The following problem and solution is taken from an actuarial exam (financial math) study manual:
I would set this problem up completely different. I think it would be necessary to consider the number of insurance policies of this particular type.
Let P be the premium and let N be the number of policies of this particular type. Then using the effective date as the comparison date, we have:
$$\frac{0.10\cdot{}N\cdot1000}{(1.04)^2} + \frac{.03\cdot{}N\cdot10,000}{(1.04)^6} + 10 + \frac{10}{1.04} = \frac{N\cdot{}P}{1.04}.$$
Using this set-up, we would have two unknown variables, the number of policies and and premium.
If we take out the $N$ in my set-up, I would have an equivalent statement to the author. But I don't understand why the author is not taking into account the number of policies.
In your PV calculation, the expense terms should also have been multiplied by $N$. Then the $N$ cancels, and a little manipulation gives the equation of the official solution.