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Foundations of finance Interest rates
Citation preview
Dr. Vinodh Madhavan
Interest Rates
Cost of Money
Factors affecting cost of money
Production Opportunities
Time preferences for consumption
Risk
Inflation
Interest rate is a function of
Producers’ expected rate of return on invested capital
Savers’ time preference for current vs. future consumption
Riskiness of loan
Expected future rate of inflation
Determination of Interest Rates
r = r* + IP + DRP + LP + MRP
r represents any nominal rate
r* represents the “real” risk-free rate of interest.
IP inflation premium
DRP is default risk premium
LP is liquidity premium
MRP and maturity risk premium
Premiums Added to r* for Different Types of
Debt
IP MRP DRP LP
S-T Treasury
L-T Treasury
S-T Corporate
L-T Corporate
Yield Curve and the Term Structure of
Interest Rates
Term structure – relationship
between interest rates (or yields)
and maturities.
The yield curve is a graph of the
term structure.
The October 2008 Treasury yield
curve is shown at the right.
0%
2%
4%
6%
8%
10%
12%
14%
0 10 20 30
Interest
Years to Maturity
March 1980
February 2000
October 2008
Constructing the Yield Curve: Inflation
N
INFL
IP
N
1t
t
N
Step 1 – Find the average expected inflation rate over Years 1 to N:
For the inflation premium to be precise and theoretically sound, it should be the geometric average of inflationary expectations for the residual life of the maturity.
Constructing the Yield Curve: Inflation
Assume inflation is expected to be 5% next year, 6% the following
year, and 8% thereafter.
Any financial security should earn atleast the estimated inflation
premium, for the holder of such a security to keep up with his/her
original purchasing power at the time of investment.
%75.720/)]18%(8%6%5[
%50.710/)]8%(8%6%5[
%00.51/%5IP
20
10
1
IP
IP
Constructing Yield Curve: Maturity Risk
Step 2 : Find the appropriate maturity risk premium (MRP).
For instance, the following simplistic equation could be a
mathematical representation of a security’s appropriate maturity
risk premium.
MRPt = 0.1% (t – 1)
Constructing Yield Curve: Maturity Risk
Using the given equation:
Notice that since the above equation is linear, the maturity risk
premium is increasing as the time to maturity increases, as it
should be.
%9.1)120(%1.0
%9.0)110(%1.0
%0.0)11(%1.0MRP
20
10
1
MRP
MPP
Step 3 – Adding the premiums to r*.
rRF, t = r* + IPt + MRPt
Assume r* = 3%,
%65.12%9.1%75.7%3
%4.11%9.0%5.7%3
%0.8%0.0%0.5%3r
20 ,
10 ,
1 RF,
RF
RF
r
r
Constructing Yield Curve: Construct Yield Curve
Hypothetical Yield Curve
An upward sloping yield
curve.
Upward slope due to an
increase in inflationary
expectations and increasing
maturity risk premium over
time.
Years to Maturity
Real risk-free rate
0
5
10
15
1
Interest Rate (%)
Maturity risk premium
Inflation premium
10 20
Treasury vs. Corporate Yield Curves
Corporate yield curves are higher than that of Treasury
securities, though not necessarily parallel to the Treasury curve.
The spread between corporate and Treasury yield curves widens
as the corporate bond rating decreases.
Bonds rated AAA (Aaa) are judged to have less default risk than
bonds rated AA (Aa), while AA bonds are less risky than bonds
rated A and so on.
Illustrating the Relationship Between
Corporate and Treasury Yield Curves
0
5
10
15
0 1 5 10 15 20
Years to Maturity
Interest Rate (%)
5.2% 5.9%
6.0% Treasury Yield Curve
BB-Rated
AAA-Rated
Pure Expectations Hypothesis
The PEH contends that the shape of the yield curve depends on
investor’s expectations about future interest rates.
If interest rates are expected to increase, long-term rates will
be higher than short-term rates, and vice-versa.
Thus, the yield curve can slope up, down, or even bow.
Assumptions of the PEH
Assumes that the maturity risk premium for Treasury securities
is zero.
Long-term rates are an average of current and future short-term
rates.
If PEH is correct, you can use the yield curve to “back out”
expected future short-term interest rates.
An Example: Observed Treasury Rates and
the PEH
If PEH holds, what does the market expect will be the interest rate on
one-year securities, one year from now? Three-year securities, two
years from now?
