Explanation

This is a fixed income valuation problem for a preferred stock with a maturity date (making it similar to a bond). The key information:

• Par value: €55
• Intrinsic value (current price): €75
• Maturity: 5 years
• Required rate of return: 7.2%
• Dividend payment frequency: Semiannual

Since this is a preferred stock with a maturity, we can value it using the bond pricing formula:

$P = \frac{C}{r} \times \left[1 - \frac{1}{(1+r)^n}\right] + \frac{FV}{(1+r)^n}$

Where:

• P = Current price = €75
• C = Semiannual dividend (what we need to find)
• r = Semiannual required return = 7.2%/2 = 3.6% = 0.036
• n = Number of semiannual periods = 5 years × 2 = 10 periods
• FV = Par value = €55

Rearranging the formula to solve for C:

$C = \frac{P - \frac{FV}{(1+r)^n}}{\frac{1}{r} \times \left[1 - \frac{1}{(1+r)^n}\right]}$

First, calculate the present value of the par value: $\frac{€55}{(1.036)^{10}} = \frac{€55}{1.4243} = €38.62$

Then calculate the annuity factor: $\frac{1}{0.036} \times \left[1 - \frac{1}{(1.036)^{10}}\right] = 27.7778 \times \left[1 - 0.7021\right] = 27.7778 \times 0.2979 = 8.275$

Now solve for C: $C = \frac{€75 - €38.62}{8.275} = \frac{€36.38}{8.275} = €4.396$

This is the semiannual dividend, which is approximately €4.40. However, looking at the options:

• A: €3.96
• B: €4.40
• C: €5.40

Our calculation gives €4.396 ≈ €4.40, which corresponds to option B.

Wait, let me double-check the calculation:

Actually, I made an error in the interpretation. The question asks for the semiannual dividend, and my calculation gave €4.40. But let me verify with the exact calculation:

1. PV of par value: €55 / (1.036)^10 = €55 / 1.4243 = €38.62
2. Present value of dividends: €75 - €38.62 = €36.38
3. Annuity factor: (1 - 1/1.036^10) / 0.036 = (1 - 0.7021) / 0.036 = 0.2979 / 0.036 = 8.275
4. Semiannual dividend: €36.38 / 8.275 = €4.396 ≈ €4.40

This matches option B. However, I need to check if there's a different approach.

Alternative approach using financial calculator logic:

• N = 10 (5 years × 2)
• I/Y = 3.6 (7.2%/2)
• PV = -75 (price paid today, negative as outflow)
• FV = 55 (par value received at maturity)
• Compute PMT = ?

Using the TVM formula: PMT = [PV - FV/(1+r)^n] / [1/r × (1 - 1/(1+r)^n)]

This gives the same result: €4.396 ≈ €4.40

Therefore, the correct answer is B. €4.40.

Why not C (€5.40)? This would be too high and would imply a much higher current price than €75 given the 7.2% required return.

Why not A (€3.96)? This would be too low and would imply a lower current price than €75 given the 7.2% required return.