Only need help with 1, 3 and 5 - these are really math questions...

Also, the trouble I'm having in question 1 is I think can't be in the expression...

For discount bonds, maturity measures the length of time that a bondholder has invested money. Namely, for an n-period discount bond, you wait n periods to obtain the future cash flow. But for coupon bonds, maturity is an imperfect measure of this length of time because much of a coupon bond's value comes from payments that are made before maturity. Macaulay (1938) came up with a clever way of measuring the investment length-it is now called duration. To understand Macaulay's duration, think of a coupon bond as a package of discount bonds. Then the duration of an n-period coupon bond is defined as a weighted average of the maturities of the underlying discount bonds: (1+ R )3 +...+n- 1 L C C . C 1+0 1 + 2 Pnit [1 + Rnt (1+ Rn,t)2 F" (1 + Rn,t)" where Rt,n represents the yield to maturity of an n-period coupon bond with a coupon rate of C and face value 1 while Pt,n represents the price of that bond as of time t. 1. Write down an expression for Pnt as a function of C and (1 + Rn.t). Hint: This is very easy! You should know this, so don't think too much or make things complicated. 2. Interpret the definition of Dn.t. In particular, explain intuitively the meaning of the weights Ga k for k = 1,2,...n) used to average the maturities of the underlying discount bonds. Do they add up to one? 3. Show dPn.it Pnit = Dnt. d(1+Rn,t) (1+Rn,t) Hint: Using the expression you wrote for Q1, take a derivative of the coupon bond price Pn,t with respect to the gross yield to maturity 1+ Rnt and do some simple algebra. 4. Interpret the expression in Q3. What does the LHS represent in words? What is the significance of this equation? How is it related to what we studied in class? 5. Show that the duration of an n-period discount bond is exactly n. For discount bonds, maturity measures the length of time that a bondholder has invested money. Namely, for an n-period discount bond, you wait n periods to obtain the future cash flow. But for coupon bonds, maturity is an imperfect measure of this length of time because much of a coupon bond's value comes from payments that are made before maturity. Macaulay (1938) came up with a clever way of measuring the investment length-it is now called duration. To understand Macaulay's duration, think of a coupon bond as a package of discount bonds. Then the duration of an n-period coupon bond is defined as a weighted average of the maturities of the underlying discount bonds: (1+ R )3 +...+n- 1 L C C . C 1+0 1 + 2 Pnit [1 + Rnt (1+ Rn,t)2 F" (1 + Rn,t)" where Rt,n represents the yield to maturity of an n-period coupon bond with a coupon rate of C and face value 1 while Pt,n represents the price of that bond as of time t. 1. Write down an expression for Pnt as a function of C and (1 + Rn.t). Hint: This is very easy! You should know this, so don't think too much or make things complicated. 2. Interpret the definition of Dn.t. In particular, explain intuitively the meaning of the weights Ga k for k = 1,2,...n) used to average the maturities of the underlying discount bonds. Do they add up to one? 3. Show dPn.it Pnit = Dnt. d(1+Rn,t) (1+Rn,t) Hint: Using the expression you wrote for Q1, take a derivative of the coupon bond price Pn,t with respect to the gross yield to maturity 1+ Rnt and do some simple algebra. 4. Interpret the expression in Q3. What does the LHS represent in words? What is the significance of this equation? How is it related to what we studied in class? 5. Show that the duration of an n-period discount bond is exactly n