at Y)

Valuation of Bonds (cont’d)
Where: = Price of the bond; = Interest payments; = Principal payment at maturity; t = Number corresponding to a period (running from 1 to n); n = Number of periods; Y = Yield to maturity (or required rate of return) Assuming interest payments ( ) = \$100; principal payments at maturity ( ) = \$1,000; yield to maturity (Y) = 10% and total number of periods (n) = 20. Thus, the price of bonds ( );

Present Value of Interest Payments
To determine the present value of a \$100 annuity for 20 years, with a discount rate of 10% We have:

Present Value of Principal Payment (Par Value) at Maturity
Principal payment at maturity is used interchangeably with par value or face value of the bond Discounting \$1,000 back to the present at 10%, we have: The current price of the bond, based on the present value of interest payments and the present value of the principal payment at maturity: Here, the price of the bond is essentially the same as its par, or stated value to be received at maturity of \$1,000

Concept of Yield to Maturity
The yield to maturity or the discount rate is the required rate of return required by bondholders Three factors influence the required rate of return: Required real rate of return Demanded by the investor against current use of the funds on a non-adjusted basis Inflation premium Compensation towards the negative effect of inflation on the value of a dollar Risk free rate of return compensates for the use of funds and loss due to inflation Risk premium Towards special risks of an investment

Concept of Yield to Maturity (cont’d)
Business Risk: inability of the firm to retain its competitive position and stability and growth Financial risk: inability of the firm to meet its debt obligations as and when due Assuming: The real rate of return 3%, inflation premium 4% and risk premium is 3%, an overall required rate of return of 10% can be computed;

Price of a Bond with Increase in Inflation Premium
Assume Inflation premium goes up from 4 to 6%, with everything else being constant Present value of interest payments: \$100 annuity for 20 years at a discount rate of 12%;

Price of a Bond with Increase in Inflation Premium (cont’d)
Present value of principal payment at maturity: Present value of \$1,000 after 20 years at a discount rate of 12%; Total present value: Assuming that increased inflation increases required rate of return and decreases the bond price by \$150 approximately

Price of a Bond with Decrease in Inflation Premium
Assuming that the inflation premium declines: The required rate of return decrease to 8%, where the 20 year bond with a 10% interest rate would now sell for; Present value of interest payments Present value of principal payment at maturity Total present value

Bond Price Table The further the yield to maturity on a bond changes from the stated interest rate on the bond, the greater the price change effect will be. The following table illustrates the impact of differences between yield to maturity and coupon rates on bond prices.

Time to Maturity Influences the impact of a change in yield to maturity on valuation Longer the maturity, the greater the impact of changes in yield

Impact of Time to Maturity on Bond Prices
The amount (premium) above par value is reduced as the number of years to maturity becomes smaller and smaller The amount (discount) below par value is reduced with progressively fewer years to maturity The following table shows the critical effect of time to maturity on bond price sensitivity

Relationship Between Time to Maturity and Bond Price

Determining Yield to Maturity from the Bond Price
The yield to maturity (Y), that will equate the interest payments ( ) and the principal payments ( ) to the price of the bond ( )

Determining Yield to Maturity from the Bond Price (cont’d)
Assuming that a 15 year bond pays \$110 per year (11%) in interest and \$1,000 after 15 years in principal repayment Choosing 13% as an initial discount rate, we have: Present value of interest payments: Present value of principal payment at maturity Total present value

Determining Yield to Maturity from the Bond Price (cont’d)
Present value of interest payments at 12% Present value of principal payment at maturity Total present value

Formula for Bond Yield Weighted average is used to get the average investment over 15 year holding period, principal payment \$1,000, with annual interest payment \$110 and price of the Bond is \$932.21 * *This formula is developed by Gabriel A. Hawawini and Ashok Vora, “Yield Approximations: A Historical Perspective,” Journal of Finance 37 (March 1982), pp. 145–56.

Semiannual Interest and Bond Prices
A 10% interest rate may be paid as \$50 twice a year in the case of semiannual payments To make the conversion: Divide the annual interest rate by two Multiply the number of years by two Divide the annual yield to maturity by two Assuming a 10%, \$1,000 par value bond has a maturity of 20 years, the annual yield at 12% 10%/2 = 5% semiannual interest rate; hence 5% × \$1,000 = \$50 semiannual interest 20 × 2 = 40 periods to maturity 12%/2 = 6% yield to maturity, expressed on a semiannual basis

Semiannual Interest and Bond Prices (cont’d)
At a present value of a \$50 annuity for the 40 periods, at discount rate of 6%: Present value of interest payments Present value of principal payment at maturity Total present value

