financial economics

                             Financial Economics (FS6061)

Net present value and capitalization (valuation) models

Introduction

Valuation model is a mechanism to convert a set of forecasts of, or to observe, a series of company and economic variables into a forecast of market value for the company`s stock. In this model we take input in terms of economic variables such as dividends, futures earning, variability of earnings, interest rate etc. and the output is in terms of expected market value or expected return from the stock variables. Valuation model helps to describe the relationship that is assumed to exist between a set of economic and corporate factors and the valuations of these factors in market.

The use of valuation models in investment decisions are based upon the perception that markets are inefficient and assumption about how and when these inefficiencies will get corrected. Nowadays every financial organization employs a valuation model and with this model they make their financial decisions.

                                                Valuation Models

         

    Assets Based            Discounted Cash flow model           Relative valuation              Contingent claim model

1)       Liquidation value                                         1) Equity and firm                    1) Option to delay

2)       Replacement Cost                                         2) Sector and market               2) Option to expand

                                                                                                                       3) Option to liquidate

                                                                                                                                      

1)       Equity valuation model(Dividend and free cash flow to firm)

2)       Firm valuation model (Cost of capital, AVP approach, and excess return model.)

Discounted cash flow model (DCF)

According to discounted cash flow approach the value of share of stock is equal to the present value of the cash flow that shareholders are expected to receive from it. The concept of DCF valuation is based on the principle that the value of a business is inherently based on its ability to generate cash flows for the shareholders or investors.

Key components of DCF model

1)      Free cash flow (cash generated by the assets of business available for shareholders)

2)      Terminal value (value at the end of the free cash flow projection)

3)      Discount rate (the rate used to discount the projected free cash flow and terminal value to their present values)

Let’s see one example to be clear on above discussions:

According to DCF approach value of shares is equal to the present value of its expected cash flows.

Let us assume if a stock is held for one period (t), then cash flows received from stock are dividend (d) and the value of stock when sold is P_t+1, then

                        P_t   =       D_t+1           +         P_t+1

                                             (1+K)                 (1+K)

                        Where, P_t= the price of share at time t/ current price of share

                                    D_t+1=the dividend received at time t₊₁/ last year dividend

                                    P_t+1= the price at period _t+1

                                    K= the appropriate discount rate

This is the case for one period if we will hold stock more than a period suppose for 3 year, then

                        P_t    =       D_t+1     +    D_t+2            +    D_t+3    +   P_t+3                    

                                                (1+K)          (1+K) ²        (1+K) ³    (1+K)³

The DCF equation can be employed in three ways:

₁₎        P_t  can be treated as an unknown

₂₎        K can be treated as an unknown

₃₎        Converted to a price earnings ratio.

Dividend stream: This model is typically the DCF using dividend forecasts over several stages. Under this model it is necessary to forecast the growth rate in dividend each year. A number of different assumptions about the growth rate patterns are made in this model. In particular we examine three sets of growth assumption. They are:

a)      Zero growth (assume constant dividend).

b)      Constant growth in dividend

c)      Super normal growth (growth for a finite number of years at a constant rate, followed by a period during which growth declines to a steady and remains the same.)

Now, we will examine these three DCF dividend model in brief.

Zero growth

In the case of zero growth in dividend, the company pays a fix and constant amount of dividend every year, there will be no change in dividend paid in 1^st year and 2^nd year or any year.

Let us assume the constant dividend amount be $ D, then,

dividend paid last year(D_o)= $D

dividend expected at the end of year 1(D₁)= $D

dividend expected at the end of 2^nd year(D₂)= $ D

Since the dividend is always the same with cash flow equal to $D, therefore price of share is given as:

                                    P_o= D/r

where, r= required rate of return

            D= constant dividend per year

            P_o= Price of share

Example, CASIO Company has policy of paying $ 20 per share as dividend every year. If this policy is to be continued indefinitely, what is the value of the share of the stock if required rate of return of the investment is 18%.

Solution:

We are given that,

                        D=$ 20

                        r= 20% or (0.2)

Present value per share (P_O) =   20/0.2

                                           = $ 100

Constant growth in dividend

In the case of constant growth rate the dividend for company always grow at a steady rate. Suppose the dividend growth rate of company is g% then it will be the same rate for indefinite future. The price of share in constant growth rate can be calculated as:

                        P_O= D₁/r-g

                        We can write D₁= D_O (1+g)

                        D₁=dividend at year 1

                        g= growth rate of dividend

                        D_O= dividend paid at year 0 (just paid dividend)

The above equation of constant growth model can be interpret as, the price of share should be equal to next year`s dividend divided by the difference between appropriate discount rate for the stock and its expected long term growth rate.

Example, A company has just paid dividend of $0.8 and is expected to exhibit of growth rate of 12% into the indefinite future and expected rate of return on investment is 15%. What is the value of stock?

Solution

We are given that,

D_O= 0.8$

g= 12%

r= 15%

The present value of share (P_O) = D₁/r-g

                                                = D_O (1+g)/r-g

                                                =0.8(1+0.12)/0.15-0.12

                                                =$29.867

Super normal growth model (two- period growth)

In this case, it is assumed that irregular amounts of dividend are paid during the first few years and then followed by a constant growth rate. Assumption of growth to be constant after some period of time is followed by the reasoning like after some period of time the firms are not differentiate on the basis of growth. The firms with higher growth will no longer have high growth and firm with less growth can generate high growth in future. Thus after some years, it is sensible not to differentiate between firms but simply to assume they all grow at same rate.

