Finance 4000

Money and Capital Markets

Third class

• Duration and Interest-rate Risk of Bonds

  
  
  

  • Other things the same, the price of a bond with a longer term to maturity decreases more if interest rates increase

    
    
    

  • The price of a zero-coupon bond decreases more than the price of a coupon bond when the yields increase by the same amount

    
    
    

    • Example

      
      
      

  • Term to maturity does not capture some aspects of interest-rate riskiness

    
    
    

    • Example

      
      
      

  • A bond really consists of many intervening payments

    
    
    

    • Initial logic in terms of a loan

      
      
      

    • Variable payments such as zero coupon payment bond mean that such a structure need never be true

      
      
      

    • Term to maturity of each payment?

• • How measure term to maturity of combined payment streams and implications for risk of bonds?

    
    
    
    
    

  • Duration

    
    
    

    • Term to maturity of each payment times fraction of present value of all payments at that term

      
      
      

    • Duration is the average lifetime of a stream of payments

      
      
      

  • Equation

    
    
    

    • Let f_t equal the fraction of the present value of all payments at time t,
      • t=1, 2, 3, ..., N years in the future
      • N is term to maturity

        
        
        

    • 

      
      
      

    • f_t is from

• • Duration and Interest Rate Risk

    
    
    

    • The longer the term to maturity, the longer the duration of a stream of payments

      
      
      

    • When the interest rate increases, the duration of a stream of payment decreases

      
      
      

    • The greater the coupon rate on a bond, the shorter the duration of a bond

      
      
      

    • Formula for effect of interest rate on price of a security

      
      
      

    • Let
      • %P_b be the percentage change in the price of a bond
      • i be the change in the interest rate

        
        
        

    • 

      
      
      

  • The greater the duration of a security, the greater the decrease in price for a given increase in the interest rate
  • Duration compared to term to maturity

    
    
    

    • Both -- longer associated with greater risk

      
      
      

    • Duration more effectively summarizes the relationship between the time to payment and interest-rate risk because it reflects the effect of intervening coupon payments

      
      
      

    • Can compute the duration of a portfolio of assets

      
      
      

      • Given same stream of payments, doesn't matter if coupon bonds or zeroes or whatever
        • Would not be true if used securities' terms to maturity

          
          
          

• Nominal and Real Interest Rates

  
  
  

  • A nominal interest rate is an interest rate in terms of dollars

    
    
    

  • A real interest rate is an interest rate in terms of commodities

    
    
    

  • Example

    
    
    

  • Treasury Inflation Protected Securities -- TIPS

    
    
    

    • The number of dollars paid increases with the level of the Consumer Price Index

      
      
      

    • Example

      
      
      

    • Recent data

• Portfolio Choice

  
  
  

  • An asset is a possession having value
    • due to anticipated payments or higher value for a security

      
      
      

  • At a general level, the demand for securities is straightforward

    
    
    

  • Demand for securities depends on

    
    
    

    • Wealth

      
      
      

    • Expected return on the security relative to other securities

      
      
      

    • Risk of the security
      • a chance or possibility of danger, loss, injury or other adverse consequences

        
        
        

    • Liquidity
      • the ability to be cheaply and quickly converted into cash

        
        
        

    • Not as informative as it could be because it's not very specific

      
      
      
      
      

  • Theories of the demand for securities
    • Capital Asset Pricing Model (CAPM)
      • Investors are risk averse and must be compensated for bearing risk
      • The expected return on securities will tend to be higher, the higher the co-movement of their expected returns with market returns

        
        
        

    • Arbitrage Pricing Theory (APT)
      • Investors are risk averse and must be compensated for bearing risk
      • The expected return on securities will tend to be higher, the higher the co-movement of their expected returns with factors that affect all securities

• • Risk Generally Measured by the Variance of the Expected Return

    
    
    

    • Expected return

Two possible outcomes

• • • • 
        • where
        • p₁ is the probability of event 1 with return r₁
        • p₂ is the probability of event 2 with return r₂
        • p₁+p₂=1
      • Average return expected in the future

        
        
        

    • Variance
      • 

        
        
        

    • Standard Deviation

      
      
      

      • 
      • Same units as expected return
      • A rough rule of thumb for securities is that an expected return more than one standard deviation from the expected return happens about 1/3 of the time

        
        
        

    • When is the standard deviation a good measure of risk? When not?
  • How Reduce Risk?

    
    
    

    • Modern Portfolio Theory: Diversify

      
      
      

    • Example

      
      
      

    • Lessons

      
      
      

      • Diversification can reduce risk if outcomes independent

        
        
        

      • Diversification can reduce risk more if bad outcomes are negatively related
        • With symmetry, bad outcomes negatively related implies good outcomes negatively related

          
          
          

    • International Diversification -- handout

      
      
      

    • The efficient portfolio frontier is the set of the most attractive combinations of the expected return and standard deviation of the expected return on the securities

      
      
      

  • As add more securities, the risk due to holding the securities decreases

• • Decrease in risk is limited by the variance common to the securities

    
    
    

  • For United States, get close to market risk with just 20 securities

    
    
    

    • Market risk for stocks is the variance of expected return on all stocks weighted by their relative values

      
      
      

  • The expected return will include compensation for systematic (non-diversifiable) risk but not for nonsystematic, or idiosyncratic, risk

    
    
    

    • Can diversify away nonsystematic risk

      
      
      

    • Results in nonsystematic risk being unpriced

• • Capital Asset Pricing Model

    
    
    

    • Systematic risk consists of changes in expected return associated with changes in the market expected return

      
      
      

    • Suppose there is a linear relationship between the expected return on individual assets and the market return

      
      
      

      • 

        
        
        

      • Systematic variance in expected return can be represented by r_m

        
        
        

      • Unsystematic variance in expected return is represented by _i

        
        
        

    • Suppose that there is a risk-free asset with a risk-free expected rate of return

      
      
      

    • Combinations of the risk-free asset and diversified portfolios are possible

      
      
      

    • The efficient portfolio frontier shows the set of most preferred portfolios of risky securities

• • • If there is a zero-variance security, then the efficient portfolio frontier including the risk-free asset is a straight line between expected return on the portfolio and the standard deviation of the portfolio

      
      
      

    • Knowing what will happen requires specifying more about investors' preferences and equilibrium

      
      
      

    • If everyone is identical, they must hold the market portfolio

      
      
      

    • In equilibrium,

      
      
      

      • is the marginal contribution of security i to the risk in the portfolio

        
        
        

      • Er_m-Er_f is called the market price of risk because it is the expected marginal return from holding the market portfolio instead of the risk-free asset

        
        
        

    • What's wrong with the Capital Asset Pricing Model?

• • Arbitrage Pricing Theory

    
    
    

    • Basically examines nondiversifiable risk in terms of factors that affect the expected return on each security

      
      
      

    • May prove to be more stable or informative empirically