Exam 9 Study Notes - Ferrari
Craig A. Sloss
Introduction
These study notes are based on the Exam 9 syllabus reading The Relationship of Underwriting, Investment, Leverage, and Exposure to Total Return on Owners’ Equity by J. Robert Ferrari and a discussion of the paper by R. J. Balcarek. The reading explains how loss reserves induce leverage in an insurance company, and how the consideration of insurance leverage can be used to determine the optimal capital structure for the firm. This paper corresponds to learning objective D2 on the syllabus.
easypackages::packages("dplyr", "ggplot2", "DT", "data.table")The ideas in the paper will be illustrated using the following example from the reading, which looks at 4 hypothetical companies with different levels of reserves:
ferrari.examples = read.csv("./Data/ferrari_example.csv")
datatable(ferrari.examples)investment.return = 0.05Key Formulas and Terminology
Basic Equation
The following notation is used in the paper:
\(I\) is the after-tax investment income
\(U\) is the underwriting profit (or loss)
\(T\) is the total after-tax return to the insurer; note that \(T = I+U\)
\(P\) is the premium income
\(A\) is the total assets
\(R\) is the reserves and other liabilities, excluding equity in the unearned premium reserves. (Equity in the net UPR is the amount by which the net UPR and unearned commissions exceeds the net policy liabilities in connection with unearned premium.)
\(S\) is the stockholders’ equity; note that \(S = A - R\). This includes equity in the unearned premium reserves
One of the main ideas in the paper is the relationship between views of the insurer’s performance from three perspectives:
From the investor’s perspective, the key metric is return on equity, or \(T/S\)
From society’s perspective, the key metric is investment return on total assets, or \(I/A\)
From the regulator’s perspective, the key metric is return on sales, or \(U/P\).
To obtain a relationship between these three, first split the total income into underwriting and investment components: \[ \frac{T}{S} = \frac{I}{S} + \frac{U}{S} \] The strategy is to relate the first term to the return on total assets, and the second to return on sales. The first term can be expressed as \[ \frac{I}{S} = \frac{I}{A}\frac{A}{S} = \frac{I}{A}\frac{R+S}{S} = \frac{I}{A}\left(1+\frac{R}{S}\right) \] The ratio of reserves to surplus, \(R/S\) is the insurance leverage ratio. The second term can be expressed as \[ \frac{U}{S} = \frac{U}{P}\frac{P}{S} \] The ratio of premium to surplus, \(P/S\) is the insurance exposure. Combining these, we obtain: \[ \frac{T}{S} = \frac{I}{A}\left(1+\frac{R}{S}\right) + \frac{U}{P}\frac{P}{S} \] In other words, the three perspectives on the insurer’s performance are related via these leverage ratios. The quantity \[ 1 + \frac{R}{S} \] is referred to as the insurance leverage factor.
Assuming a fixed underwriting profit margin of 6%, we can use this formula to relate the investment return and the underwriting profit as follows. Note that a premium-to-surplus ratio was not given explicitly in the reading, premium can be calculated based on the footnote indicating that the given UW profit is given as a percentage of reserves, not premium.
uw.profit.margin = 0.06
ferrari.examples.formula.1 = ferrari.examples %>% mutate(surplus = assets - reserve, premium = reserve * (1 + uw.profit.margin), P_to_S = premium / surplus, R_to_S = reserve / surplus, ROE = ifelse(premium > 0, investment.return * (1 + R_to_S) + uw.profit.margin * reserve / premium * P_to_S, investment.return * (1 + R_to_S)))
ferrari.examples.formula.1## company assets reserve surplus premium P_to_S R_to_S ROE
## 1 A 20000000 0 20000000 0 0.00 0.0 0.050
## 2 B 20000000 6666667 13333333 7066667 0.53 0.5 0.105
## 3 C 20000000 10000000 10000000 10600000 1.06 1.0 0.160
## 4 D 20000000 13333333 6666667 14133333 2.12 2.0 0.270Note a subtlety in the use of the formula: for the unlevered Company A, there is no premium so technically the formula as stated is not valid. However, since the \(P\) cancels out in the formula, we can effectively ignore the second term for a company that writes no business since \(U=0\).
