# Valuation and Rates of Return

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Valuation and Rates of Return
Chapter 10 Valuation and Rates of Return

Growing Characteristic of Money
In 1624, the Indians sold Manhattan Island for \$24. If the Indians had invested the \$24 at 7.5% compounded annually, they would now have over \$15 trillion - sufficient to repurchase most of the New York City. Due to the growing property of money (i.e., interest may be earned on the current money) a dollar today is worth more than a dollar tomorrow Was the price really ridiculous?

Winner of \$10 million The \$10 million is paid out as \$500K the first year, \$250K for the next 29 years and a final payment of \$2.25 million. Assuming a discount rate of 10%, the present value of the rewards is less than \$3 million the longer you have to wait to receive the dollar, the less it is worth \$10 million is a future value while the \$3 million is a present value.

Time Value of Money The value of \$1 in different points in the future are not the same In order to make comparison, we have to convert the money back to the same point in time Normally we will use the present as the reference point in time

Valuation of Securities
Most of the securities will promise the holders a certain form of cash flow in the future E.g. a corporate bond guarantees a yearly payment of interest (\$100) and a preferred stock provides an biannual interest payment of \$200, which one will worth more?

Valuation of Securities cont’
To answer this question, we need to know the present value of the interest payments. That is, the equivalent amount of current money for the future income.

PV Formula FV = PV (1+i )^n PV = FV/(1+i )^n In table form,
FV = PV x FVIF PV = FV x PVIF

Present Value Calculation
Assume an compound interest rate of 10% p.a. Corporate Bond: PV of \$100 in 1 yr = \$90.91= \$100/(1+10%) PV of \$100 in 2 yrs = \$82.64= \$100/(1+10%)^2 Total = \$ \$82.64 = \$173.55 Preferred Stock: PV of \$200 in 2 yrs= \$200/(1+10%)^2 = \$165.2

PV of an annuity Assuming the same annual compound int. rate
Further assume that the corporate bond offer the same annual interest (i.e. \$100 p.a.) for a term of three years only (each interest payment is made at the end of each year) The total PVa is: \$100/(1+10%) + \$100/(1+10%)^2 + \$100/(1+10%)^3 = \$ = \$248.74

PV of an annuity formula
PVa = \$100 { 1/(1+10%)+ 1/(1+10%)^2+ 1/(1+10%)^3} (1+10%)PVa = \$100(1+10%) { 1/(1+10%) /(1+10%)^2+ 1/(1+10%)^3} = \$100{1+1/(1+10%)+1/(1+10%)^2} (1+10%)PVa-PVa = \$100{1-1/(1+10%)^3)} (10%)PVa = \$100{1-1/(1+10%)^3)} PVa = \$100{1-1/(1+10%)^3)}/(10%) PVa for n years = \$100{1-1/(1+10%)^n)}/(10%)

PV of an annuity formula
PV a=A{1- 1/(1+i )^n}/i compare with FV a=A{(1+i )^n - 1}/i In table form, PV a=A x PVIFA compare with FV a=A x FVIFA For an annuity in advance (or annuity due), just multiply (1+i ) to either of the formulae

Discount Cash Flow Valuation
Since most of the securities will provide certain form of future cash flow, their values are determined by the present value of their expected cash flows In order to compute the PV, we need to know the amount and timing of expected cash flows as well as the required rate of return

Required Rate of Return
The required rate of return (discount rate) is determined by the market It depends on the market’s perceived level of risk associated with the individual security E.g. If you can get a 4% annual return from Canadian Saving Bonds, will you required the same rate of return for stock investment

Required Rate of Return cont’
Strictly speaking, we can break it down into three components Real Risk-Free Rate Premium for expected Inflation (Inflation Premium) Premium for Risk associated with individual security (Risk Premium)

Real Risk-Free Rate Rate of return on a risk-free security assuming that there is no inflation it is basically the financial rent the investors charge for using their funds for risk-free investment during non-inflationary period changes over time depends on rate of return investors expect to earn on non-inflationary risk-free assets (investors’ expectation) depends on people’s trade off of current with future consumption (investors’ time preference) “Real” means without the effect of inflation

Inflation Premium investors also expect to be compensated for inflation, i.e., for expected loss in purchasing power reflects the inflation rate expected in the future inflation premium on a given security reflects the average inflation expected over the lifetime of that security real risk-free rate + inflation premium = risk-free rate e.g. yield on government bonds Risk-free rate has already considered the inflation effect

Risk Premium Of primary interest to us are two types of risks:
Business risk - inability of a firm to sustain its competitive position and growth in earnings Financial risk - inability of a firm to meet its payment obligations as they come due There are many other types of risks such as maturity risk(e.g yield curve of long-term bond), liquidity risk(e.g real estate investment), etc

Simplified Bond Valuation
Bonds are issued by firms and governments to raise debt capital. Typically, bond issuers pay interest semi-annually at a specified coupon rate. They are frequently denominated in \$1000 face values (i.e. the issuer will pay back the bearer \$1000 at the maturity date of the bond) For illustration purpose, let’s assume an annual interest paying bond with a 10% coupon rate. It will mature in 3 years and the market discount rate has just changed to 12%. Normally, the coupon rate will be the market discount rate at the time of issuance and the bond will be sold at par ( i.e. at the face value)

Simplified Bond Valuation cont’
The value of a bond is made up of two different parts: PV of the interest payments (an annuity cashflow) plus PV of the principal payment ( a lump sum cashflow)

Relationship between discount rate and coupon rate
Coupon rate is fixed at the time of issuance while the discount rate fluctuates with the market. When the discount rate changes, bond price will also change. If discount rate is > or < coupon rate, the bond will be sold at a discount (below face value) or at a premium (above face value) If discount rate is = coupon rate, the bond will be sold at par (at the face value) An understanding of the relationship between the discount rate and the coupon rate will allow you to anticipate the value of a bond prior to calculation.