Maturity Yield
1 year 6.0%
2 years 6.2%
3 years 6.4%
4 years 6.5%
5 years 6.5%
One-Year Forward Rate
(1.062)2 = (1.060) (1 + X)
1.12784/1.060 = (1 + X)
6.4004% = X
0 1 2
6.0% x%
6.2%
PEH says that one-year securities will yield 6.4004%, one year from now.
Notice, if an arithmetic average is used, the answer is still very close. Solve: 6.2% = (6.0% + X)/2, and the result will be 6.4%.
Three-Year Security, Two Years from Now
(1.065)5 = (1.062)2 (1 + X)3
1.37009/1.12784 = (1 + X)3
6.7005% = X
0 1 2 3 4 5
6.2% x%
6.5%
PEH says that three-year securities will yield 6.7005%, two years from now.
Conclusions about PEH
Some would argue that the MRP ≠ 0, and hence the PEH is incorrect.
Most evidence supports the general view that lenders prefer short-
term securities, and view long-term securities as riskier.
Hence, investors demand a premium to persuade them to hold
long-term securities (i.e., MRP > 0).
Macroeconomic Factors That Influence
Interest Rate Levels
Monetary policy
Federal budget deficits or surpluses
International factors / foreign trade deficit
Level of business activity
Bond Valuation
What is value?
The term value is used in different senses in the finance literature.
Liquidation Value vs. Going Concern Value
Liquidation value is the amount that can be realized if part of
a firm or the firm as a whole is sold separately from the
operating organization to which it belongs.
Going concern value is the amount that can be realized
should the firm be sold as a continuing operating entity.
Book Value vs. Market Value
Book Value of an asset is the carrying value of any asset, which is
calculated as the original cost-base of the asset minus the accumulated
depreciation accounted for the specific asset
Book value for a firm as a whole is the difference between book value of
all assets of the reporting entity minus the book value of all liabilities of
the reporting entity (SHE = TA-TL)
Market Value of an asset is the price at which the asset trades in the
market place. Almost always, market value of equity is higher than its
book (par) value. However this is not the case with bonds.
What is value?
Market Value vs. Intrinsic Value
The intrinsic value of an asset is the present value of all cash
flows expected from the asset, discounted at a rate of return
that is appropriate for the risk associated with the security.
Intrinsic value is economic value of an asset
Should the market be reasonably efficient, the market price of
an asset should hover around its intrinsic value.
What is value?
Bonds
A bond is a contract wherein a borrower promises to pay interest
and principal on specific dates to the holders of the bond.
In India, the principal issuers of bonds are
Central Government (Treasury Bonds)
State Government (State Government Bonds)
Public Sector Undertakings (PSU Bonds)
Private sector companies (Corporate Bonds)
Bonds issued by PSUs and private sector companies, generally
have a maturity ranging from 1 year to 15 years, and they pay
coupons on a semi-annual basis, unless stated otherwise.
Key Features of a Bond
Par value – face amount of the bond, which is paid at maturity (assume $1,000 if not specified).
Coupon interest rate – stated interest rate (generally fixed) paid by the issuer.
Multiply by par value to get dollar payment of interest.
Maturity date – years until the bond must be repaid.
Issue date – when the bond was issued.
Yield to maturity – rate of return earned on a bond held until maturity (also called the “promised yield”).
The Bond Pricing Equation
T
T
)(1
FV
R
R)(1
1-1
C Value BondR
Pure Discount Bonds
Make no periodic interest payments (coupon rate = 0%)
The entire yield to maturity comes from the difference between the purchase price and the par value.
Cannot sell for more than par value
Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs)
Treasury Bills and principal-only Treasury strips are good examples of zeroes.
Pure Discount Bonds
Information needed for valuing pure discount bonds: Time to maturity (T) = Maturity date - today’s date Face value (F) Discount rate (r)
TR
FVPV
)1(
Present value of a pure discount bond at time 0:
0
0$
1
0$
2
0$
1T
F$
T
Pure Discount Bond: Example
Find the value of a 30-year zero-coupon bond with a $1,000 par
value and a YTM of 6%.
11.174$)06.1(
000,1$
)1( 30
TR
FVPV
0
0$
1
0$
2
0$
29
000,1$
30
0
0$
1
0$
2
0$
29
000,1$
30
Level Coupon Bonds
Make periodic coupon payments in addition to the maturity value
The payments are equal each period. Therefore, the bond is just a combination of an annuity and a terminal (maturity) value.
Coupon payments are typically semiannual.
Consols
Not all bonds have a final maturity.
British consols pay a set amount (i.e., coupon) every period forever.
These are examples of a perpetuity.
