Finance 4000

Money and Capital Markets

Second class

Interest Rates

• Interest Rate -- definition

  
  
  

  • A simple interest rate is the excess of the amount to be received in the future relative to the amount paid now

    
    
    
    
    

  • Formula

    
    
    

    • 

      
      
      

    • i_t is the interest rate

      
      
      

    • V_t is the value in period t -- today

      
      
      

    • V_t is the value in period t+1 -- next period or one period in the future

      
      
      

  • Given any two of the variables in this formula, you can figure out the other one

    
    
    

  • Examples

• • Borrow $100 and pay back $105

    
    
    

    • V_t is $100

      
      
      

    • V_t+1 is $105

      
      
      

    • i_t is .05 or 5 percent -- ($105-$100)/$100

      
      
      
      
      
      
      
      
      

  • Borrow $100 at 5 percent

    
    
    

    • V_t is $100

      
      
      

    • i_t is 5 percent -- or .05

      
      
      

    • V_t+1 is $105

      
      
      

    • often say "pay back $100 and interest payment of $5"
    • 

• • If willing to pay $105 a year from now and the interest rate is 5 percent, how much can you borrow?
    • V_t+1 is $105
    • i_t is 5 percent or .05
    • V_t is $105
    • 

      
      
      

    • ads on tv
      • "If you have a structured settlement coming in a bunch of small payments, we will give you one large payment now."
      • How much?

        
        
        

    • Given some interest rate, you will get the present value of the future payments

      
      
      
      
      

  • Present value

    
    
    

    • The present value of an amount to be received in the future is the value today of that payment.

      
      
      

    • A dollar in the future is not worth as much as a dollar today

      
      
      
      
      

• What if there is more than one payment?

  
  
  
  
  

• Compound interest

  
  
  

  • Suppose that we have $100 and we lend it at 5 percent per year for two years

    
    
    
    
    

  • How much should we get at the end of two years?

    
    
    
    
    

  • At the end of one year, we are owed $105
    • the original $100 plus 5 percent interest

      
      
      
      
      

  • We are willing to lend our funds for an additional year
    • from t+1 to t+2
    • say 2000 to 2001

      
      
      
      
      

  • If the interest rate is 5 percent, then in 2001, we should get our $105 plus 5 percent interest
    • $105 + .05$105 = $105+$5.25 = $110.25
    • not $105 + $5
    • $.25 is interest on the interest from the first year
  • The formula for this is determined the following way

    
    
    

    • 

      
      
      

    • and for the next period, this is

      
      
      

    • 

      
      
      

    • For simplicity, suppose that the interest rate is constant at i

      
      
      

    • Then

      
      
      

    • And substituting the second into the first

      
      
      

    • 

      
      
      
      
      

  • The same proposition holds for this equation as for the simple one

    
    
    

    • Given any two of the variables in this formula, you can figure out the other one
  • The formula for the present value of a payment two periods from today is

    
    
    

• • Indicates how much willing to pay for a payment two years from now given an interest rate of i

    
    
    

• More than two payments in a "structured settlement"?

  
  
  

  • Just add up the present value of each of the payments

    
    
    

    • Let S_t+1 be the structured settlement payment in period t+1
    • S_t+2 be the payment in period t+2
    • and so forth

      
      
      

    • The present value of a structured settlement for 10 periods is

      
      
      

  • 

    
    
    

    • Works for any number of years -- 10, 4, or 99
    • Same payments every year, could get rid of subscripts on the payment -- just use S

      
      
      

Financial Markets -- Bonds

• Coupon bond

  
  
  

  • Price of bond -- how much is paid for it

    
    
    

  • Maturity

    
    
    

    • The maturity of a bond is the time until the last payment
      • For example, a bond might have 20 years to maturity from today

        
        
        
        
        

  • Coupon payment

    
    
    

    • The coupon payment on a bond is a periodic payment made on a bond until maturity
      • A bond might pay $500 per year until maturity

        
        
        

  • Face value

    
    
    

    • The face value on a bond is a final payment
      • A bond might have a face value of $1000

        
        
        

  • What is interest rate on funds if hold bond to maturity?
  • The interest rate on funds if the bond is held to maturity is called the yield to maturity

    
    
    

    • The yield to maturity is the interest rate that equates the present value of payments received from a credit market instrument with its value today

      
      
      

  • Example -- One year-bond
    • a price of $9000
    • a coupon payment of $500 per year
    • a face value payment of $10000
    • What is interest rate on funds?
    • i=($10000+$500-$9000)/$9000=.1666...
      • or 16.66... percent

