223.3

3. Several major companies (like Disney and AT&T) have issued "Century Bonds." These bonds pay regular semi-annual coupons, but do not mature until 100 years after they are issued. Some critics have stated that there is huge risk that you won't get the principal repaid because you just can't predict what will happen to the company in 100 years. For this question assume that the company has just issued a $1000 par, 8% coupon bond with semi-annual payments.
 

1. If current market rates are 8% compounded semi-annually, what is the current price of the bond? What is the present value of the principal repayment at maturity? Finally, what proportion of the bond's price does the principal repayment make-up? (5 pts.)

   The bond will be priced at par (market rates=coupon rates), so it will be worth $1000.

   The PV of the principal repayment is:
   

   That's less than 4/100 of a percent!

    b. What is the total value today of the last 40 years (years 61-100) of payments, including coupons and principal repayment? (8 pts.)
    
   This is a deferred annuity of $40 (half of 8% of $1000) every 6 months:
   

    c. If interest rates go down, will the bond be worth more or less? Why? (4 pts.)

The bond will be worth more. There are two ways to think about this: first, interest rates enter into the denominator of the PV calculation, so the PV goes up and PV is the price of the bond. Second, and more importantly is the economic intuition that the bond's package of cash flows hasn't changed, but it has become more attractive relative to what's being offered currently.
 
d.If interest rates go down, will the principal repayment be a larger or smaller portion of the total value of the bond? Why? (6 pts.)

The principal will become a larger proportion of the bond's value. While the bond's value will go up, the principal's value will go up more than proportionately because of the compounding effect of 200 periods of lower interest rates. Effectively, the 6 payment is not being discounted as much relative to the other payments.

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