R
CPV
Bond Concepts
Bond prices and market interest rates move in opposite directions.
When coupon rate = YTM, price = par value
When coupon rate > YTM, price > par value (premium bond)
When coupon rate < YTM, price < par value (discount bond)
YTM with Annual Coupons
Consider a bond with a 10% annual coupon rate, 15 years to
maturity, and a par value of $1,000. The current price is $928.09.
Will the yield be more or less than 10%?
YTM = {C+(MV – Price )/n}/{0.4*MV + 0.6*Price}
Effect of a Call Provision
Allows issuer to refund the bond issue if rates decline (helps the
issuer, but hurts the investor).
Borrowers are willing to pay more, and lenders require more, for
callable bonds.
Most bonds have a deferred call and a declining call premium.
What is a sinking fund?
Provision to pay off a loan over its life rather than all at maturity.
Similar to amortization on a term loan.
Reduces risk to investor, shortens average maturity.
But not good for investors if rates decline after issuance.
How are sinking funds executed?
Call x% of the issue at par, for sinking fund purposes.
Likely to be used if rd is below the coupon rate and the bond sells at
a premium.
Buy bonds in the open market.
Likely to be used if rd is above the coupon rate and the bond sells at
a discount.
Definitions
CGY Expected
CY Expected YTM return total Expected
price Beginning
price in Change (CGY) yieldgains Capital
priceCurrent
payment coupon Annual (CY) eldCurrent yi
7-38
Other Types (Features) of Bonds
Convertible bond – may be exchanged for common stock of the
firm, at the holder’s option.
Warrant – long-term option to buy a stated number of shares of
common stock at a specified price.
Putable bond – allows holder to sell the bond back to the company
prior to maturity.
Income bond – pays interest only when income is earned by the
firm.
Indexed bond – interest rate paid is based upon the rate of inflation.
Stock Valuation
The Present Value of Common Stocks
The value of any asset is the present value of its expected future cash
flows.
Stock ownership produces cash flows from:
Dividends
Capital Gains
Valuation of Different Types of Stocks
Zero Growth
Constant Growth
Differential Growth
Case 1: Zero Growth
Assume that dividends will remain at the same level forever
RP
RRRP
Div
)1(
Div
)1(
Div
)1(
Div
0
3
3
2
2
1
10
321 DivDivDiv
Since future cash flows are constant, the value of a zero growth stock is the present value of a perpetuity:
Case 2: Constant Growth
)1(DivDiv 01 g
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:
gRP
1
0
Div
Assume that dividends will grow at a constant rate, g, forever, i.e.,
2
012 )1(Div)1(DivDiv gg
3
023 )1(Div)1(DivDiv gg .
. .
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.
Case 3: Differential Growth
)(1DivDiv 101 g
Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter.
2
10112 )(1Div)(1DivDiv gg
N
NN gg )(1Div)(1DivDiv 1011
)(1)(1Div)(1DivDiv 21021 ggg N
NN
. . .
. . .
Case 3: Differential Growth
)(1Div 10 g
Dividends will grow at rate g1 for N years and grow at rate g2 thereafter
2
10 )(1Div g
Ng )(1Div 10 )(1)(1Div
)(1Div
210
2
gg
g
N
N
…
0 1 2
…
N N+1
…
Case 3: Differential Growth
We can value this as the sum of:
an N-year annuity growing at rate g1
T
T
AR
g
gR
CP
)1(
)1(1 1
1
plus the discounted value of a perpetuity growing at rate g2 that starts in year N+1
NBR
gRP
)1(
Div
2
1N
Case 3: Differential Growth
Consolidating gives:
NT
T
R
gR
R
g
gR
CP
)1(
Div
)1(
)1(1
2
1N
1
1
Or, we can “cash flow” it out.
Estimates of Parameters
The value of a firm depends upon its growth rate, g, and its discount rate, R.
Where does g come from? g = Retention ratio × Return on retained earnings
Stock Valuation Problems
1. Ezzel Corporation issued perpetual preferred stock with a 10%
annual dividend. The stock currently yields 8% and its par value
is $100.
a. What is the preferred stock’s value?
b. Should the interest rates in the broader economy increase,
and in-turn pull the preferred stock’s yield up to 12%, what is
the new market value of preferred stock?
2. Bruner Aeronautics has perpetual preferred stock outstanding
with a par value of $100. The stock pays a quarterly dividend of
$2 and its current price is $80.
a. What is its nominal annual rate of return?
b. What is its effective annual rate of return?
