Exam 9 Study Notes - McClenahan
Craig A. Sloss
Introduction
These study notes are based on the Exam 9 syllabus reading Insurance Profitability by Charles L. McClenahan. The reading explains how to measure insurance company profitability in the context of rate regulation, and presents an argument that the appropiate measure is return-on-sales. This paper corresponds to learning objective D3 on the syllabus.
easypackages::packages("dplyr", "ggplot2")
options(scipen = 999)Defining Profitability
Opportunity Cost to Policyholder
When considering insurance company profitability from the perspective of the policyholder, the following should be considered:
Investment income from policy-holder supplied funds should be included, but investment income from shareholder-supplied funds should not. Rationale is that even though the surplus protects policyholders, it does not belong to them; greater surplus would provide more protection but also more income, so it is not equitable to credit this to policyholders.
The insured incurs an opportunity cost in paying for the policy up front, even though the insurer does not require funds until claims are paid. The premium could have theoretically been invested by the policyholder until it was needed.
Not all cash flows go through invested assets; for example, some supports the insurer’s infrastructure cost. Treat this as though the infrastructure is purchased at the beginning of the policy, and sold at the end.
Risk-free rates of return should be used; rationale is that if the insurer were to engage in a speculative investment resulting in the loss of policyholder-supplied funds, they would not be reimbursed by the policyholders.
The opportunity cost to the policyholder can be calucated as follows:
For each time period, calculate the total net cash flows resulting from premiums, losses, and expenses.
Discount all cash flows to time zero at the risk-free rate of return.
McClenahan provides an example using the following parameters:
premium = 100000
loss.ratio = 0.65
expense.ratio = 0.35
loss.payout = c(0.25, 0.35, 0.20, 0.12, 0.08)
risk.free.rate = 0.06The premium is assumed to be paid at policy inception, the expenses in the middle of the policy term, and the losses in the middle of each year.
PH.opportunity.cost = data.frame(time = c(0, 0.5, 1.5, 2.5, 3.5, 4.5), premium = c(premium, rep(0, 5)), loss = premium * loss.ratio * c(0, loss.payout), expense = c(0, expense.ratio*premium, rep(0, 4)))
PH.opportunity.cost## time premium loss expense
## 1 0.0 100000 0 0
## 2 0.5 0 16250 35000
## 3 1.5 0 22750 0
## 4 2.5 0 13000 0
## 5 3.5 0 7800 0
## 6 4.5 0 5200 0Calculate the total cash flow and its present value:
PH.opportunity.cost = PH.opportunity.cost %>% mutate(total_cash_flow = premium - loss - expense, PV_cash_flow = total_cash_flow / (1 + risk.free.rate)^time)
PH.opportunity.cost## time premium loss expense total_cash_flow PV_cash_flow
## 1 0.0 100000 0 0 100000 100000.000
## 2 0.5 0 16250 35000 -51250 -49778.400
## 3 1.5 0 22750 0 -22750 -20845.994
## 4 2.5 0 13000 0 -13000 -11237.732
## 5 3.5 0 7800 0 -7800 -6360.981
## 6 4.5 0 5200 0 -5200 -4000.617Sum the final column to obtain the opportunity cost:
print(paste0("The opportunity cost to the policyholder is ", sum(PH.opportunity.cost$PV_cash_flow), " which represents ", sum(PH.opportunity.cost$PV_cash_flow) / premium * 100, " percent of premium."))## [1] "The opportunity cost to the policyholder is 7776.27592737545 which represents 7.77627592737545 percent of premium."This is explicitly only relevant to the policyholder. It is distinct from the insurer’s overall profitability (which would involve discounting at a rate higher than the risk-free rate) and from the shareholder’s perspective on profitability (which would include investment income on surplus).
Return on Equity
In order to convert the dollar-value profits into a rate of return, we need to determine a denominator for calculating rates. McClenahan uses the following example to illustrate the problem with using equity as the denominator by comparing four companies with different writing-to-surplus policies and premium:
mcclenahan.example = read.csv("./Data/mcclenahan_example.csv")
mcclenahan.example## company loss_and_expense rate writings_to_surplus
## 1 A 95 100 4
## 2 B 95 100 1
## 3 C 95 110 4
## 4 D 95 110 1Calculate the profit and surplus requirement for each company:
mcclenahan.example = mcclenahan.example %>% mutate(profit = rate - loss_and_expense, surplus = rate / writings_to_surplus, return_on_equity = profit / surplus)
mcclenahan.example## company loss_and_expense rate writings_to_surplus profit surplus
## 1 A 95 100 4 5 25.0
## 2 B 95 100 1 5 100.0
## 3 C 95 110 4 15 27.5
## 4 D 95 110 1 15 110.0
## return_on_equity
## 1 0.2000000
## 2 0.0500000
## 3 0.5454545
## 4 0.1363636Problems with the Return on Equity approach include:
In the above example, suppose that the regulator used a 15% ROE benchmark for approving rates. In this example, companies B and D would get their rates approved, even though they are different. Companies A and C would not get their rates approved, even though other companies with the same rates did get approval. This clarifies the distinction between rate equity and rate of return equity.
