Introduction

This notebook illustrates the concepts in the Exam 7 reading P&C Insurance Company Valuation by Richard Goldfarb, which relates to learning objectives B1 to B3 of the syllabus. These concepts are illustrated by reproducing the examples in the paper; however, due to rounding not all numbers will match exactly. The following packages are used in this notebook:

easypackages::packages("data.table", "DT", "ggplot2")
options(scipen = 999)

Dividend Discount Model

Mechanics of the Method

The Dividend Discount Model involves calculating the value of a firm as the present value of all future dividends. Typically, this involves a projection of a dividend for each year in a forecast period, as well as a terminal value representing the value of all future dividends at the end of the forecast period. This method requires the following assumptions:
  • The expected dividend \(D_i\) for each year in the forecast period
  • An appropriate discount rate \(k\)
  • A dividend growth rate \(g\), assumed to apply in perpetuity after the end of the forecast period

Suppose that \(n\) is the final year of the forecast period. Then the terminal value \(T\) can be calculated as \[ T = \sum_{i \geq 1} \frac{D_n (1+g)^i}{(1+k)^i} = D_n \frac{1+g}{1+k}\frac{1}{1 - \frac{1+g}{1+k}} = \frac{D_{n+1}}{k-g} \] Then the present value of all dividends is \[ \sum_{1\leq i \leq n} \frac{D_i}{(1+k)^i} + \frac{T}{(1+k)^n} \] The overall mechanic will be illustrated using the projections in the following example, followed by a discussion of the assumptions underlying the method:

goldfarb.ddm = fread("./goldfarb_ddm.csv")
datatable(goldfarb.ddm)
We will make the following assumptions:
  • The company’s dividend policy is to pay 50% of the after-tax net income as a dividend. Note that the company’s dividend policy is a key consideration when using this method.
  • The discount rate is 8.95%.
  • The growth rate beyond the forecast period is 5%.

The first step is to calculate the projected dividend in each year. Based on the company’s dividend policy, we can calculate these projections as follows:

goldfarb.ddm[,net_income_after_tax := GAAP_Net_income_before_tax - Corporate_income_tax]
goldfarb.ddm[,expected_dividend := 0.5 * net_income_after_tax]
datatable(goldfarb.ddm)

We’ll calculate the terminal value using an R function, with the assumed growth rate as an input. This will facilitate sensitivity testing on the growth rate assumption.

terminal.value = function(growth.rate, discount.rate, final.dividend) {
  return(final.dividend * (1 + growth.rate) / (discount.rate - growth.rate))
}

With the assumptions given, we can calculate the terminal value as follows:

final.dividend = goldfarb.ddm[Year == 2009, expected_dividend]
terminal.value(growth.rate = 0.05, discount.rate = 0.0895, final.dividend = final.dividend)
## [1] 161367.7

To perform sensitivity testing on the terminal value, first create a dataset containing the range of assumptions to be tested. This can be visualized using the “geom_raster” function in the ggplot package.

discount.range = 0.090 + 0:40*0.001
growth.range = 0.04 + 0:40*0.001
TV.sensitivity.test = as.data.table(merge(discount.range, growth.range))
colnames(TV.sensitivity.test) = c("discount.rate", "growth.rate")
TV.sensitivity.test$terminal.value = apply(TV.sensitivity.test, 1, function(y) terminal.value(growth.rate = y['growth.rate'], discount.rate = y['discount.rate'], final.dividend = final.dividend))
ggplot(data = TV.sensitivity.test, aes(x = discount.rate, y = growth.rate, fill = terminal.value)) + geom_raster() + labs(title = "Terminal Value Projections - Sensitivity Analysis")
## Warning in plyr::split_indices(scale_id, n): '.Random.seed' is not an
## integer vector but of type 'NULL', so ignored

Notice that the most extreme values occur in the upper left corner, where the discount rate is close to the growth rate. This is not surprising given that the value \(k-g\) appears in the denominator of the terminal value calculation. Let’s zoom in on the areas where the discount rate is above 0.1, and the growth rate is below 0.07:

ggplot(data = TV.sensitivity.test[discount.rate > 0.1 & growth.rate < 0.07], aes(x = discount.rate, y = growth.rate, fill = terminal.value)) + geom_raster() + labs(title = "Terminal Value Projections - Sensitivity Analysis")

An additional consideration is that there is likely correlation between the two assumptions: high growth rates are associated with higher discount rates. Because of this, the top left and bottom right corners of this table, where the values are most extreme, are the more unlikely regions for the terminal value.

