Can you help me solve these question A B C ?

A. In a recent trading session, the benchmark 30-year Treasury bonds market price went up A1 dollars per $1000 face value to A2 dollars, while its yield fell from .07 to .068. The bonds market price then went up another A3 dollars per $1000 face value as the yield fell further to .0665 from .068. Report answers for the following (using average of latest two bond prices for duration and average of latest three bond prices for convexity):

1. Convexity, using the average of all three market prices of the bond in the denominator of the formula.

2. First modified duration, using average of first two market prices of the bond in the denominator of the formula.

3. Second modified duration, using average of second and third prices of the bond.

4. The expected rise in bond price if the interest rate were to fall another 20 basis points (.002) from .0665, using the average of two modified durations and convexity.

B. Estimate term structure of discount factors, spot rates and forward rates by using data on five semi-annual coupon paying bonds with $100 face value each: The bonds, respectively, have 1.25, 5.35, 10.4, 15.15 and 20.2 years to maturity; pay coupon at annual rates of B1, B2, B3, B4, and B5 percent of face value; and are trading at quoted spot market prices in dollars of B6, B7, B8, B9 and B10. Specify the discount factor function d(t) by a third degree polynomial with unknown parameters a, b, and c, as done in class. Using estimated d(t) function, determine spot rate and forward rate functions by assuming half-year compounding. Then write the values of the following based on your estimation.

5. Coefficient of parameter a in first bond price equation.

6. Coefficient of parameter b in first bond price equation.

7. Coefficient of parameter c in first bond price equation.

8. Coefficient of parameter a in second bond price equation.

9. Coefficient of parameter b in second bond price equation.

10.Coefficient of parameter c in second bond price equation.

11.Coefficient of parameter a in third bond price equation.

12.Coefficient of parameter b in third bond price equation.

13.Coefficient of parameter c in third bond price equation.

14.Coefficient of parameter a in fourth bond price equation.

15.Coefficient of parameter b in fourth bond price equation.

16.Coefficient of parameter c in fourth bond price equation.

17.Coefficient of parameter a in fifth bond price equation. 1

18.Coefficient of parameter b in fifth bond price equation.

19.Coefficient of parameter c in fifth bond price equation.

20.Parameter a.

21.Parameter b.

22.Parameter c.

23.Current price of a dollar at 5th year.

24.Current price of a dollar at 7th year.

25.Current price of a dollar at 10th year.

26.Current price of a dollar at 15th year.

27.Spot rate for term 2 year.

28.Spot rate for term 5 year.

29.Spot rate for term 10 year.

30.Spot rate for term 17 year.

31.Forward rate for half year period 2.5 to 3.0 years.

32.Forward rate for half year period 5.5 to 6.0 years.

33.Forward rate for half year period 10.5 to 11.0 years.

34.Forward rate for half year period 15.5 to 16.0 years.

C. Estimate the 2-year, 5-year, and 10-year key rate durations of a 20-year bond carrying a coupon of C1 percent on face value $100 paid semi-annually. The given term structure starts with C2 percent spot rate of interest at time zero and rises at a rate of 0.002 (.2%) per half year thereafter. Take a 20 basis point (.002) move in each key interest rate to calculate the key rate durations by the method done in class and given in textbook. Report answers for the following:

35. Current fair price of the bond with the given term structure.

36. Price change needed to calculate 2-year key rate duration.

37. Price change needed to calculate 5-year key rate duration.

38. Price change needed to calculate 10-year key rate duration.

39. 2-year key rate duration.

40. 5-year key rate duration.

41. 10-year key rate duration.

