Valuation of Bonds and Stock First Principles

Valuation of Bonds and Stock

• First Principles:

• Value of financial securities = PV of expected future cash flows

• To value bonds and stocks we need to:

• Estimate future cash flows:
• size (how much) and timing (when)
• Discount future cash flows at an appropriate
• rate

Pure-Discount (Zero-Coupon) Bonds

• Information needed for valuing pure discount bonds:

• Time to maturity (T):
• T = Maturity date - today’s date
• Face value (F)
• Discount rate (r)

• 0 1 2 … T

• |-------------------|-------------------|------ … ------|

• F

• Value of a pure discount bond:

• PV = F / (1 + r)T

Level-Coupon Bonds

• Information needed to value level-coupon bonds:

• Coupon payment dates and Time to maturity (T)
• Coupon (C) per payment period and Face value (F)
• Discount rate

• 0 1 2 … T

• |----------------|------------------|------- … ------|

• Coupon Coupon Coupon + F

• Value of a Level-coupon bond:

• PV = C/(1+r) + C/(1+r)2 + .. + C/(1+r)T + F/(1+r)T
• = C (1/r){1 - [1 / (1 + r)T]} + F/(1 + r)T
• = PV of coupon payments + PV of face value

Four theories:

• Four theories:

• I. Expectations theory

• e.g. if investors expect next years yield to be 12%, then

• the forward rate will also be 12%: 1f2 = 12%

• Example: two alternative investments

• B1: a zero coupon bond with: T = 1, YTM: 0r1 = 8%

• B2: a zero coupon bond with: T = 2, YTM: 0r2 = 9%,

This implies the following forward rate for year-2:

• This implies the following forward rate for year-2:

• The general case:

• Note: This formula can be used only for zero-coupon bonds

• Example: 3 alternative zero-coupon bonds, with the following spot rates:

• 0r1 = 8%

• 0r2 = 10%

• 0r3 = 12%

• Calculating 1f2 and 2f3:

II. Liquidity Premium Theory

• II. Liquidity Premium Theory

• If you invest for (t+1) years, you commit to reinvest in every

• year after the 1st year, and thereby lose liquidity and ask

• for a liquidity premium:

• III. Augmented Expectations Theory

• Combines the pure expectations theory with the liquidity premium theory:

• Example - Suppose: 0r1 = 8% and 0r2 = 9%. By the Expectations Theory:

• By the Liquidity Premium Theory, when L2 =1%, we get:

• 1f2= E[1r2] + L2. = E[1r2] + 1%

• Both theories together, give:

Common Stock Valuation

• The value of a stock = PV of all expected future cash flows

• Thus, the information needed to value common stocks:

• Common Stock Dividends (Dt)
• Discount rate (r)

• PV0 = D1/(1 + r)1 + D2/(1 + r)2 + D3/(1 + r)3 + . . . forever. .

• We have to estimate future dividends

Case 1: Zero Growth

• Assume that dividends will remain at the same level forever, i.e. D1 = D2 =…= Dt = D

• Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:

• Pt = Dt+1 / r

Case 2: Constant Growth

• Assume that dividends will grow at a constant rate, g, forever, i. e.,

• D1 = D0 x (1+g)

• D2 = D1 x (1+g) = D0 x (1+g)2

• 

• Dt = D0 x (1+g)t

• 

• Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:

• Pt = Dt+1 / (r - g)

Case 3: Differential Growth

• Assume that dividends will grow at different rates in the foreseeable future and then will grow at a constant rate thereafter.

• To value a Differential Growth Stock, we need to:

• Estimate future dividends in the foreseeable future.
• Estimate the future stock price when the stock becomes a Constant Growth Stock (case 2).
• Compute the total present value of the estimated future dividends and future stock price at the appropriate discount rate.

Examples - Differential Growth

• Q1. Whizzkids Inc. is experiencing a period of rapid growth. Earnings and dividends are expected to grow at a rate of 8 percent during the next three

• years, and then at a constant rate of 4% thereafter. Whizzkids’ last dividend,

• which has just been paid, was $2 per share. If the required rate of return on

• the stock is 12 percent, what is the price of the stock today?

• A1. It is given that:

• r = 12%, D0 = $2, g1 = g2 = g3 = 8%, and g4 = g* = 4% (forever)
• We calculate:

• D1=$2 =$ , D2= =$ ,

• D3= =$

• With g4=g* =4%, we have:

• D4= =$

• Since constant growth rate applies to D4, we use Case 2 (constant growth) to compute P3:

• P3 = = $

Expected future cash flows of this stock:

• Expected future cash flows of this stock:

• 0 1 2 3

• |----------|---------|---------| (r = 12%)

• D1 D2 D3 + P3

• 2.16 2.33 2.52 + 32.75

• The current (time 0) value of the stock:

• P0 = D1/(1+r) + D2/(1+r)2 + (D3+P3)/(1+r)3

• = + +

• = $