Next, discount the dividends during the projection period, and add the discounted value of the terminal value:

ddm.valuation.details = function(dataset, discount.rate) {
  first.year = min(dataset$Year)
  last.year = max(dataset$Year)
  pv.dividends = apply(dataset, 1, function(x){x['expected_dividend']} / ((1 + discount.rate)^(x['Year'] - first.year + 1)))
  discount.factor = apply(dataset, 1, function(x){1 / (1 + discount.rate)^(x['Year'] - first.year + 1)})
  return(as.data.table(cbind(dataset$Year, discount.factor, pv.dividends)))
}

The projection is based on years 2005 and later:

datatable(ddm.valuation.details(goldfarb.ddm[Year >= 2005], 0.0895))

Finally, we sum the present value of the dividends and add the present value of the terminal value:

ddm.valuation = function(dataset, discount.rate, growth.rate) {
  total.pv.dividends = sum(ddm.valuation.details(dataset, discount.rate)$pv.dividends)
  first.year = min(dataset$Year)
  last.year = max(dataset$Year)
  final.dividend = dataset[Year == last.year, expected_dividend]
  pv.terminal.value = terminal.value(growth.rate = growth.rate, discount.rate = discount.rate, final.dividend = final.dividend) / (1 + discount.rate)^(last.year - first.year + 1)
  return (total.pv.dividends + pv.terminal.value)
}
ddm.valuation(goldfarb.ddm[Year >= 2005], 0.0895, 0.05)
## [1] 126434.9

Note that the present value of the terminal value in this case is

terminal.value(growth.rate = 0.05, discount.rate = 0.0895, final.dividend = 6070) / (1 + 0.0895)^5
## [1] 105110.2

This represents a significant fraction of the value of the company, indicating that this method is particularly sensitive to the selection of assumptions underlying the terminal value. Earlier, we performed sensitivity testing on the terminal value; we can perform similar tests on the equity value as well.

EV.sensitivity.test = as.data.table(merge(discount.range, growth.range))
colnames(EV.sensitivity.test) = c("discount.rate", "growth.rate")
EV.sensitivity.test$equity.value = apply(EV.sensitivity.test, 1, function(y) ddm.valuation(goldfarb.ddm[Year >= 2005], discount.rate = y['discount.rate'], growth.rate = y['growth.rate']))
ggplot(data = EV.sensitivity.test, aes(x = discount.rate, y = growth.rate, fill = equity.value)) + geom_raster() + labs(title = "Equity Value Estimates - Sensitivity Analysis")

Let’s zoom in on the areas where the discount rate is above 0.1, and the growth rate is below 0.07:

ggplot(data = EV.sensitivity.test[discount.rate > 0.1 & growth.rate < 0.07], aes(x = discount.rate, y = growth.rate, fill = equity.value)) + geom_raster() + labs(title = "Equity Value Estimates - Sensitivity Analysis")

Assumptions Underlying the Method

The method requires assumptions about the following.

Expected Dividends during the Forecast Horizon

Based on:
  • Forecasts of revenue, expenses, investments, etc.
  • Prior business written, renewals, and new business written

Dividend Growth Rates Beyond Forecast Horizon

The plowback ratio represents the percentage of ROE that is re-invested in the firm in order to achieve growth. Therefore, we can estimate the growth rate as \[ g = \text{plowback} \times \text{ROE} \] Considerations in selecting a growth rate include:
  • It must represent steady state growth in perpetuity. In particular, it is not reasonable to assume a growth rate higher than the growth rate for the entire economy, since this implies that the firm’s share of the economy will grow to unreasonable levels.
  • High growth rates are likely to be offset by lower dividend amounts.
  • High growth rates are likely to be associated with higher risk, and therefore higher discount rates.

