To calculate the current price of the Cinque Terra bond, we can use the present value formula for bonds. The price of the bond is the present value of its future cash flows, which consist of the semi-annual coupon payments and the face value at maturity.
Given:
- Coupon Rate = 7.375% APR
- Face Value (FV) = $1,000
- Annual Payment (C) = \( \text{Coupon Rate} \times \text{Face Value} = 0.07375 \times 1000 = 73.75 \) (This is for a year)
- Semi-Annual Coupon Payment = \( \frac{C}{2} = \frac{73.75}{2} = 36.875 \)
- Total Maturity Period = 10 years
- Time elapsed = 3 years
- Remaining Maturity = 10 - 3 = 7 years = 14 semi-annual periods
- Required Return (r) = 6.44% APR (or 0.0644), which needs to be converted for semi-annual periods:
\( \text{Semi-Annual Required Return} = \frac{0.0644}{2} = 0.0322 \)
Now, we can find the present value of the future cash flows:
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Present Value of Coupon Payments (annuity): \[ PV_{\text{coupons}} = C \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] where \( n \) is the number of periods (14) and \( r \) is the semi-annual required return (0.0322).
Plugging in the values: \[ PV_{\text{coupons}} = 36.875 \left( \frac{1 - (1 + 0.0322)^{-14}}{0.0322} \right) \]
Let's calculate that: \[ (1 + 0.0322)^{-14} \approx 0.636578 \quad \text{(Using a calculator)} \] So, \[ 1 - 0.636578 \approx 0.363422 \] Thus, \[ PV_{\text{coupons}} = 36.875 \left( \frac{0.363422}{0.0322} \right) \approx 36.875 \times 11.2904 \approx 416.77 \]
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Present Value of Face Value (lump sum): \[ PV_{\text{face}} = \frac{FV}{(1 + r)^n} = \frac{1000}{(1 + 0.0322)^{14}} \]
Again let's calculate: \[ (1 + 0.0322)^{14} \approx 1.537679 \quad \text{(Using a calculator)} \] So, \[ PV_{\text{face}} = \frac{1000}{1.537679} \approx 651.05 \]
Finally, we add the present values of the coupon payments and the face value to find the current price of the bond:
\[ \text{Current Price} = PV_{\text{coupons}} + PV_{\text{face}} \approx 416.77 + 651.05 \approx 1067.82 \]
Therefore, the price of the Cinque Terra bond today is approximately $1067.82.