Important issues
723g28
The Term Structure of Interest Rates
• The term structure is the set of interest rates
on the same risk instrument for various time
to maturity.
• The longer the time to maturity of a bond, the
higher the expected return of the bond in
normal time. (YTM)
• We usually use the different maturity of zerocoupon bond´s yield to represent the term
structure.
Term structure
• We can define spot rates rn as the yield to maturity on
zero-coupon bonds issued today and maturing at time
n. The term structure of interest rates is given by the
set of rates, r1, r2, r3, ...
• Spot Rate - The actual interest rate today (t=0)
• Forward Rate - The interest rate, fixed today, but refer
to a time period in the future.
• If forward rate is low, investors want to wait to make
new investment. Companies withhold investment due
to the deem outlook in the future. It is cheaper to
borrow in the future! Lots of foregone investment
opportunities!
The Yield
curve
• The yield curve is the graph of the term
structure of interest rate. It is the graphic
representation of the relation between the
interest rate (or cost of borrowing) and the
time to maturity of the debt.
A typcal yield curve
How to explain an upward sloping or a
downward sloping yield curve?
A upward sloping yield curve is the normal shape of a
yield curve: liquidity premium theory.
The longer time to maturity, the more sensitive the bond
price to the interest rate changes. To compensate the risk
of interest rate changes.
If the expected future rate is up, investors investing in
longer term bonds need to be compensated on the yield
now in order to be tied up in the bond contract.
The risks of future uncertainty needs to be compensated:
a risk premium rp. (1+r2)2 = (1+r1)(1+f2) + rp2 =
(1+r1)(1+E(r2)) + rp2
Explain a downward sloping yield
curve:
• The reason of a downward sloping yield curve is because
that investors expect short term interests rate to fall.
(1+r2)2 = (1+r1)(1+f2) = (1+r1)(1+E(r2)).
• Since the forward rate f is an unbiased future spot rate. As f
rises, market believes the short term interest rate is about
to rise, and vice versa. If f falls, the market believes the
short term interest rate is about to fall.
• A downward sloping yield curve indicates a recession to
come on the horizon, or perhaps it is well into the recession
already. When r2 is less than r1, f2 is also less than r1.
E(r2)<r1.
Interest rate falls in the near future.
• Risk premium is negative under downward sloping yield
curve. Investors prefer to hold on longer term bonds at bad
times.
Efficient Market Hypothesis
• State the definitions of EMH.
• If the market is not efficient then there will be
sure profit to be made. The profit will soon
disappear as it shows up, since the investors
will buy the stocks up until the expected value
is reached.
• What are the evidence that the market is semi
strong form efficent?
Efficient Market Theory
Efficient Market – well functioning capital market in which
prices reflect all available information.
In an efficient market, one cannot expect to make
persistent abnormal return over the risk adjusted return.
The risk adjusted return is determined by the CAPM.
Abnormal return  actual stock return - expected stock return
~
r  (   ~
rm )
• Weak Form Efficiency
– Market prices reflect all historical information
• Semi-Strong Form Efficiency
– Market prices reflect all publicly available information
• Strong Form Efficiency
– Market prices reflect all information, both public and private
Random Walk Theory
Market
Index
1,300
T1 value
Discounted value of t1
1,200
1,100
Cycles
disappear
once
identified
Last
Month
This
Month
Next
Month
Security Market Line
Return
.
r
Market Return = m
Market Portfolio
Risk Free Return =
rf
(Treasury bills)
1.0
BETA
One can create any return and risk on the SML by holding a varying proportion of the
risky assets rm and a risk free loan.
11
Portfolio theory
Mean: weighted average value.
Variance - Average value of squared deviations
from mean. A measure of volatility.
Standard Deviation - Average value of squared
deviations from mean. A measure of volatility.
Unique Risk - Risk factors affecting only that firm. Also called
“diversifiable risk.”
Market Risk - Economy-wide sources of risk that affect the overall stock
market. Also called “systematic risk.”
Efficient portfolio provides the highest return for a
given level of risk, or least risk for a given level of
return. The market portfolio is the one that has the
highest Sharpe ratio with the expected return and
risk. Sharpe ratio=(ri-rf)/σ
Why do company issue bonds, what
are the characteristics of bond?
• The pecking order theory
• lower cost of capital
• the monitoring role of debt holders, signaling
mechanism,
• interest tax shields
• No ownership dilution
What is the moral hazard problem
associated with debt financing?
• In real option terms, the limited liability is a put
option (abandonment option)
• The firm is bankrupt when the value of the firm is
less than the debt.
• Shareholders have an incentive to increase the
volatility if the firm (σ ) when the value of the
firm is low. Since the benefit is to the
shareholders, also the lower the value of the firm
becomes the higher the value of the put option at
strike price D.
Equity Value of the firm as options
• Shareholders value = Equity Value of the firm
+ a put option on the firm at strike price D
• There are talks about the Draghi put.
• Whenever the market interest rate on shart
term bonds is high (say over 7%), ECB will do
something to ease the yield. Quantitative
easing 1, 2 and 3. What are the consequescies
of the QE?
Option Payoffs at Expiration
• Long Position in an Option Contract
The value of a put option at expiration is
P  max (K  S , 0)
• Where S is the stock price at expiration, K is the
exercise price, P is the value of the put option, and max
is the maximum of the two values in the bracket.