        
        
        

  • Two-year bond -- two years to maturity

    
    
    

    • 

      
      
      

    • Solve for yield to maturity i

Other kinds of bonds

• Treasury bills

  
  
  

  • Issued as discount securities
    • Only one payment

      
      
      

  • Term to maturity when issued of 90, 180 or 360 days

    
    
    

• STRIPS

  
  
  

  • Separately Traded Registered Interest and Principal Securities

    
    
    

  • Take a coupon bond and break it into single payments

    
    
    

  • For example, two year bond (often called a note)
    • two coupon payments of $1000
      • one in January 2000
      • one in January 2001
    • face value of $10,000 paid in January 2001

      
      
      

  • Take bond apart into 3 discount bonds
    • One paying $1000 in January 2000
    • One paying $1000 in January 2001
    • One paying $10,000 in January 2001
  • Why create a strip?

    
    
    

    • For example, someone may have contract requiring that they pay $10,000 in January 2002

      
      
      

    • Can buy bond and guarantee that they can make payment

      
      
      

    • More generally, can put together any stream of receipts that is convenient

      
      
      

    • Called zeroes

      
      
      
      
      

• State and local bonds

  
  
  

  • Typically not subject to federal income tax or state income tax if state of residence

    
    
    

• Private bonds

  
  
  

  • Higher risk than federal government

    
    
    

  • The higher the risk, the higher the yield

• Return from holding a bond

  
  
  

  • Really applies to any asset

    
    
    

  • Often called the holding period return

    
    
    

  • The holding-period return is the rate of return received over the period that the asset is held including any capital gains or losses on the asset

    
    
    

  • Let h be the holding-period return from holding a bond from t to t+1

    
    
    

    • The holding period return for one period is

      
      
      

    • For one period, it is just
      • payment received
      • less payments made
      • initial payments made

• • Example of holding-period return

    
    
    

    • Buy a 20 year bond today for $10,000

      
      
      

      • coupon payment of $1000 per year
      • face value of $10,000

        
        
        

      • yield to maturity is 10 percent

        
        
        

      • Suppose sell it a year from now for $9,000

        
        
        

      • The holding period return for one period is

        
        
        

      • Means that holding period return is 0

        
        
        

        • 

          
          
          

    • If sold bond for $11,000 a year from now, h is

      
      
      

      • 

or 20 percent

• Bonds, Changes in Interest Rates and Risk

  
  
  

• Higher bond prices are associated with lower yields

  
  
  

  • The price of a bond equals the present value of payments

    
    
    

  • For a one-year bond

    
    
    

  • The bond contract specifies the coupon payments and the face value

    
    
    

  • Changes in the price are associated with changes in the interest rate (or yield to maturity)

    
    
    

    • Example -- one-year bond

      
      
      

    • Coupon payment equal to zero

      
      
      

    • Face value equal to $1,000

      
      
      

    • If the price of the bond is $900, the interest rate is 11.11... percent

      
      
      

    • If the price of the bond is $950, the interest rate is about 5.26 percent

      
      
      

• If interest rates increase, the yield to maturity on existing bonds increases

  
  
  

• As a result, increases in interest rates lower prices of existing bonds

  
  
  

  • • Bonds are subject to interest rate risk

      
      
      

    • The prices of longer term bonds decrease more when interest rates increase

      
      
      

    • Example

Prices of two bonds

Face value $10,000 and coupon payment $0

Interest Rate Term to maturity (years)

1 2

5	9524	9070
10	9091	8264

Change in price -$433 -$806

• For a given increase in the interest rate, longer-term bond prices change more

  
  
  

• Empirically, longer-term bond prices are more variable

• Does that mean that one should hold short-term bonds for less risk?

  
  
  

  • Future short-term interest rates are unknown

    
    
    

  • Because of short-term interest rates are uncertain,

    
    
    

Short-term securities are subject to reinvestment risk

• • Reinvestment risk is the risk that proceeds from securities will have to be reinvested at uncertain interest rates

    
    
    

• Long-term bonds are subject to interest-rate risk because prices change

  
  
  

• Short-term bonds are subject to interest-rate risk because of reinvestment risk

  
  
  

• Next time

  
  
  

  • Nominal and real interest rates

    
    
    

  • Duration as a measure of the maturity of a bond

    
    
    

  • Chapter 4 -- portfolio choice