Explanation

To solve this bond pricing problem, we need to find the price of the 4½% coupon bond using linear interpolation between the other two bonds.

Given:

• 2⅞% coupon bond: Price = 94.40
• 6¼% coupon bond: Price = 101.30
• 4½% coupon bond: Price = ?

Step 1: Convert coupons to decimals

• 2⅞% = 2.875%
• 4½% = 4.5%
• 6¼% = 6.25%

Step 2: Use linear interpolation We can interpolate linearly between the coupon rates:

Price of 4.5% bond = Price of 2.875% bond + [(4.5 - 2.875) / (6.25 - 2.875)] × (Price of 6.25% bond - Price of 2.875% bond)

Let's calculate: Numerator: 4.5 - 2.875 = 1.625 Denominator: 6.25 - 2.875 = 3.375 Ratio: 1.625 / 3.375 = 0.48148

Price difference: 101.30 - 94.40 = 6.90

Price of 4.5% bond = 94.40 + 0.48148 × 6.90 = 94.40 + 3.322 = 97.722 ≈ 97.72

Step 3: Verify with cash flow approach Since all bonds mature in exactly one year with semi-annual coupons, they have the same cash flow structure:

• One coupon payment in 6 months
• Final coupon + principal payment in 12 months

Let d₁ = discount factor for 6 months Let d₂ = discount factor for 12 months

For 2.875% bond (face value 100): Coupon = 2.875/2 = 1.4375 1.4375 × d₁ + 101.4375 × d₂ = 94.40

For 6.25% bond: Coupon = 6.25/2 = 3.125 3.125 × d₁ + 103.125 × d₂ = 101.30

Solving these equations simultaneously: From first equation: 1.4375d₁ = 94.40 - 101.4375d₂ d₁ = (94.40 - 101.4375d₂)/1.4375

Substitute into second equation: 3.125 × [(94.40 - 101.4375d₂)/1.4375] + 103.125d₂ = 101.30

3.125/1.4375 = 2.1739 2.1739 × (94.40 - 101.4375d₂) + 103.125d₂ = 101.30 205.22 - 220.43d₂ + 103.125d₂ = 101.30 205.22 - 117.305d₂ = 101.30 117.305d₂ = 103.92 d₂ = 0.8859

d₁ = (94.40 - 101.4375 × 0.8859)/1.4375 d₁ = (94.40 - 89.86)/1.4375 = 4.54/1.4375 = 3.159

Now for 4.5% bond: Coupon = 4.5/2 = 2.25 Price = 2.25 × d₁ + 102.25 × d₂ = 2.25 × 3.159 + 102.25 × 0.8859 = 7.108 + 90.58 = 97.688 ≈ 97.69

Both methods give approximately 97.71, which matches option C.

Therefore, the correct answer is C. 97.71.