Firm A issues a zero-coupon bond with a face value of $100,000 at the beginning of the first year (t=0) which is purchased by investor 

B. The bond matures at the end of 20 years (t=20). The prevailing interest rate at t=0 for bonds with similar risk is 5%.

Consider the following for Investor :

(a)  How much does Investor B pay for this 20-year $100,000 zero-coupon bond  at t=0? What is the yield to maturity?

(b)  Suppose that when markets open on the first day of the sixth year, the  interest rate is still 5%. What would be the price of the bond that  morning?

(c)  Suppose that by the time the markets close on the first day of the sixth year, interest rates have fallen to 4%. What would be the price of Investor B's bond at the end of the day? (For calculations, ignore the fact that one day has passed, just assume we are still at the beginning of the sixth year).

(d)  Suppose Investor C buys Investor B's bond at the beginning of the ninth year for $60,000. Assuming this was a fair exchange, what does this tell you about the movement of interest rates between the beginning of the sixth year and the beginning of the ninth year?

(e) In the case that the bond that Firm A issued at t=0 was actually a 30-year bond rather than a 20-year bond. 

What does your answer

tell you about the relationship between time to maturity and price  volatility?

Problem:

Prove mathematically that for a level coupon bond, the discount  rate equals to the coupon rate, if the bond price is the face value?