Risk-adjusted Discount Rate

The Capital Asset Pricing Model (CAPM) can be used to determine an appropriate risk-adjusted rate of return. In this model, the required rate of return is \[ k = r_f + \beta(E[r_m] - r_f) \] In this formula:
  • \(r_f\) is the risk-free rate of return. This may be selected using the rate on a risk-free investment, such as a 20-year T-bond, with the liquidity (term) premium netted out.
  • \(\beta\) is a measure of the degree to which rates of return for the firm vary relative to the rate of return for a market portfolio. It is a measure of systemic risk (risk that cannot be diversified away).
  • \(E[r_m]\) is the expected market return. The difference \(E[r_m] - r_f\) is referred as the equity market risk premium, and is often used rather than using the market return directly. Considerations in estimating this value include:
    • Short-term vs. long-term yields: should match the term used in selecting the risk-free rate.
    • Arithmetic averages are preferred for single-period forecasts, while geometric are preferred for multiple periods
    • Historical averages vs. risk premium implied by current prices (e.g. by applying DDM to an aggregate market index).
    Considerations in selecting a discount rate include:
    • In a private valuation, the discount rate is based on the individual investor’s portfolio, and different investors can have a different view of the value of a firm. In an equilibrium valuation, the marginal investor’s risk and cash flow assumptions determine the market price of the investment.
    • When performing valuation of P&C companies, the all-equity \(\beta\) is used without adjustments for leverage. The rationale is that policyholder liabilities are not fully accounted for when adjusting for leverage, so there is an assumption that the total leverage of all firms in the industry is similar. Considerations when selecting a \(\beta\) include:
      • Consider firms with a comparable mix of business to the firm being valued
      • Consider firms with a similar amount of financial leverage
    • If the mix of business or degree of financial leverage is expected to change over time, then there may be a need for different discount rates for different time periods.

    A simple R function can be used to calculate the CAPM discount rate:

    CAPM.discount = function(risk.free.rate,  beta, equity.risk.premium) {
  return(risk.free.rate + beta * equity.risk.premium)
}

    Using the values in Goldfarb’s paper, we obtain the discount rate used in the examples above:

    CAPM.discount(risk.free.rate = 0.0433, beta = 0.84, equity.risk.premium = 0.0550)
    ## [1] 0.0895

    We can use this function as one of the inputs to the DDM valuation function to obtain the results:

    ddm.valuation(goldfarb.ddm[Year >= 2005], discount.rate =  CAPM.discount(risk.free.rate = 0.0433, beta = 0.84, equity.risk.premium = 0.0550), growth.rate = 0.05)
    ## [1] 126434.9

    By combining the functions in this way, we can now perform sensitivity testing on the final equity value relative to changes in \(\beta\) and the equity risk premium, keeping the dividend growth rate and risk-free rate fixed:

    beta.range = 0.70 + 0:40*0.01
equity.risk.range = 0.04 + 0:40*0.001
CAPM.sensitivity.test = as.data.table(merge(beta.range, equity.risk.range))
colnames(CAPM.sensitivity.test) = c("beta", "equity.risk")
CAPM.sensitivity.test$equity.value = apply(CAPM.sensitivity.test, 1, function(y) { ddm.valuation(goldfarb.ddm[Year >= 2005], discount.rate =  CAPM.discount(risk.free.rate = 0.0433, beta = y['beta'], equity.risk.premium = y['equity.risk']), growth.rate = 0.05)})
ggplot(data = CAPM.sensitivity.test, aes(x = beta, y = equity.risk, fill = equity.value)) + geom_raster() + labs(title = "Equity Value Projections - Sensitivity Analysis")

Free Cash Flow to Equity

The Free Cash Flow to Equity method is based on the present value of all the cash that could have been paid out as dividends, after making adjustments to support current operations and expected growth. Considerations when using this approach:

