Important issues
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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)

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Important issues


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 zero-coupon 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 Yieldcurve

  • 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 downwardslopingyieldcurve:

  • 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.

  • 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

T1 value

Discounted value of t1




Cycles disappear once identified

Last Month

This Month

Next Month

Market Return = rm

Security Market Line



Market Portfolio


Risk Free Return =

(Treasury bills)



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.

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

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.

The higher the volatility the higher the call option value


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.

Put-Call Parity


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.

Example 1


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?

Example 1


Put-Call Parity states:

Put-Call Parity

If the stock pays a dividend, put-call parity becomes

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


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.

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.

A Two-State Single-Period Model

The payoffs can be summarized in a binomial tree.

A Two-State Single-Period Model

Let D be the number of shares of stock purchased, and let Bbe 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.

A Single-Period Model

In the up state, the value of the portfolio must be $10.

In the down state, the value of the portfolio must be $0.

A Single-Period Model

Two equations, two variables, D and B can be solved for.

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.

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

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=

The result is the same as in the risk neutral method.

The Binomial Pricing Formula


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.

The Binomial Pricing Formula

Given the above assumptions, the binomial tree would look like:

The payoffs of the replicating portfolios could be written as:

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

The value of the option is the value of the portfolio today: Option Price in the Binomial Model

A put option valuation: the same method as call

K=60, rf=3%, u=20%, d=-10%, put price?

  • P=(3%+10%)/(20%+10%)=43,33%

  • 1-p=56,67%

    P(t)=43,33%*0 +56,67%*6=3,4$


Put value=0




Put value=6

The replicating portfolio methods should give the same result, try yourself!

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