• Clearly the value of the put is high, when K-S is high.
16
The higher the volatility the higher
the call option value
D
Equity Value of the firm
Put-Call Parity
• Consider the two different ways to construct
portfolio insurance discussed above.
– Purchase the stock and a put
– Purchase a bond and a call
• Because both positions provide exactly the
same payoff, the Law of One Price requires
that they must have the same price.
18
Put-Call Parity
• Therefore,
S  P  PV (K )  C
Where K is the strike price of the option (the price you
want to ensure that the stock will not drop below), C is
the call price, P is the put price, and S is the stock price
This relationship between the value of the stock, the
bond, and call and put options is known as put-call
parity.
19
Example 1
Assume:
• You want to buy a one-year call option and put option
on Dell.
• The strike price for each is $15.
• The current price per share of Dell is $14.79.
• The risk-free rate is 2.5%.
• The price of each call is $2.23
Using put-call parity, what should be the price of
each put?
20
Example 1
• Solution
– Put-Call Parity states:
S  P  PV (K )  C
$15
$14.79  P 
 $2.23
1.025
P  $2.07
21
Put-Call Parity
• If the stock pays a dividend, put-call
parity becomes
C  P  S  PV (K )  PV (Div)
22
Option pricing
Binomial Option Pricing Model
• A technique for pricing options based on the
assumption that each period, the stock’s return
can take on only two values
A Single-Period Model
• Assume
 A European call option expires in one period and has
an exercise price of $50.
 The stock price today is equal to $50 and the stock
pays no dividends.
 In one period, the stock price will either rise by $10 or
fall by $10.
 The one-period risk-free rate is 6%.
We can construct the binomial tree based on these
information.
24
Using Risk neutral method
• The risk neutral probability is
• P=(6%+20%)/(20%+20%)=65%
• 1-p=35%
• The expected call value=10*65%+0*35%=6,5
• The call price at the starting period t0=
C= 6,5/(1+0,06)=6,13$
A Single-Period Model
• Replicating Portfolio
is a portfolio consisting of a stock and a risk-free
bond that has the same value and payoffs in one
period as an option written on the same stock
• The Law of One Price implies that the current value of
the call and the replicating portfolio must be equal.
26
A Two-State Single-Period Model
• The payoffs can be summarized in a
binomial tree.
27
A Two-State Single-Period Model
• Let D be the number of shares of stock
purchased, and let B be an investment in bonds.
• To create a call option using the stock and the
bond, the value of the portfolio consisting of the
stock and bond must match the value of the
option in the two possible state (u, d).
• Create a call: Hold the stock and short the bond.
28
A Single-Period Model
• In the up state, the value of the portfolio must
be $10.
60D  1.06B  10
• In the down state, the value of the portfolio
must be $0.
40D  1.06B  0
29
A Single-Period Model
• Two equations, two variables, D and B can be
solved for.
60D  1.06B  10
40D  1.06B  0
D = 0.5
B = –18.8679
The call payoff
at period 1
Replicating portfolio
Note that by using the Law of One Price, we
are able to solve for the price of the option
without knowing the probabilities of the states
in the binomial tree.
30
A Single-Period Model
• A portfolio that is long 0.5 share of stock and
short approximately $18.87 worth of bonds
will have a value in one period that exactly
matches the value of the call.
60 × 0.5 – 1.06 × 18.87 = 10
40 × 0.5 – 1.06 × 18.87 = 0
31
A Single-Period Model
• By the Law of One Price, the price of the call
option today must equal the current market
value of the replicating portfolio
• The value of the portfolio today is the value
of 0.5 shares at the current share price of $50,
less the amount borrowed (the short
position). Thus, the call premium is c=
50D  B  50(0.5)  18.87  6.13
• The result is the same as in the risk neutral
method.
32
The Binomial Pricing Formula
• Assume:
• S is the current stock price, and S will either
go up to Su or go down to Sd next period.
• The risk-free interest rate is rf .
• Cu is the value of the call option if the stock
goes up and Cd is the value of the call
option if the stock goes down.
33
The Binomial Pricing Formula
• Given the above assumptions, the binomial
tree would look like:
• The payoffs of the replicating portfolios could
be written as:
Su D  (1  rf )B  Cu and,
Sd D  (1  rf )B  Cd
34
The Binomial Pricing Formula
• Solving the two replicating portfolio equations for the two
unknowns D and B yields the general formula for the replicating
formula in the binomial model.
Replicating Portfolio in the Binomial Model
Cu  Cd
Cd  Sd D
D 
and B 
Su  Sd
1  rf
The value of the option is the value of the portfolio
today: Option Price in the Binomial Model
C  SD  B
35
A put option valuation: the same method as call
K=60, rf=3%, u=20%, d=-10%, put
price?
Put value=0
72
60
54
• P=(3%+10%)/(20%+10%)=43,33
%
• 1-p=56,67%
Put value=6
P(t)=43,33%*0 +56,67%*6=3,4$
P(t-1)=3,4/1,03=3,3$
The replicating portfolio methods should
give the same result, try yourself!
Su D  (1  rf )B  Cu and,
Sd D  (1  rf )B  Cd