Adjustments to FCFE for a P&C Company

For a firm in general, the FCFE is calculated by adjusting net income as follows:
  • Add non-cash charges. (Expenses deducted under GAAP that do not represent actual cash expenditures.) For a P&C company, the most significant non-cash charge is changes in the loss and expense reserves; however, these cancel out with a later adjustment.
  • Deduct net working capital investment: short-term assets held to facilitate operations (inventory, etc.). Generally not significant for P&C companies.
  • Deduct capital expenditures. For a P&C company, this includes changes in loss and expense reserves, as well as changes in capital required to meet regulatory or rating agency targets. The adjustments to loss and reserve expenses cancel out with corresponding adustements based on non-cash charges, so the changes in captial requirements are primarily of concern here. The most binding constraint on capital should be used here, whether it be a regulatory target, the amount needed to maintain a rating consistent with the business plan, or management’s own internal target.
  • Add net borrowing
Therefore, the adjustments to net income specific to a P&C company are:
  • Add non cash-charges, excluding changes in reserves
  • Deduct net working captial investment
  • Deduct the increase in required capital
  • Add net borrowing

Alternatively, FCFE can be calculated as the difference between the minimum capital required and ending GAAP equity before dividends for the year.

Mechanics of the FCFE Valuation Method

Broadly speaking, the mechanics of valuing a firm using discounted FCFE is similar to the dividend discount method, with the following adjustments:
  • The growth rate used in the terminal value calculation is based on the percentage of net income that is used to increase the capital base of the firm, i.e. the amount that is re-invested to satisfy regulatory or rating agency constraints. Specifically:
    • Reinvested capital is the difference between net income and FCFE
    • The reinvestment rate is the ratio of reinvested capital to net income
    • The ROE is the ratio of net income to beginning capital
    • The horizon growth rate in the final year is the product of the reinvestment rate and ROE in the final year in the forecast period. Note that this is equivalent to the ratio of reinvested capital to beginning capital.
    • Within the forecast period, growth can be calculated directly as the ratio of FCFE in the current time period to the FCFE in the prior time period.
  • There is theoretically a different discount rate between the DDM and FCFE methods, since the risk under DDM is influenced by the risk associated with the securities in which the non-dividend income is invested; however, this difference is difficult to quantify and is generally not material.

The FCFE method can be illustrated using the following dataset. Sensitivity testing can also be performed on these results, but it is not significantly different from the approach illustrated for the DDM method, so I will omit it for simplicity.

goldfarb.fcfe = fread("./goldfarb_fcfe.csv")
datatable(goldfarb.fcfe)

First, derive the required capital investment and FCFE in each year:

goldfarb.fcfe[, reinvested.capital := Minimum_capital - GAAP_equity_start]
goldfarb.fcfe[, FCFE := GAAP_net_income - reinvested.capital]
datatable(goldfarb.fcfe)

First, we will calculate the growth rate, reinvestment rate, and ROE in each year. We calculate it in two ways: as the ratio of successive values of FCFE, and as the product of ROE and the reinvestment rate. Note that due to cancellation of net income in the numerator and denominator, the second calculation is equal to the ratio of reinvested capital to starting GAAP equity.

goldfarb.fcfe[, growth_rate := as.double(FCFE)/as.double(shift(FCFE, type = "lag")) -1]
goldfarb.fcfe[, ROE := GAAP_net_income / GAAP_equity_start]
goldfarb.fcfe[, reinvestment_rate := reinvested.capital / GAAP_net_income]
goldfarb.fcfe[, horizon_growth_rate := ROE * reinvestment_rate]
goldfarb.fcfe[, horizon_growth_rate_2 := reinvested.capital / GAAP_equity_start]
datatable(goldfarb.fcfe[, .(Year, growth_rate, horizon_growth_rate, horizon_growth_rate_2)])

The growth rate beyond the forecast horizon is:

perpetual.growth.rate = goldfarb.fcfe[Year == 2009, horizon_growth_rate]
perpetual.growth.rate
## [1] 0.03943736