Suppose that each company had equity in the unearned premium reserve equal to 5% of premium. In this case, the amount would be deducted from the reserve and added to the surplus:
ferrari.examples.formula.1b = ferrari.examples %>% mutate(premium = reserve * (1 + uw.profit.margin), eupr = 0.05 * premium, reserve = reserve - eupr, surplus = assets - reserve, P_to_S = premium / surplus, R_to_S = reserve / surplus, ROE = ifelse(premium > 0, investment.return * (1 + R_to_S) + uw.profit.margin * reserve / premium * P_to_S, investment.return * (1 + R_to_S)))
ferrari.examples.formula.1b## company assets reserve premium eupr surplus P_to_S R_to_S
## 1 A 20000000 0 0 0.0 20000000 0.0000000 0.0000000
## 2 B 20000000 6313334 7066667 353333.4 13686666 0.5163176 0.4612762
## 3 C 20000000 9470000 10600000 530000.0 10530000 1.0066477 0.8993352
## 4 D 20000000 12626666 14133333 706666.6 7373334 1.9168172 1.7124773
## ROE
## 1 0.0500000
## 2 0.1007404
## 3 0.1489269
## 4 0.2383725Reserves as Non-Equity Capital
An alternative perspective can be obtained by re-arranging the formula for \(T/S\) as follows: \[ \frac{T}{S} = \frac{I}{A} + \frac{R}{S}\left(\frac{I}{A} + \frac{S}{R}\frac{U}{S}\right) = \frac{I}{A} + \frac{R}{S}\left(\frac{I}{A} + \frac{U}{R}\right) \] In this form, \(R\) is viewed as capital that has been provided by individuals other than the owners, which earns both investment income and underwriting profit (expressed, in this case, as a percentage of reserves as opposed to premium). This view is analogous to the classical view of financial leverage associated with debt capital, and underwriting losses can be viewed as “interest” on the reserve capital. This perspective is key to assessing the insurer’s profitability: provided that the sum of \(I/A\) and \(U/R\) is positive, then the insured is incentivized to write policies, even if the underwriting profit is negative.
As before, we can use this formula to calculate the total return on equity for a fixed profit assumption, noting that as leverage increases, the UW profit has a greater impact and ROE increases as a result:
uw.profit.margin = 0.06
ferrari.examples.formula.2 = ferrari.examples %>% mutate(surplus = assets - reserve, R_to_S = reserve / surplus, ROE = investment.return + R_to_S * (investment.return + uw.profit.margin))
ferrari.examples.formula.2## company assets reserve surplus R_to_S ROE
## 1 A 20000000 0 20000000 0.0 0.050
## 2 B 20000000 6666667 13333333 0.5 0.105
## 3 C 20000000 10000000 10000000 1.0 0.160
## 4 D 20000000 13333333 6666667 2.0 0.270Impact on Stockholders’ Equity and Optimal Captial structure
The amount of leverage determines the sensitivity of the total ROE to changes in the UW profit. This can be demonstrated by calculating the ROE under a wide range of profit assumptions from -6% to +6% for each of the four companies in the example.
uw.profit.range = data.frame(uw_profit = -60:60/1000)
ferrari.ROE.range = merge(ferrari.examples, uw.profit.range)
ROE.leverage.formula = function(return.on.assets, UW.profit, assets, reserve) {
surplus = assets - reserve
return(return.on.assets + reserve / surplus * (return.on.assets + UW.profit))
}
ferrari.ROE.range$ROE = apply(ferrari.ROE.range %>% select(uw_profit, assets, reserve), 1, function(y) {ROE.leverage.formula(return.on.assets = investment.return, UW.profit = y['uw_profit'], assets = y['assets'], reserve = y['reserve'])})Here are the ROE values calculated in Ferrari’s paper:
ferrari.ROE.range %>% filter(uw_profit %in% c(-0.06, 0, 0.06)) %>% select(company, uw_profit, ROE)## company uw_profit ROE
## 1 A -0.06 0.050
## 2 B -0.06 0.045
## 3 C -0.06 0.040
## 4 D -0.06 0.030
## 5 A 0.00 0.050
## 6 B 0.00 0.075
## 7 C 0.00 0.100
## 8 D 0.00 0.150
## 9 A 0.06 0.050
## 10 B 0.06 0.105
## 11 C 0.06 0.160
## 12 D 0.06 0.270The results can also be plotted to compare the sensitivity of ROE to insurance leverage:
ggplot(data = ferrari.ROE.range, aes(x = uw_profit, y = ROE, col = company)) + geom_line()
Note that as the leverage of the company increases, the ROE exhibits greater sensitivity to the underwriting profit margin. As a result, the insurance leverage ratio is an indicator of the level of riskiness to which the owners’ equity is exposed.