Relationship between discount rate and coupon rate cont’
Annual coupon payment = 1000x10% = 100 Bond price = 100 x PVIFA (n=3, i=12%) x PVIF (n=3, i=12%) = 100 x x 0.712 = = 952.2

Actual Bond Valuation: An Application of non-Annual Discounting
A firm issued bonds three years ago with a \$1000 face value at a coupon rate of 12 percent. The current yield or yield to maturity on such bonds (i.e. the discount rate to be used, varies with economic conditions) is 10 percent. The bonds have five more years to maturity. What is the current price of such a bond? The purchaser of the bond gets: \$1000 in 5 years (a lump sum) coupon payments of 12%/2 x \$1000 =\$60 every six months

Bond valuation cont’ Discount rate for a (half yearly) period
= 10% / 2 = 5% PV0 = \$60 PFIVA{5%, 10 periods} PVIF{5%, 10 periods} = 60 x x 0.614 = = \$

Bond valuation cont’ Six months pass. Interest rates have fallen such that the yield on the bond is now 8 percent. What would the bond be worth? New price = \$60 PVIFA{4%, 9 periods} PVIF{4%, 9 periods} = \$1,149.42 Note first bond theorem: Bonds prices vary inversely with interest rates. Increases in rates lead to decreased prices; decreases in rates lead to bond price increases.

Preferred share valuation - application of a perpetual annuity
Preferred shares are issued by firms to raise capital. These shares pay a fixed expected dividends but not binding as interest on debt. The principal is not expected to be paid. In essence, the buyer receives dividends in perpetuity How to price these securities?

Valuation of Preferred shares
PV of a perpetual cash flow = A{1- 1/(1+i )^n}/i with n = infinity = A{1- 0}/i = A/ i Hence PV of a perpetual preferred shares = D/Kp where Kp is the required rate of return for the preferred shares

Valuation of Common Stock
In theory, the value of a share of common stock is equal to the present value of an expected stream of future dividends Assume a constant growth (g) in dividends And Ke is the required rate of return for the common shares

Constant Dividend Growth Model
Multiple both sides with (1+Ke)/(1+g) and subtract P again After munipulation, Share price (P) = D0(1+g)/(Ke-g) = D1/(Ke-g) (see proof on p.338)

What have you learnt? How to calculate the present value of a lump sum investment or saving How to calculate the present value of a series of equal payments at uniform intervals for a number of periods ( including annuity in advance) Apply the discount cash flow technique in the valuation of corporate bond (lump sum + annuity), preferred stock (perpetual annuity) and common stock (assuming constant dividend growth)

Effective Annual Rate (EAR)
Often interest is quoted at a nominal rate (an annual rate) but compounded semi-annually or even more frequently The higher the frequency of compounding, the greater will be the rate of return That is, the real interest rate is higher than the nominal rate The real interest rate is named effective annual rate.

Effective Annual Rate cont’
Suppose an investment account offers a nominal interest rate of 8 percent, compounded quarterly. What is its EAR? Since 8 percent is an annual rate, interest rate of a quarter will be 8 percent divided by 4 As it is compounded quarterly, there will be 4 compounding period in a year EAR = (1+2%)^4 - 1 = 8.24%

General formula for EAR
EAR = (1+i/m)^m - 1 i = nominal interest rate (an annual rate) m = the number of compounding periods per year

Real Estate Investment
An illustration of using other people’s money to make money for yourself Borrow money from the bank - mortgage (lowest interest rate among all types of personal loans) High gearing return (leverage) The first thing to buy after you get your first job A car or a house?

Buy a house at a price of \$100,000
Own money \$30,000 Mortgage loan at a rate of 5% p.a. \$70,000 Total investment \$100,000 Return in a year (rise in house price) % Gross gain \$10,000 Interest expense 70,000 x 5% \$3,500 Net gain \$6,500 Return on your own money \$6,500/30,000 = 22%

Mortgage Amortization
Assume the bank lends you \$70,000 for 5 years at a fixed rate of 5% (compound annual rate) Further assume that you are required to pay back the mortgage by annual installments What will be the annual payment? How much of each annual payment is interest and how much goes to pay down the mortgage?

How much will be the annual payment?
What type of cash flow is the mortgage from banker’s perspective? Initial investment = PV = \$70,000 Annual payment = A PV = A x PVIFA (5 years, 5%) 70,000 = A x 4.329 A = 70000/4.329 = 16170

How much of each annual payment is interest?

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