Using the same discount rate as in the DDM example, undiscounted terminal value is:

final.FCFE = goldfarb.fcfe[Year == 2009, FCFE]
discount.rate = 0.0895
terminal.value.FCFE = final.FCFE * (1 + perpetual.growth.rate) / (discount.rate - perpetual.growth.rate)
terminal.value.FCFE
## [1] 293564.3

At present value, the terminal value is:

TV.discount = 1/(1+discount.rate)^5
PV.terminal.value.FCFE = terminal.value.FCFE * TV.discount
PV.terminal.value.FCFE
## [1] 191234.9

Note that the growth and discount rate assumptions can be combined into a single assumption that the terminal value is a fixed multiple of the final projected FCFE, namely,

(1 + discount.rate) / (discount.rate - perpetual.growth.rate)
## [1] 21.76273

Finally, discount the projected FCFE values to present:

goldfarb.fcfe[, discount_factor := 1 / (1 + discount.rate)^(Year - 2004)]
goldfarb.fcfe[, PV_FCFE := FCFE * discount_factor]
datatable(goldfarb.fcfe[, .(Year, FCFE, discount_factor, PV_FCFE)])

The total value of the company is therefore:

PV.terminal.value.FCFE + goldfarb.fcfe[, sum(PV_FCFE)]
## [1] 241887.6

Abnormal Earnings Method

In the Abnormal Earnings approach, the value of a firm is equal to its book value, plus the present value of abnormal earnings. A key consideration in this approach is that normal earnings growth does not add value – only abnormal earnings add value. Abnormal earnings are earnings in excess of the required return on equity: \[ \mathrm{AE} = \mathrm{NI} - k \times \mathrm{BV} = (\mathrm{ROE} - k) \times \mathrm{BV} \] These can be discounted using the same mechanics as the dividend discount method. Main differences between this method and the other methods include:

This technique can be illustated using the same data underlying the FCFE example, and the same required rate of return.

goldfarb.ae = fread("./goldfarb_fcfe.csv")
k = 0.0895
goldfarb.ae[, normal_earnings := GAAP_equity_start * k]
goldfarb.ae[, abnormal_earnings := GAAP_net_income - normal_earnings]
goldfarb.ae[, PV_abnormal_earnings := abnormal_earnings / (1 + k)^(Year - 2004)]
datatable(goldfarb.ae)

In a simple case, we can assume that there are no abnormal earnings beyond the forecast period. In this case, the valuation is

goldfarb.ae[Year == 2005, GAAP_equity_start] + goldfarb.ae[, sum(PV_abnormal_earnings)]
## [1] 133546.5

Terminal Value: Constant Abnormal Earnings

One approach to calculating the terminal value is to assume that the abnormal earnings persist in perputuity, at the same level (growth rate of 0). In this case, the terminal value is:

terminal.value.AE = goldfarb.ae[Year == 2009, abnormal_earnings] / (k - 0)
terminal.value.AE
## [1] 89499.79

In this case, the present value of the equity is

goldfarb.ae[Year == 2005, GAAP_equity_start] + goldfarb.ae[, sum(PV_abnormal_earnings)] + terminal.value.AE / (1 + k)^5
## [1] 191848.8

For reasons discussed above, this is not a reasonable result since we do not expect abnormal earnings to persist in perpetuity.

Terminal Value: Exponential Decay in Abnormal Earnings

Although not mentioned in Goldfarb’s paper, an easy-to-implement assumption is that the abnormal earnings will decay exponentially beyond the forecast period. For example, if we assume that abnormal earnings will decay by 20% per year, then the terminal value is:

decay.factor = 0.2
terminal.value.AE = goldfarb.ae[Year == 2009, abnormal_earnings] * (1-decay.factor) / (k + decay.factor)
terminal.value.AE
## [1] 22135.35

