Under cash accounting, a person or firm counts income when it is received and claims deductions when money is paid. Annual cash flows may consist of income items: premium income, interest and dividends on assets, maturity proceeds of assets, and reinsurance recoveries in respect of claims. The outflows may consist of: claims settle or amounts paid on account, reinsurance premiums, expenses, tax, and dividends. In cash accounting, the profit, or net income, is simply the excess of money received over money paid. Following cash makes life easy – one simply has to look at the checkbook to see when money is coming in and when it goes out. Projecting annual cash flows as they emerge, or emerging cash flows, is an important task.
However, firms cannot simply follow cash to determine profitability. Consider the case of a savings bank that receives deposits from their customers without making any payments. This money received is not a profit to the bank – rather the bank receives the deposit as money coming in but also has an obligation to return the money when demanded by the customer. That is, the bank’s obligations have increased with the customer deposit.
Under accrual accounting, revenues and costs are recognized when they are incurred, not necessarily when related payments are received or made. In many businesses, the timing of incurral is easy to identify. Typically, a company records revenue as incurred when a product or service is shipped or billed to a customer, not when payments are actually made (they could be much later if you are buying on a payment plan). To assess profitability, a company’s costs need to be matched to the revenue.
Matching costs to revenue is difficult in life insurance because of the potential time lag between premiums (one type of revenue) and benefits. To understand the cost of a life insurance product, let us begin by considering a net basis, ignoring expenses from doing business as well as any profit motivation.
Suppose that we have 1,000 identical policyholders age (x) who purchase a policy with a generic premium payment schedule {(P_h)} and benefit schedule {(b_h)}. At some future time point, say (h) years later, we expect there to be (1000 ~_h p_x) policyholders alive with each policy having value (_h V). Thus, the company’s total obligation is (1000 ~_h p_x ~_h V). One year later, the company’s obligation will be (1000 v ~_{h+1} p_x ~_{h+1} V). The increase in this obligation is a cost to the company. Using the recursive reserve formula, this cost may be expressed as
begin{eqnarray*}1000 ~v _{h+1} p_x ~_{h+1} V &-& 1000 ~_h p_x ~_h V \
&=& 1000 ~_h p_x left( v p_{x+h}~_{h+1} V – ~_h V right) \
&=& 1000 ~_h p_x left( v p_{x+h}~_{h+1} V – left{ v q_{x+h} b_{h+1} – P_h +v p_{x+h}~_{h+1} V right}right) \
&=& 1000 ~_h p_x left( P_h – v q_{x+h} b_{h+1} right) ,end{eqnarray*}
with the relation (~_{h+1} p_x = ~_h p_x ~p_{x+h}). Thus, the cost is simply the expected cash inflow, premiums in excess of benefits. This is anticipated as we are working on a net basis which implies zero profits.