3. A stock is expected to pay a dividend of $0.50 one year hence, and it should continue to grow at a constant rate of 7% a year. If its required rate is 12%, what is the stock’s expected price 4 years from today?
4. Microtech Corporation is expanding rapidly and currently needs to retain all of its earnings, hence it does not pay dividends. However investors expect Microtech to begin paying dividends, beginning with a dividend of $1.00 coming 3 years from today. The dividend should grow rapidly –at the rate of 50% per year- during years 4 and 5; but after year 5, growth should be a constant 8% per year. If the required return on Microtech is 15%, what is the value of the stock today?
Stock Valuation Problems
5. Mitts Cosmetics Co’s stock price is $58.88, and it recently paid a
$2.00 dividend. This dividend is expected to grow by 25% for the
next 3 years , then grow forever at a constant rate, g and r = 12%. At
what constant rate is the stock expected to grow after year 3?
6. Your broker offers to sell you some shares of Bahnsen and Co.
common stock that paid a dividend of $2.00 yesterday. Bahnsen’s
dividend is expected to grow at 5% per year for the next 3 years.
a. If you buy the stock, plan to hold it for 3 years, and then sell it at
$34.73, what is the most you should be willing to pay for this
stock, assuming a discount rate of 12%.
b. If the holding period is 5 years rather than 3 years, would this
affect the value of stock today?
Stock Valuation Problems
7. Taussing Technologies Corporation (TTC) has been growing at a
rate of 20% per year in recent years. This same growth rate is
expected to last for another 2 years, and then decline to 6%. If
current dividend (at t=0) is $1.60, and if discount rate is 10%
a. What is TTC’s stock worth today?
b. What are the expected dividend yields and capital gains yield
for years of supernormal growth (years 1 and 2)?
c. Should TTC’s supernormal growth rate last for 5 years rather
than 2 years, calculate the price of TTC’s stock today, and the
dividend yield and capital gains yield for the years of
supernormal growth.
Stock Valuation Problems
Q7 continued: d: Suppose TTC recently introduced a new line of products
that has been wildly successfully. On the basis of this success and anticipated
future success, the following free cash flows (in millions) were projected.
After the tenth year, TTC’s financial planners anticipate FCF to grow by 6%
every year. Further, this new project has reduced overall enterprise risk,
which in-turn has reduced enterprise cost of capital to 9%.
Assuming (a) market value of TTC’s debt to be 1200 million, and (b) 20
million common shares outstanding (no preferred shares), what is value of
TTC’s stock as of today (use corporate valuation model).
Stock Valuation Problems
Year FCF Year FCF
1 5.5 6 88.8
2 12.1 7 107.5
3 23.8 8 128.9
4 44.1 9 147.1
5 69.0 10 161.3
8. A company’s annual dividends have increased from $1.25 for 1990 to $1.75 for 1995.
a. What is the average annual rate of growth of dividends from 1990 to 1995?
b. If an investor’s required rate of return is 12%, how much should he be willing to pay for a share of the company’s stock at the beginning of 1996, assuming that the rate of growth will continue at the same rate as during the preceding five years?
c. What would be the required rate of growth of the annual dividends for the stock to be worth a selling price of $40 per share at the beginning of 1996?
Stock Valuation Problems
9. Barrett Industries invests a large sum of money in R&D, as a result,
it retains and reinvests all of its earnings. In other words, Barrett
does not pay any dividends and it has no plans to pay any dividends
in the near future. A major pension fund is interested in purchasing
Barrett’s stock. The pension fund manager has estimated Free Cash
Flows for the next four years as follows: $3 million, $6 million, $10
million, and $15 million. After the fourth year, Barrett’s cash flow is
projected to grow at a constant rate of 7%.
I. If Barrett’s enterprise cost of capital is 12%, what is the firm’s
value as of today?
II. What is the estimate of Barrett’s price per share, if Barrett
Industries has (a) cumulative debt and preferred stock totaling
$60 million and (b) 10 million outstanding common shares
Stock Valuation Problems
10. Assume that today is December 31st, 2008 and that the following
information (estimation) applies to Vermeil Airlines:
• After-tax operating income for 2009 is expected to be $500
million
• Depreciation expense for 2009 is $100 million
• Capital Expenditures for 2009 are expected to be $200 million
• No change in net working capital
• FCFs are expected to grow at a constant rate of 6% going forward
• Enterprise cost of capital is 10%
• Market Value of company’s debt is $3 billion.
• Number of common shares outstanding: 200 million
• What should be the company’s stock price today?
Stock Valuation Problems