In this case, the present value of the equity is

goldfarb.ae[Year == 2005, GAAP_equity_start] + goldfarb.ae[, sum(PV_abnormal_earnings)] + terminal.value.AE / (1 + k)^5
## [1] 147966

Terminal Value: Linear Decay

Goldfarb suggests an assumption that the abnormal earnings decay linearly over a horizon of \(n\) years. Below, this approach is illustrated for \(n=5\), and then as a sensitivity test, the terminal value is calculated for a range of \(n\) from 1 to 20.

linear.decay.table = function(initial.AE, decay.duration, discount.rate) {
  horizon.year = 1:decay.duration
  result.table = as.data.table(horizon.year)
  result.table[, AE := initial.AE * (decay.duration - horizon.year + 1) / (decay.duration + 1)]
  result.table[, PV_AE := AE / (1 + discount.rate)^horizon.year]
  return (result.table)
}
datatable(linear.decay.table(initial.AE = goldfarb.ae[Year == 2009, abnormal_earnings], decay.duration = 5, discount.rate = k))

We can calculate the terminal value as the sum of the present value of the abnormal earnings under this projection:

linear.decay.TV = function(initial.AE, decay.duration, discount.rate) {
  AE.table = linear.decay.table(initial.AE, decay.duration, discount.rate)
  return(AE.table[,sum(PV_AE)])
}
linear.decay.TV(initial.AE = goldfarb.ae[Year == 2009, abnormal_earnings], decay.duration = 5, discount.rate = k)
## [1] 16487.31

Finally, let’s calculate it under each value of \(n\) between 1 and 20, and plot the results.

AE.duration.assumption = 1:20
AE.TV.sensitivity = as.data.table(AE.duration.assumption)
AE.TV.sensitivity$terminal.value = apply(AE.TV.sensitivity, 1, function(x){linear.decay.TV(initial.AE = goldfarb.ae[Year == 2009, abnormal_earnings], decay.duration = x['AE.duration.assumption'], discount.rate = k)})
ggplot(data = AE.TV.sensitivity, aes(x = AE.duration.assumption, y = terminal.value)) + geom_line() + labs(title = "Sensitivity Test: Terminal Value under Linear Decay Assumption")

Relative Valuation Using Multiples

The general approach in this section is to express the value of the firm in terms of a ratio to some other metric, such as earnings, book value, sales, or cash flow. Multipyling the ratio by the value of this metric gives a quick estimate of the value of the firm. These ratios can either be trailing (expressed in terms of prior period’s actual results) or leading (expressed in terms of expected future results). An advantage of forward ratios is that they are not skewed by unusual events; trailing ratios may need to be adjusted or smoothed out to compensate for these events. Uses of these ratios include: Relative valuation may be performed using either market market multiples (based on market value) or transaction multiples (based on actual merger and acquisition transactions). Key differences between the two include:

Relationship Between P-E Ratio and Dividend Discount Model

The P-E ratio is equal to the ratio of the price per share to earnings per share (or equivalently, the ratio of total value to total earnings). This ratio is closely related to the dividend discount model, using the following notation:
  • There is a constant dividend payout rate \(d\)
  • Earnings growth is constant in perpetuity, at a rate \(g\)
  • Expected total earnings in the next period are denoted by \(E\)
  • The discount rate is \(k\)
  • \(P\) is the total value of the firm

Then the total value of the firm is \[ P = \frac{dE}{k-g}. \] Dividing by \(E\), we get the following: \[ \frac{P}{E} = \frac{d}{k-g} \] In other words, the P-E ratio summarizes the combined effect of three assumptions: discount rate, growth rate, dividend payout rate, into a single number. Alternately, since the plowback ratio is equal to \(1-d\), we can express this in terms of the assumed ROE rather than growth rate, using \[ g = (1-d)\mathrm{ROE} \] We can visualize the effect that different assumptions about the dividend payout ratio and discount rate have on the P-E ratio, assuming a constant ROE of 15%:

