Let's solve the problem step by step: --- ## **a) Calculate zero rates for 6, 12, 18, and 24 months** **Zero-coupon bonds:** - 6 months: Price = $98, Principal = $100, Coupon = $ - 12 months: Price = $95, Principal = $100, Coupon = $ ### **Step 1: 6-Month Zero Rate** For a zero-coupon bond, the price formula is: \[ P = \frac{F}{(1 + r)^{n}} \] Where: - \( P \) = Price now - \( F \) = Face value ($100) - \( r \) = periodic rate (for 6 months) - \( n \) = number of periods (here, just 1 for 6 months) \[ 98 = \frac{100}{1 + r_{.5}} \] \[ 1 + r_{.5} = \frac{100}{98} = 1.02041 \] \[ r_{.5} = 2.041\% \] Annualized (compounded semiannually): \[ \text{Zero rate for 6 months (annualized)} = 2 \times 2.041\% = 4.08\% \] --- ### **Step 2: 12-Month Zero Rate** \[ 95 = \frac{100}{(1 + r_{1})^2} \] \[ (1 + r_{1})^2 = \frac{100}{95} = 1.05263 \] \[ 1 + r_{1} = \sqrt{1.05263} = 1.02596 \] \[ r_{1} = 2.596\% \ \text{per 6 months} \] Annualized: \[ \text{Zero rate for 12 months (annualized)} = 2 \times 2.596\% = 5.19\% \] --- ### **Step 3: 18-Month Zero Rate** The 1.5-year bond has a 6.2% coupon (paid semiannually: 3.1 every 6 months). \[ \text{Cash flows: } 3.1 \ (\text{at 6, 12 months}), \ 103.1 \ (\text{at 18 months}) \] The price is: \[ 101 = \frac{3.1}{1.02041} + \frac{3.1}{(1.02596)^2} + \frac{103.1}{(1 + r_{1.5})^3} \] Calculate present values of the first two coupons using earlier zero rates: \[ \frac{3.1}{1.02041} = 3.038 \] \[ \frac{3.1}{(1.02596)^2} = \frac{3.1}{1.05263} = 2.946 \] So: \[ 101 = 3.038 + 2.946 + \frac{103.1}{(1 + r_{1.5})^3} \] \[ 101 - 3.038 - 2.946 = \frac{103.1}{(1 + r_{1.5})^3} \] \[ 95.016 = \frac{103.1}{(1 + r_{1.5})^3} \] \[ (1 + r_{1.5})^3 = \frac{103.1}{95.016} = 1.0852 \] \[ 1 + r_{1.5} = (1.0852)^{1/3} = 1.02764 \] \[ r_{1.5} = 2.764\% \text{ per 6 months} \] Annualized: \[ \text{Zero rate for 18 months (annualized)} = 2 \times 2.764\% = 5.53\% \] --- ### **Step 4: 24-Month Zero Rate** 2-year bond: 8.% coupon (4. every 6 months), price = $104 \[ \text{Cash flows: } 4., 4., 4., 104. \text{ at 6, 12, 18, 24 months} \] \[ 104 = \frac{4}{1.02041} + \frac{4}{(1.02596)^2} + \frac{4}{(1.02764)^3} + \frac{104}{(1 + r_2)^4} \] \[ \frac{4}{1.02041} = 3.922 \] \[ \frac{4}{1.05263} = 3.801 \] \[ \frac{4}{1.0852} = 3.687 \] \[ 104 = 3.922 + 3.801 + 3.687 + \frac{104}{(1 + r_2)^4} \] \[ 104 - 3.922 - 3.801 - 3.687 = \frac{104}{(1 + r_2)^4} \] \[ 92.59 = \frac{104}{(1 + r_2)^4} \] \[ (1 + r_2)^4 = \frac{104}{92.59} = 1.1232 \] \[ 1 + r_2 = (1.1232)^{1/4} = 1.02945 \] \[ r_2 = 2.945\% \text{ per 6 months} \] Annualized: \[ \text{Zero rate for 24 months (annualized)} = 2 \times 2.945\% = 5.89\% \] --- ### **Summary Table** | Maturity | Zero Rate (annualized, comp. semiannual) | |------------|------------------------------------------| | 6 months | 4.08% | | 12 months | 5.19% | | 18 months | 5.53% | | 24 months | 5.89% | --- ## **b) Forward Rates** The formula for the forward rate \( f \) from time \( t_1 \) to \( t_2 \) is: \[ (1 + z_2)^{t_2} = (1 + z_1)^{t_1} \cdot (1 + f)^{t_2 - t_1} \] Where \( z_1 \) and \( z_2 \) are the zero rates for \( t_1 \) and \( t_2 \) (in periods of 6 months). - \( z_1 \) for 6 months = 2.041% (per 6 months) - \( z_2 \) for 12 months = 2.596% (per 6 months) **Forward Rate from 6 to 12 months:** \[ (1 + .02596)^2 = (1 + .02041)^1 \cdot (1 + f_{.5,1})^1 \] \[ 1.05263 = 1.02041 \cdot (1 + f_{.5,1}) \] \[ (1 + f_{.5,1}) = \frac{1.05263}{1.02041} = 1.0316 \] \[ f_{.5,1} = 3.16\% \text{ per 6 months} \] Annualized: 6.32% --- **Forward Rate from 12 to 18 months:** - \( z_1 \) for 12 months = 2.596% - \( z_2 \) for 18 months = 2.764% \[ (1 + .02764)^3 = (1 + .02596)^2 \cdot (1 + f_{1,1.5})^1 \] \[ 1.0852 = 1.05263 \cdot (1 + f_{1,1.5}) \] \[ (1 + f_{1,1.5}) = \frac{1.0852}{1.05263} = 1.031 \] \[ f_{1,1.5} = 3.10\% \text{ per 6 months} \] Annualized: 6.20% --- **Forward Rate from 18 to 24 months:** - \( z_1 \) for 18 months = 2.764% - \( z_2 \) for 24 months = 2.945% \[ (1 + .02945)^4 = (1 + .02764)^3 \cdot (1 + f_{1.5,2})^1 \] \[ 1.1232 = 1.0852 \cdot (1 + f_{1.5,2}) \] \[ (1 + f_{1.5,2}) = \frac{1.1232}{1.0852} = 1.0351 \] \[ f_{1.5,2} = 3.51\% \text{ per 6 months} \] Annualized: 7.02% --- ### **Summary Table** | Period | Forward Rate (per 6 mo) | Annualized | |--------------------|------------------------|-------------| | 6 to 12 months | 3.16% | 6.32% | | 12 to 18 months | 3.10% | 6.20% | | 18 to 24 months | 3.51% | 7.02% | --- ## **c) Par Yields (6, 12, 18, 24 months)** Par yield is the coupon rate that makes the bond price equal to face value ($100). ### **Formula:** \[ P = \sum_{i=1}^{n} \frac{c}{(1 + z_i)^i} + \frac{100}{(1 + z_n)^n} \] Set \( P = 100 \) and solve for \( c \). **Let’s do for each:** --- ### **6-month Par Yield** \[ 100 = \frac{c}{1.02041} + \frac{100}{1.02041} \] \[ 100 = \frac{c + 100}{1.02041} \] \[ c + 100 = 100 \times 1.02041 = 102.041 \] \[ c = 2.041 \] Annual coupon = \( 2 \times 2.041 = 4.08\% \) --- ### **12-month Par Yield** \[ 100 = \frac{c}{1.02041} + \frac{c}{(1.02596)^2} + \frac{100}{(1.02596)^2} \] \[ (1.02596)^2 = 1.05263 \] \[ 100 = \frac{c}{1.02041} + \frac{c}{1.05263} + \frac{100}{1.05263} \] \[ 100 = .980 + .950c + 95. \] Calculate each term: \[ \frac{c}{1.02041} = .980c \] \[ \frac{c}{1.05263} = .950c \] \[ \frac{100}{1.05263} = 95. \] So, \[ 100 = .980c + .950c + 95. \] \[ 100 - 95. = 1.93c \] \[ c = \frac{5.}{1.93} = 2.59 \] Annual coupon = \( 2 \times 2.59 = 5.18\% \) --- ### **18-month Par Yield** \[ 100 = \frac{c}{1.02041} + \frac{c}{1.05263} + \frac{c}{1.0852} + \frac{100}{1.0852} \] \[ \frac{c}{1.02041} = .980c \] \[ \frac{c}{1.05263} = .950c \] \[ \frac{c}{1.0852} = .921c \] \[ \frac{100}{1.0852} = 92.18 \] \[ 100 = .980c + .950c + .921c + 92.18 \] \[ 100 - 92.18 = 2.851c \] \[ 7.82 = 2.851c \] \[ c = \frac{7.82}{2.851} = 2.74 \] Annual coupon = \( 2 \times 2.74 = 5.48\% \) --- ### **24-month Par Yield** \[ 100 = \frac{c}{1.02041} + \frac{c}{1.05263} + \frac{c}{1.0852} + \frac{c}{1.1232} + \frac{100}{1.1232} \] \[ \frac{c}{1.02041} = .980