The price of share in super normal growth can be calculated as:

P_O=      D₁          +     D₂              + …………              D_n          +       P_n

          (1+r)             (1+r)²                                     (1+r)ⁿ            (1+r) ⁿ  

Where P_n=   D_n(1+g)

                        r-g

Example, A company is expected to pay dividend $ 0.10 per share in 1^st year, $0.20 in 2^nd year and $0.25 per share in third year. After third year the dividend will grow at a constant rate of 5% per year. The required rate of return for your investment is 10% per year. What is price of share now?

Solution,

From question we are given that,

Dividend for 1^st year (D₁) = $0.10

Dividend for 2^nd year (D₂) = $0.20

Dividend for 3^rd year (D₃) = $0.25

Required rate of return(r) =10% (0.10)

Growth after third year (g) = 5% (0.05)

First we calculate the share price at third year,

                                    P_n  =      D_n(1+g)

                                                     (r-g)

                                    P₃   =        D₃ (1+g)      =   0.25(1+0.05)     = $ 5.25

                                                        (r-g)                 0.10-0.05

Now calculating share price,

P_O  =       D₁             +            D₂                 +      D₃               +    P₃

                  (1+r)                         (1+r)²                  (1+r)³            (1+r³)

            =      0.10         +     0.20            +    0.25            +    5.25

                   (1+0.10)         (1+0.10)²           (1+0.10)³         (1+0.10)³

            =    0.09    +       0.17        +   0.19     +    3.94

            =   $4.93

Model based on Price- Earnings ratios

This model argues common stocks are possible to analyze by applying price-earnings ratio to either present or forecasted earnings. The major factor that separates this model from DCF model is the selection of a terminal P/E ratio. Under this model we have to compare the price and earnings per share for a company. By comparing these two factors we can analyze the markets stock valuation of company and its shares relative to the income company is generating. Stocks with higher or certain forecasts earnings growth will usually have a higher P/E and those expected to have lower or riskier earnings growth will usually have a lower P/E. To use this model one should at least explore growth rate implicit in the use of terminal P/E ratio.

Basically P/E ratio is calculated as:

                                                P/E= MPS/EPS

                                        Or, MPS= EPS× P/E ratio

                                                P­_O = EPS× Multiplier (m)

                                                Where, MPS= market price of share

                                                            EPS= Earnings per share

                                                           

 Under this model we examine two sets of growth assumption. They are:

1)      Zero growth

2)      Constant growth

Zero growth: In zero growth all earnings of firms are distributed as a dividend. Here,

                                    Price of share (P_O)= EPS × m

                                    Where m= multiplier and calculated as

                                                = dividend payout ratio/ required rate of return

Example: A company has a earning per share of $4 and required rate return is 16%. If company decided to pay all earnings as a dividend then what is the price of share now?

Solution,

We are given that,

EPS= $4

r= 16% or 0.16

First calculating multiplier,

                        m= DPR/r

                           = 1/0.10

                           = 6.25

(Since company have decided to pay all earnings as a dividend, DPR=100% or 1)

P_O= EPS× m

     = 4× 6.25

      = 25$

Constant growth: in constant growth the company has a policy of constant retained earnings for future investment. Here,

                                    Price of share (P_O) = EPS × m

                                    Where, m= multiplier= DPR (1+g)/r-g

Example: A firm has decided to pay 60% of their earnings as a dividend and remaining 40% is retained for future investment. Earnings per share are $4 and required rate of return is 16%. What is the price of share?

Solution,

We are given that

Dividend payout ratio (DPR)= 60% or 0.60

Required rate of return(r) = 16% or 0.16

Earnings per share (EPS) = $4

Now, multiplier (m) = DPR/ r

                                    = 0.60/0.16

                                    =3.75

Price of share (P_O) = EPS ×m

                                    = 3.75× 4

`                                   = $15

The theoretical Price earnings ratio can be estimated by calculating an average historical value, using an explanatory model assigning weights to certain factors by qualitative reasoning and using regression analysis to estimate the weights.

In 1963 Whitbeck- Kisor model was introduced which attempt to use multiple regressions to explain price earnings ratios. According to this model,

Price earnings ratio= 8.2 +1.50(earnings growth rate) +0.067 (dividend payout rate) – 0.200 (standard deviation in growth rate)

This equation represents the estimate of stock at a point of time and the impact of three variables, earnings growth rate, dividend payout and standard deviation on the price earnings ratio. The numbers represent the weight that the market placed on each variable at the point of time and the signs represent the direction of impact of each variable on the price earnings ratio.

 There are three reasons suggesting Whitbeck- kisor model does not work. They are:

i)                    Market tastes changes and with change in tastes the weight on each variable also changes over time.

ii)                  Inputs like dividend and growth in earning changes over time

iii)                This model has failed to show the firm effects.

Capital assets pricing model (CAPM)

Capital assets pricing model is tool used to analyze the relationship between risk and rates of return. CAPM model states that the relevant riskiness of an individual stock is its contribution to the riskiness of a well diversified portfolio.  Beta coefficients are key elements in this model, which measures the extent to which return of a given stock move with the stock market.

Under CAPM,

                  E (R_A) = R_f + [E (R_M) – R_{F] ×} B_A

The above equation is also known as Security market line (SML).

Where, R_f = risk free rate

              R_M = market return

               B_A= Beta coefficients

     

 DCF and the CAPM

Wells Fargo stock valuation system attempts to implement stock valuation and selection systems that incorporate the DCF approach and capital assets pricing theory. There normally three steps in Wells Fargo stock valuation system,

Step1

·         Estimation of expected rate of return implied by the market price.

·         Find the discount rate which equates the present value of all future dividends with price.

Step2

·         Estimation of the beta of each security.

Step 3

·         Using the expected return and expected beta from each of the stocks.

·         A graph is plotted and the straight line which fits these points is used as an estimated SML

Result: If the stock has return above the SML, given its risk, it should be a good buy and if it has a lower return, it should not be bought.