discount.rate = 0.1 + 1:50*0.001
dividend.payout = 0.4 + 1:40*0.005
assumption.table = as.data.table(merge(discount.rate, dividend.payout))
colnames(assumption.table) = c("discount.rate", "dividend.payout")
PE.ratio = function(discount.rate, dividend.payout, ROE) {
  return(dividend.payout/(discount.rate - (1 - dividend.payout) * ROE))
}
assumption.table$PE.ratio = apply(assumption.table, 1, function(x){PE.ratio(discount.rate = x['discount.rate'], dividend.payout = x['dividend.payout'], ROE = 0.15)})
ggplot(data = assumption.table, aes(x = discount.rate, y = dividend.payout, fill = PE.ratio)) + geom_raster() + labs(title = "PE Ratio - Sensitivity Analysis")

Notice that the PE ratio is constant when the discount rate is equal to ROE:

assumption.table[discount.rate > 0.1495, PE.ratio]
##  [1] 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667
##  [8] 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667
## [15] 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667
## [22] 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667
## [29] 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667 6.666667
## [36] 6.666667 6.666667 6.666667 6.666667 6.666667

This is consistent with the philosophy underlying the abnormral earnings approach, that the growth rate does not affect the value of the firm if it does not generate earnings in excess of the discount rate.

Relationship Between P-BV Ratio and Abnormal Earnings Model

The price-to-book value ratio is often preferred over the price-to-earnings ratio when valuing financial services firms, given their substantial holdings in marketable securities. It is related to the abnormal earnings method, using the following notation and assumptions:
  • \(B\) is the book value of the firm
  • ROE is constant in perpetuity – this is not a realistic assumption, as discussed in the section on abnormal earnings. (It’s still possible to derive expressions under alternate assumptions.)
  • Book value grows at a constant rate \(g\)
  • \(k\) is the required return on equity

Then we have: \[ P = B + \sum_{i\geq 1}\frac{AE_i}{(1+k)^i} = B + \sum_{i\geq 1} \frac{(\mathrm{ROE}-k) B}{(1+k)^i} \] Incorporating the growth rate in book value, this becomes \[ P = B + \sum_{i\geq 1}\frac{(\mathrm{ROE}-k)B(1+g)^{i-1}}{(1+k)^i} \] Simplifying the geometric series, \[ P = B + (\mathrm{ROE} - k)B\frac{1}{1+k}\frac{1}{1 - \frac{1+g}{1+k}} = B + \frac{(\mathrm{ROE} - k)B}{k-g} \] Therefore, the price-to-book value ratio is \[ \frac{P}{B} = 1 + \frac{\mathrm{ROE}-k}{k-g} \] We can visualize the relationship between fundamentals and the P-B ratio as follows:

discount.rate = 0.1 + 1:50*0.001
growth.rate = 1:40*0.001
assumption.table = as.data.table(merge(discount.rate, growth.rate))
colnames(assumption.table) = c("discount.rate", "growth.rate")
PB.ratio = function(discount.rate, growth.rate, ROE) {
  return(1 + (ROE - discount.rate)/(discount.rate - growth.rate))
}
assumption.table$PB.ratio = apply(assumption.table, 1, function(x){PB.ratio(discount.rate = x['discount.rate'], growth.rate = x['growth.rate'], ROE = 0.15)})
ggplot(data = assumption.table, aes(x = discount.rate, y = growth.rate, fill = PB.ratio)) + geom_raster() + labs(title = "P-BV Ratio - Sensitivity Analysis")

As before, the growth rate has no impact on the valuation of the firm if the expected ROE is equal to the discount rate. It fact, it clearly demonstrates that growth adds nothing to the book value unless earnings are in excess of the required rate of return.

assumption.table[discount.rate > 0.1495, PB.ratio]
##  [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [36] 1 1 1 1 1

Relative Valuation for Multiline Firms

A single-business entity is called a pure play firm. To value a multiline firm using multiples for pure play firms, use the following procedure.
  1. Separate data for the multiline firm by line of business.
  2. Calculate the average P-E and P-BV ratios for the pure play firms in each line of business. A weighted average may be used if the peer firms are not roughly the same size.
  3. Apply the average multiples to the multiline firm by line of business. Take the average of the results produced using the P-E and P-BV approaches.
  4. Sum the results across lines of business.
  5. Results can be validated by comparing to valuation produced by comparison to other diversified insurers.