This leads naturally to the question of determining the optimum capital structure for the firm. Some considerations when performing this analysis include:
Non-equity financing (i.e. reserves) may impact the magnitude and variance of the insurer’s earnings.
The cost of debt capital is fixed, while the cost of reserve capital is an expected cost with a variance.
Taking on increasing debt capital generally results in an increase in the cost of financing; however, increasing insurance writings may not necessarily increase costs (e.g. by reducing relative variance through diversification).
Need to consider the quality and earning capacity of the assets; if these earnings are more variable, then a lower reserve amount is indicated.
Dynamic Relationship Among Formula Variables
Balcarek observes that the formulas in Ferrari’s paper assume a static state, and do not reflect the potential for dynamic relationships between the variables. Examples of these relationships include:
As additional business is written, it may be of lower quality. Therefore, as \(P/S\) increases, \(U/P\) may decrease. (This could be avoided by the company provided it had adequate controls over expansion.)
As \(P/S\) increases, \(I/A\) will decrease because a greater proportion of assets are tied up in insurance operations (e.g. agents’ balances), and the risk associated with a higher \(P/S\) ratio will need to be offset by a more conservative investment policy.
If underwriting results are favourable, the insurer may be willing to write more business – so, if \(U/P\) is high then \(P/S\) will tend to rise as well.
If underwriting results are favourable, the insurer may be able to adopt a more aggressive investment policy, so \(I/A\) tends to move in the same direction as \(U/P\).
To illustrate these ideas, let’s assume that the insurer adopts a strategy in which additional premium is written when underwriting profits are good, and restricted when underwriting profits are negative. We’ll use a 0% profit as the baseline, and assume that the premium grows / shrinks by 10% for each percentage change in the profit margin. (That is an extreme assumption, but it will assist with visualizing the impact of the change.) As a result, the reserves will change by the same percentage, but we will assume that the surplus is fixed, so assets will change as well.
ferrari.ROE.range = ferrari.ROE.range %>% mutate(surplus = assets - reserve, revised_reserve = reserve * (1 + 10*uw_profit), revised_assets = revised_reserve + surplus)
ferrari.ROE.range$ROE_v2 = apply(ferrari.ROE.range %>% select(uw_profit, revised_assets, revised_reserve), 1, function(y) {ROE.leverage.formula(return.on.assets = investment.return, UW.profit = y['uw_profit'], assets = y['revised_assets'], reserve = y['revised_reserve'])})Here is how the ROE changes under this new assumption:
ferrari.ROE.range %>% filter(uw_profit %in% c(-0.06, 0, 0.06)) %>% select(company, uw_profit, ROE, ROE_v2)## company uw_profit ROE ROE_v2
## 1 A -0.06 0.050 0.050
## 2 B -0.06 0.045 0.048
## 3 C -0.06 0.040 0.046
## 4 D -0.06 0.030 0.042
## 5 A 0.00 0.050 0.050
## 6 B 0.00 0.075 0.075
## 7 C 0.00 0.100 0.100
## 8 D 0.00 0.150 0.150
## 9 A 0.06 0.050 0.050
## 10 B 0.06 0.105 0.138
## 11 C 0.06 0.160 0.226
## 12 D 0.06 0.270 0.402The graphs of ROE versus UW profit now show some curvature as a result of the interaction between these two parameters:
ggplot(data = ferrari.ROE.range, aes(x = uw_profit, y = ROE_v2, col = company)) + geom_line()