The approach requires that equity be allocated to line of business and jurisdiction; however, the entire surplus of the insurer supports every risk. Even if surplus were artificially allocated, the method ignores the value of unallocated surplus. For example, a national company with $100M of surplus that allocates $M to a jurisdiction is offering more protection than an insurer that operates only in that jurisdiction with a surplus of $1M.
Return on Sales
Continuing the above example, we can determine return on sales by dividing profit by the rate:
mcclenahan.example %>% mutate(return_on_sales = profit / rate)## company loss_and_expense rate writings_to_surplus profit surplus
## 1 A 95 100 4 5 25.0
## 2 B 95 100 1 5 100.0
## 3 C 95 110 4 15 27.5
## 4 D 95 110 1 15 110.0
## return_on_equity return_on_sales
## 1 0.2000000 0.0500000
## 2 0.0500000 0.0500000
## 3 0.5454545 0.1363636
## 4 0.1363636 0.1363636In this example, companies A and B would be treated the same by the regulator, as would companies C and D. Advantages of this approach include:
It provides meaningful information to the consumer, e.g. as the percentage of premium that represents profit to the insurer.
The approach is independent of the relationship between premium and equity, and results in true rate regulation.
Benchmark Rates of Return
Premium-to-Surplus ratios
One possible approach to addressing the problems with the return-on-equity approach would be to set benchmark premium-to-surplus ratios, rather than using the company’s actual premium-to-surplus ratio. Use the following notation:
\(P\) is the premium
\(S\) is the surplus
\(x = P/S\) is the premium-to-surplus ratio
\(U\) is the underwriting profit
\(r_e\) is the fixed return-on-equity target used by the regulator.
\(r_s = U/P\) is the return-on-sales.
Under the assumption that \(r_e\) is fixed, then \(r_s\) is completely determined by the premium-to-surplus ratio. The reason is because \(S = U / r_e\), so \[ x = \frac{Pr_f}{U} = \frac{r_f}{r_s} \] Rearranging this, \[ r_s = \frac{r_f}{x} \] In other words, the benchmark premium-to-surplus ratio approach reduces the return-on-equity approach to a return-on-sales approach. This can be illustrated as follows:
return.on.equity = 0.125
P.to.S.benchmark.data = data.frame(premium_to_surplus = 5:60 / 10)
P.to.S.benchmark.data = P.to.S.benchmark.data %>% mutate(return_on_sales = return.on.equity / premium_to_surplus)
ggplot(data = P.to.S.benchmark.data, aes(x = premium_to_surplus, y = return_on_sales)) + geom_line(colour = "blue") + geom_hline(yintercept = return.on.equity, colour = "red")
Now, suppose we select a benchmark premium-to-surplus ratio of 3:1, but calculate return on equity using the insurer’s actual premium-to-surplus ratio. In this situation, the return on sales is fixed by the above equation:
premium.to.surplus = 3
return.on.sales = return.on.equity / premium.to.surplus
print(paste0("The fixed return on sales is ", return.on.sales))## [1] "The fixed return on sales is 0.0416666666666667"Now, calculate the return on equity using the true premium-to-surplus ratio:
P.to.S.benchmark.data = P.to.S.benchmark.data %>% mutate(return_on_equity = premium_to_surplus * return.on.sales)
ggplot(data = P.to.S.benchmark.data, aes(x = premium_to_surplus, y = return_on_equity)) + geom_line(colour = "red") + geom_hline(yintercept = return.on.sales, colour = "blue")
In this case, the return on equity is only equal to 12.5% when the true premium-to-surplus ratio equals the benchmark.
Considerations when Setting Return-on-Sales Benchmarks
When setting a profitability standard based on return on sales, there is no consensus on what constitutes an “appropriate” return. Considerations include:
Providing a buffer for uncertainty in the estimation of the various components of the rate
Inadequate rates may result in insurers tightening underwriting or reducing volume; market conditions such as the size of the residual market, the number of insurers in the voluntary market, and the degree of product diversity provide the industry’s view as to whether the returns are reasonable.
The benchmark for when a rate is considered “excessive” should be determined based on the market conditions desired by the regulator.