Two (default-free) government bonds, A and B, are trading at a current market price of $82 and $76, respectively. Bond A is a zero-coupon bond with 1 year to maturity. Bond B is a 12% coupon bond with 2 years to maturity. Both bonds have a face value of $100. Assume any coupons are paid annually.

1. b) Answer the following:

   1. What is the Macaulay Duration of Bond B at its yield to maturity? You may

      find the following formula helpful: the solution to the quadratic equationx2 +x+c=0isgivenbyx=(-b! 4)/2a

   2. Using the duration approach, approximate the dollar change in the price of the bond if its yield increases by 1 percentage point. NOTE: you MUST Show FULL workings to score well.

   3. Would the magnitude of this price change be larger or smaller if Bond B was currently selling at par? Explain clearly you may do this mathematically or without the aid of any calculations.

   4. A third government bond, Bond C, is currently trading for $87. Bond C has a 18% coupon rate, a face value of 100, and matures in 2 years. Is there an arbitrage opportunity at this price? If yes, show how to construct an arbitrage portfolio that delivers a guaranteed positive cash-flow today and zero net cash flows in future years. Assume you can go long or short in all assets.

   5. Consider a default-free coupon bond with market price P. The bond matures in N years time, has face value F, a constant coupon rate c > 0, and yield to maturity r > 0. Coupons are paid annually, and all rates are expressed assuming annual compounding.

   6. i. Show that if a bond sells at a discount to face value, then c < r .

   7. The current yield of a coupon bond is defined as the annual coupon payment expressed as a fraction of the bond price. Show that the following relationship holds for all discount bonds and provide an intuitive explanation:

      yield to maturity > current yield > coupon rate