Estimating Term Structure The current price of a future dollar is the discount factor needed to determine the current value of future cash flows. Term structure refers to the discount factor as a function of the time or term of future cash flows. A zero coupon bond which pays the face value at its maturity can be valued if the true discount factor as a function of time (maturity term) is known. The true discount factor function can be estimated or extracted from historical bond price data. Such a term structure of discount factors will, however, is specific to the risk rating of bonds used for estimation, like AAA-rated or Treasury bonds. Once the discount factor function is estimated, the corresponding spot interest rate function as well as the forward rate function can be determined. These are useful functions deployed in the real world to value fixed income instruments. If the interest rate is continuously compounded, d (t ) = e r(t )t r(t ) = ln(d (t )) , t where d(t) is the current price of a future dollar and r(t ) is the spot rate of interest. If the interest rate is semi-annually compounded, d (t ) = 1 r(t ) 1 + 2 2t 1 1 2t r(t ) = 2 1 d (t ) Steps to estimate term structure Step-1: Select bonds with different times to maturities and record their market prices, coupon rates, payment frequencies, times to maturity and face values. This should be a random sample of all bonds in a given risk rating category. Step-2: Specify d(t) as a polynomial function of degree three as follows: d (t ) = 1 + at + bt 2 + ct 3 where a, b, and c are the parameters to be estimated from bond price data. Higher degree polynomials can be specified for greater accuracy. The exact degree of the polynomial can also be chosen by simulation. Step-3: Add the accrued interest rate to the quoted market price for each bond to obtain the cash market price of the bond. Then assume that the cash market price (CMP) of each bond is equal to its fair value given by the right side of the following equation: CMP = CFi d (ti ) = CFi (1 + ati + bti2 + cti3 ) n n i =1 i =1 where the specified functional form of d(t) has been used, and CFi denotes the cash flow of bond i at time ti and n is the number of bonds in the sample. Since CMP and cash flows of the bond are known values, the above equation can be simplified to take the following form: pi = ax1i + bx2i + cx3i , i = 1,2,..., n, where a, b, and c are the parameters to be estimated. We can use the method of least squares by assuming that the above equation may have an error ei : pi = ax1i + bx2i + cx3i + ei , i = 1,2,..., n, to estimate a, b and c, using data on a number of bonds i=1,2,3,...,n. In the above bond equation, xi1, xi2 and xi3 are the coefficients (with definite values based on cash flows of a bond) of the unknown parameters a, b and c being estimated. The least square errors (e1, e2, ...,en) are automatically generated by the regression. Example of deriving a bond equation: Consider for example a 4 percent bond maturing in 1.5 years, paying coupon at 6 months, 1 year and 1.5 years from now. Suppose that the bond is selling now for $103. Then 103 = 2 d (.5) + 2 d (1) + 102 d (1.5) 2 3 = 2 1 + a (.5) + b (.5) + c (.5) + 2 1 + a 1 + b 12 + c 13 + 2 3 102 1 + a (1.5) + b (1.5) + c (1.5) = (2 + 2 + 102) + a 1 + 2 + 153 + b .5 + 2 + 229.5 + c .25 + 2 + 344.25 103 = 106 + 156 a + 232 b + 346.5 c 3 = 156 a 232 b 346.5 c In the above equation for bond i, xi1 = -156, xi2 = -232 and xi3 = -346.5 and a least squares error ei is used for estimation after such equations for all bonds in the sample are constructed. After constructing such bond equations for all bonds in the data set, we use least squares to estimate a, b, and c. Writing the bond equations in a matrix form: p1 x11 p2 = x21 . . pn xn1 x12 x13 e1 a x22 x23 e2 b + . . . c xn 2 xn3 en and then using P to denote the column vector of the values of the dependent variable, X to denote the matrix of coefficients of the parameters to be estimates and e to represent the column vector of least squares errors, we have P = X +e, where is the vector of coefficients that can be estimated by ordinary least squares regression using any standard statistical package. The values of the estimated coefficients are exactly given by the following matrix expression: a 1 b = = ( X ' X ) X ' P . c Step-4: After a, b, and c are estimated as a vector , d(t) function is known. We can then use this d(t) to determine the spot rate function and the forward rate function