Rather than thinking of an asset swap as a traded instrument, it can be useful to think of the asset swap spread as a relative value indicator. The asset swap PV has two parts, the bond part (valued at $100 - P$) and the swap, which pays the bond's coupon on the fixed leg, and ${\rm LIBOR} + s$ on the floating leg.
The spread $s$ accounts for two properties of the bond that make it distinct from a risk-free bond trading at par -
- The bond's coupon may be higher than a risk-free bond (reflecting credit risk when the bond was issued)
- The bond may not trade at par (reflecting changes in credit risk since the bond was issued)
These components together reflect the current credit-worthiness of the bond. In order for the asset swap spread to correctly reflect the second component, it is necessary that in the initial exchange of bond for capital, the capital is $100 i.e. the face value of the bond, not its market value.
If the capital exchanged was the market value of the bond, the asset swap spread would only reflect the coupon of the bond, which isn't a good indicator of its current credit risk, and would render the asset swap spread useless as a tool for comparing relative value of two related bonds.