Valuation and Preferred Stock
Preferred stock represents a perpetuity, having no maturity date It has a fixed dividend payment It has no binding contractual obligation of interest on debt Being a hybrid security, it does not have: The ownership privilege of a common stock The legal provisions that could be enforced on debt

Perpetuity of a Preferred Stock
Where, = the price of the preferred stock; = the annual dividend for the preferred stock (constant); = required rate of return (discount rate) applied to preferred stock dividends A more usable formula is: Assuming, the annual dividend is \$10, and the stockholder requires a 10% rate of return, the price of the preferred stock would be:

Perpetuity of a Preferred Stock (cont’d)
If the rate of return required by security holders change, the value of the preferred stock also changes The longer the period of an investment, the greater the impact of a change in the require rate of return With perpetual security, the impact is at a maximum Assuming that the required rate of return has increased to 12%. The value of the preferred stock would be: If it were reduced to 8%, the value of the preferred stock would be:

Determining the Rate of Return (Yield) from the Market Price
Assuming the annual preferred dividend ( ) is \$10 and the price of the preferred stock ( ) is \$100, the required rate of return (yield): A higher market price provides quite a decline in the yield:

Valuation of Common Stock
Interpreted by the shareholder as the present value of an expected stream of future dividends The ultimate value of any holding lies with: The distribution of earnings in the form of dividend payments The earnings must be translated into cash flow for the stockholder

Dividend Valuation Model
Where, = Price of stock today; D = Dividend for each year; = the required rate of return for common stock (discount rate) This formula, with modifications is generally applied to three different situations: No growth in dividends Constant growth in dividends Variable growth in dividends

No Growth in Dividends The common stock pays a constant dividend as in the case of a preferred stock This is not a very popular option Where, = Price of the common stock; = Current annual common stock dividend (constant); = Required rate of return for common stock Assuming = \$1.86 and = 12%, the price of the stock would be:

Constant Growth in Dividends
The general valuation process is shown: Where, = Price of common stock today = Dividend in year 1, = Dividend in year 2, , and so on g = Constant growth rate in dividends = Required rate of return for common stock (discount rate)

Constant Growth in Dividends (cont’d)
Assuming: = Last 12 month’s dividend (assume \$1.87) = First year, \$2.00 (growth rate, 7%) = Second year, \$2.14 (growth rate, 7%) = Third year, \$2.29 (growth rate, 7%) etc = Required rate of return (discount rate), 12%

Constant Growth Dividend Valuation Model
The formula shown can be modified to a simple form if: The firm must have a constant dividend growth rate (g) The discount rate (Ke) must exceed the growth rate (g) Where: P0 = Price of the stock today D1 = Dividend at the end of the first year Ke = Required rate of return (discount rate) g = Constant growth rate in dividends Based on the current example; D1 = \$2.00; Ke = .12; g = .07. P0 is computed as:

Stock Valuation Based on Future Stock Value
Assumption: To know the present value of an investment Stock is held on for three years and then sold Adding the present value of three years of dividends, and the present value of the stock price after three years gives the present value of the benefits The appropriate formula to be used is:

Determining the Required Rate of Return from the Market Price
Determining the required rate of return, knowing the first year’s dividend, the stock price, and the growth rate (g): Assuming; = Required rate of return (to be solved) = Dividend at the end of the first year, \$2.00 = Price of the stock today, \$40 g = Constant growth rate 7%, we have: = \$ % = 5% + 7% = 12% \$40

Determining the Required Rate of Return from Market Price (cont’d)
The stockholder is receiving a current dividend plus anticipated growth in the future If the dividend yield is low, the growth rate must be high to provide the necessary return If the growth rate is low, a high dividend yield will be expected The first term represent the dividend yield the stockholder will receive The second represents the anticipated growth in dividends, earnings, and stock price

Price-Earnings Ratio Concept and Valuation
A multiplier applied to current earnings to determine the value of a share of stock in the market Influenced by: Earnings and sales growth of a firm Risk (or volatility in performance) The debt-equity structure of the firm The dividend policy The quality of management

Variable Growth in Dividends – Supernormal Growth
In evaluating a firm with an initial pattern of supernormal growth: First, take the present value of dividends during the exceptional growth period Then determine the price of the stock at the end of the supernormal growth period by taking the: present value of the normal, constant dividends that follow the supernormal growth period Discount this price to the present Add it to the present value of the supernormal dividends This gives the current price of the stock

Variable Growth in Dividends – No Dividends
Approach 1: though no dividend is paid currently The stockholders will be paid a cash dividend at a later date The present value of their deferred payments may be used Approach 2: Take the present value of earnings per share for a number of periods Add that to the present value of the future anticipated stock price The discount rate applied to future earnings is generally higher than the discount rate applied to future dividends

Stock Valuation under Supernormal Growth Analysis

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