Finance 4000

Money and Capital Markets

Fourth class

• Duration and Interest-rate Risk of Bonds

  
  
  

  • What is the relationship between duration and the price of a bond?

    
    
    

  • 

• • • ProportionalP_b is the proportional change in the price of a bond

      
      
      

    • i be the change in the interest rate

      
      
      

    • Example

      
      
      

      • Suppose that the duration of a bond is 10 years and the yield to maturity is 5 percentage points per year

        
        
        

      • A increase in i by 1 percentage point is associated with a proportional decrease in the price of the bond by 0.097 or 9.7 percent [100.01/1.05 multiplied by 100 to get the percentage change]
        • The bond loses the fraction 0.097 of its value or 9.7 percent

          
          
          

• • Simplify by using modified duration

    
    
    

    • Modified duration = duration / (1+i)
    • 
    • If i in percentage points then can think of percentage change in price

      
      
      

      • If modified duration is 10 years, an increase in i by 1 percentage point per year is associated with a 10 percent fall in price

        
        
        

      • Need not concern ourselves with level of the interest rate to compute effect of change in yield on price if given modified duration
• The Determination of the Interest Rate

  
  
  

  • Loans and bonds
  • Borrowing and lending
  • Demand and supply

    
    
    

• The demand for loans and the supply of loans determine the equilibrium interest rate and quantity of loans

  
  
  

• The Demand for Loans -- borrowers

  
  
  

  • The quantity demanded of loans is the amount that borrowers want to borrow

    
    
    

    • Households, firms and the government

      
      
      

  • The demand for loans is the relationship between the amount that borrowers want to borrow and the interest rate on loans

    
    
    

    • In short, the demand for loans is the relationship between the quantity demanded of loans and the interest rate

      
      
      

    • The demand curve for loans is the curve showing the relationship between the quantity demanded of loans and the interest rate on loans

• • We will suppose that a higher interest rate on loans implies that households want to borrow less

    
    
    

  • This implies that the demand curve is downward sloping

    
    
    

  • What else affects the demand for loans?

    
    
    

    • Variable Effect on demand

      
      
      

    • Expected inflation rate +

      
      
      

    • Expected return on other assets +
      • Investment opportunities

        
        
        

    • Government deficits +

      
      
      

    • Other variables could be added such as
      • Income in the future relative to today

        
        
        

  • L^d = f^Ld( i, Expected inflation,

- +

investment opportunities, government deficit,..)

+ +

L^d is the quantity of loans demanded

• Supply of loans -- lenders

  
  
  

  • In the economy, ultimately households

    
    
    

    • Households supply funds to banks and other financial intermediaries

      
      
      

    • Households own firms

      
      
      

    • Households pay taxes that fund government

      
      
      

  • The quantity supplied of loans is the amount that lenders want to lend

    
    
    

  • The supply of loans is the relationship between the amount of loans that lenders want to lend and the interest rate on loans

    
    
    

    • In short, the supply of loans is the relationship between the quantity supplied of loans and the interest rate

      
      
      

    • The supply curve of loans is the curve showing the relationship between the quantity supplied of loans and the interest rate on loans

      
      
      

  • We will suppose that, the higher the interest rate, the more lenders want to lend

    
    
    

  • This implies that the supply curve is upward sloping

    
    
    

  • What else affects the supply of loans?

    
    
    

Variable Effect on demand

• • • Wealth +

      
      
      

    • Expected return on other assets -

      
      
      

    • Expected inflation rate -

      
      
      

    • Riskiness of loans -

      
      
      

    • Other variables could be added such as
      • Income in the future relative to today
      • Taxes

        
        
        

  • L^s = f^Ls( i, Wealth, Expected return on other assets,

-+ -

Expected inflation, Riskiness of loans, ...)

- -

L^s is the quantity of loans demanded

• Equilibrium

  
  
  

  • Definition: Quantity of loans demanded equals the quantity supplied

L^d=L^s

• • Only at that interest rate can borrowers and lenders both be doing what they want

    
    
    

• Effect of changes in demand and supply

  
  
  

• Effect of an increase in the expected inflation rate

  
  
  

• Effect of an increase in the government deficit

• The demand and supply of bonds

  
  
  

  • Fixed promised payments

    
    
    

    • 

      
      
      

    • CP_t+1 is constant, P_bt and i change

      
      
      

    • A higher interest rate implies a lower price

      
      
      

• The Demand for Bonds -- buyers of bonds -- lenders

  
  
  

  • The quantity demanded of bonds is the amount that buyers want to hold

    
    
    

    • In the economy, ultimately households

      
      
      

      • Households supply funds to banks and other financial intermediaries

        
        
        

      • Households own firms

        
        
        

      • Households pay taxes that fund government

• • The demand for bonds is the relationship between the amount of bonds that households want to hold and the price of bonds

    
    
    

    • In short, the demand for bonds is the relationship between the quantity demanded of bonds and the price of bonds

      
      
      

    • The demand curve for bonds is the curve showing the relationship between the quantity demanded of bonds and the price of the bonds

      
      
      

  • We will suppose that a higher price of bonds implies that households want to hold fewer bonds

    
    
    

    • A higher price of bonds means that you pay more now for the same number of dollars in the future

      
      
      

  • This supposition means that the demand curve is downward sloping

• • What else affects the demand for bonds?

    
    
    

    • Variable Effect on demand

      
      
      

    • Wealth +

      
      
      

    • Expected return on other assets -

      
      
      

    • Expected inflation rate -

      
      
      

    • Riskiness of bonds -

      
      
      

    • Other variables could be added such as
      • Income in the future relative to today
      • Taxes

        
        
        

  • B^d = f^Bd( P_b, Wealth, Expected return on other assets,

-+ -

Expected inflation, Riskiness of bonds, ...)

- -

B^d is the quantity of bonds demanded

• Supply of bonds -- issuers, borrowers

  
  
  

  • households, firms and the government

    
    
    

  • The quantity supplied of bonds is the amount that borrowers want to create

    
    
    

  • The supply of bonds is the relationship between the amount of bonds that borrowers want to create and the price of the bonds

    
    
    

    • In short, the supply of bonds is the relationship between the quantity supplied of bonds and the price of bonds

      
      
      

    • The supply curve of bonds is the curve showing the relationship between the quantity supplied of bonds and the price of bonds

      
      
      

  • We will suppose that, the higher the price, the more issuers want to create

    
    
    

  • This implies that the supply curve is upward sloping

• • What else affects the supply of bonds?

    
    
    

    • Variable Effect on supply

      
      
      

    • Expected inflation rate +

      
      
      

    • Expected return on other assets +
      • Investment opportunities

        
        
        

    • Government deficits +

      
      
      

    • Other variables could be added such as
      • Income in the future relative to today

        
        
        

  • B^s = f^Bs( P_b, Expected inflation,

+ +

investment opportunities, deficits,..)

+ +

B^s is the quantity of bonds supplied

• Equilibrium

  
  
  

  • Definition: Quantity demanded equals quantity supplied

B^d=B^s

• • Only at that price can buyers and sellers of bonds both do what they want

    
    
    

• The supply of bonds in the short run