To illustrate this approach, import the data for the peers and the company whose valuation is being calculated:

RV.peers <- fread("./Goldfarb_RV_peers.csv")
RV.subject <- fread("./Goldfarb_RV_subject.csv")
datatable(RV.peers)

We need the average ratios for each of the pure play lines. Note that the P-E ratio may be missing if earnings were negative, so it is important to remove missing values when calculating the mean.

pure.play.averages = RV.peers[Line != "Diversified", .(avg_PE = mean(`P-E`, na.rm = TRUE), avg_PBV = mean(`P-BV`)), by = Line]
datatable(pure.play.averages)

Join the firm-specific financial measures:

firm.with.pp.averages = merge(RV.subject, pure.play.averages, by = "Line")

Calculate the valuation, by line, using both the P-E and P-BV ratios:

firm.with.pp.averages[, PE_valuation := Earnings * avg_PE]
firm.with.pp.averages[, PBV_valuation := Book_Value * avg_PBV]
firm.with.pp.averages[, avg_valuation := (PE_valuation + PBV_valuation) / 2]
datatable(firm.with.pp.averages)

We can now determine the total firm valuation by adding the results by line:

total.firm.valuation = firm.with.pp.averages[, sum(avg_valuation)]
total.firm.valuation
## [1] 36024.56

To compare against diversified firms, first calculate average ratios for these firms:

diversified.averages = RV.peers[Line == "Diversified", .(avg_PE = mean(`P-E`, na.rm = TRUE), avg_PBV = mean(`P-BV`)), by = "Line"]
datatable(diversified.averages)

We sum the subject companye financial results before applying the ratios.

firm.total = RV.subject[, .(total_earnings = sum(Earnings), total_BV = sum(Book_Value), Line = "Diversified")]

Join as before, then calculate the average valuation using the P-E and P-BV methods:

firm.with.diversified.averages = merge(firm.total, diversified.averages, by = "Line")
firm.with.diversified.averages[, PE_valuation := total_earnings * avg_PE]
firm.with.diversified.averages[, PBV_valuation := total_BV * avg_PBV]
firm.with.diversified.averages[, avg_valuation := (PE_valuation + PBV_valuation) / 2]
datatable(firm.with.diversified.averages)

Option Pricing

To understand how equity can be valued as a call option, assume that the firm has a debt \(D\) that must be repaid at time \(T\). Let \(V_T\) denote the value of the firm at time \(T\). Then the value of the firm to the equityholders at time \(T\) is \[ E_T = \max(V_T-D, 0) \] This is the same as the value of a call option with a strike price of \(D\). However, this approach is not generally used for insurance companies, since the notion of debt is not well-defined due to the existince of policy liabilities, and there is no single expiration time for all of the insurer’s debt.

Options can be used as a way to price managerial flexibility to abandon, expand, contract, defer, or extend projects. This approach is only valid when there is the possibility that assets could be purchased at less than their fair value, and the firm has exclusive access to the opportunity at a fixed price. (A future opportunity to purchase assests at market prices has no value.) Goldfarb provides an example in which an insurance company is presented with a new business opportunity, with the following characteristics:

This can be valued as a European call option, expiring in 3 years, with a current asset value of $500M and a strike price of $500M. The value of this option can be calculated using the Black-Scholes formula:

A = 500000000
I = 500000000
sigma = 0.2
T = 3
r = 0.0455
d1 = (log(A/I) + T * (r + sigma^2/2)) / sqrt(T) / sigma
d2 = d1 - sigma * sqrt(T)
option.value = A * pnorm(d1) - I * exp(-r*T) * pnorm(d2)
option.value
## [1] 101141917
Therefore, we can add $101.1M to the value of the firm to reflect the value attributed to the flexibility associated with this project. Potential drawbacks to this approach include: