(*) The Venn diagram below shows symbolically a portfolio at two successive underwriting years. The shaded area (intersection) ℘′ ∩℘ is made of the contracts that have been renewed; the area ℘−℘′ is made of the contracts that have lapsed;

℘’−℘ is made of new business.

i. Prove that the change in premium adequacy between ℘ and ℘’ can be approximately written as:
´ ´ PAI R (℘ ℘→ ′ ′ ) ≈ + % % C PAI ( ) ℘℘ ℘℘ ∩ → in in × − ´PAI o ( ) ∩ → ut × %out ’ ’
where:
• δ ℘ PAI P (℘→℘′ ′ ) = ℘ AI’( ) − PAI( ) is the difference in premium adequacy between the two portfolios;
• %RC is the percentage rate change for the matching portfolio;
• δ ℘ PAI i ’ : ( ) ℘∩℘ →’ n P = AI’ ’ (in) − ∩ PAI ( ) ℘’ is the difference in premium adequacy between the new business and the common portfolio;
• ´PAI o ( ) ℘℘ ℘℘ ∩ → ut PA = I o( ut PA ) − ∩ I( ) ’ ’ is the difference in premium adequacy between the lapsed portfolio and the common portfolio;

• % / in = A A P ' ′(in) P’(℘) is the percentage (in premium) of the current portfolio that is due to new business;
• % / out A= P A (out) P(℘) is the percentage (in premium) of the expiring portfolio that has lapsed.
ii. Give an interpretation of the formula in (i) and explain why it is not an exact relationship.