Math 479 / 568
Casualty Actuarial Mathematics
Fall 2014
University of Illinois at Urbana-Champaign
Professor Rick Gorvett
Session 15: Finance I
November 4, 2014
1
Agenda
• Basics of finance
• Applications of financial models to
insurance
2
The Most Important Concepts
in Finance
•
•
•
•
•
•
•
Net present value
Capital asset pricing model
Efficient capital markets
Value additivity / conservation of value
Capital structure theory
Option theory
Agency theory
Source: Brealey and Myers, 1996, Principles of Corporate
Finance, 5th Edition, Chapter 35
3
Capital Asset Pricing Model
E(Ri) = Rf + i [E(Rm) - Rf]
where:
E = expected value operator
Ri = return on an asset
Rf = risk free rate
Rm = return on market portfolio
i = Cov(Ri,Rm) / 2(Rm)
= systematic risk
4
Example
• A stock has a beta of 1.5
• The risk free rate is 5 percent
• The expected market risk premium is 9
percent
• What is the expected return on this stock?
E(Ri) = 5% + 1.50 [9%] = 18.5%
5
CAPM Assumptions
1)
2)
3)
4)
5)
6)
Investors maximize expected return and
minimize risk
Trades do not affect prices
Expectations are homogeneous
No taxes or transaction costs
All borrowing and lending is at the risk
free rate
Assets are infinitely divisible
6
Problems with the CAPM
• How can beta be measured?
• What is the market portfolio?
• Does the CAPM explain stock returns?
– Fama and French, 1992, “The Cross-Section of
Expected Stock Returns,” Journal of Finance: Size and
book-to-market ratios explain stock returns better than
beta over the period 1941-1990
7
What We Do Not Know About
Finance...
10 Unsolved Problems
1) How are major financial decisions made?
2) What determines project risk and present
value?
3) Risk and return - have we missed something?
4) Are there important exceptions to the efficient
market theory?
5) Is management an off-balance sheet liability?
8
What We Do Not Know About
Finance...
10 Unsolved Problems (cont.)
6) How can we explain the success of new
securities and new markets?
7) How can we resolve the dividend controversy?
8) What risks should a firm take?
9) What is the value of liquidity?
10) How can we explain merger waves?
Source: Brealey and Myers, 1996, Principles of Corporate Finance, 5th Edition, Ch. 35
9
“New Facts in Finance”
“New Facts in Finance,” by John H. Cochrane,
Economic Perspectives, Federal Reserve Bank of
Chicago, 1999 (third quarter), pp. 36-58
• Discusses the “revolution” that occurred in what
we know (or think we know) about finance over
the last approximately 15 years of the 20th
century.
10
“New Facts in Finance” (cont.)
• Mid-1980s understanding of financial world
– “The CAPM is a good measure of risk and thus a good
explanation of the fact that some assets… earn higher average
returns than others.”
– “Returns are unpredictable, like a coin flip.”
•
•
•
•
Stock prices: random walk
Bond returns: expectations model of term structure
FX: investment rate – exchange rate “balance”
Stock volatility: relatively constant over time
– “Professional managers do not reliably outperform simple
indexes and passive portfolios once one corrects for risk
(beta).”
• Overall  asset markets are “informationally efficient”
11
“New Facts in Finance” (cont.)
• “Now (1999), we know that:”
– “There are assets whose average returns can not be explained
by their beta. Multifactor extensions of the CAPM
dominate…”
– “Returns are predictable. In particular: Variables including
the dividend/price (d/p) ratio and term premium can predict
substantial amounts of stock return variation. This
phenomenon occurs over business cycle and longer
horizons.”
• Bond returns, FX also somewhat predictable.
• Stock volatility: changes over time
– “Some mutual funds seem to outperform simple indexes, even
after controlling for risk through market betas.”
12
Applications of Finance
to
Insurance Pricing
13
Financial Economics
• Has developed tools that could be useful in
insurance
• Initial application in Massachusetts
– confrontation-style hearings
– used to depress rate levels
– tended to tarnish insurers’ perception of this
field
• The models need to be adapted for
insurance applications
14
Caveat
• Competitive markets establish appropriate
profit margins, not models
• A useful model can approximate
appropriate profit margins and serve as an
aid to management decisions
• A useful model could reduce cyclicality of
insurance pricing by providing advance
warning of market reactions to various price
levels
15
Financial Models Applied
To Pricing
• Total rate of return
• Capital asset pricing model
• Discounted cash flow models
– Myers-Cohn
– NCCI
• Option pricing model (to be discussed later)
16
Total Rate of Return
TRR = (IA/S)(IR) + (P/S)(UPM)
TRR =
IA =
S
=
IR =
P
=
UPM=
Target total rate of return
Investable assets
Surplus
Investment return
Premium
Underwriting profit margin
17
Example
An insurer with $120 million in investable assets
and $40 million in surplus writes $60 million in
premium for the year. The company’s investment
return is 8 percent. What is the appropriate
underwriting profit margin if the company has a
target rate of return of 21 percent?
18
Class Problem
A small, stockholder-owned insurer that specializes in nonstandard auto is using the Total Rate of Return method to establish a
benchmark underwriting profit margin. The company intends to
write $24 million in premium for the year, which would lead to
having $48 million in investable assets.
Assets are invested three-quarters in mid-maturity bonds,
with a current yield of 8 percent, and one-quarter in stocks, which
pay a dividend of 4 percent and have an expected capital gain of 10
percent. Last year, the net investment income yield on the
company’s portfolio was 7 percent and the capital gains were
negative 4 percent.
The company has $8 million in surplus. The market value of
the company based on its current stock price is $12 million. The
company’s own stock has historically had a beta of 1.5.
1) Select and justify a reasonable Target Total Rate of Return for this
company
2) Calculate the required Underwriting Profit Margin to achieve this
19
target
Capital Asset Pricing Model
Application to Insurance
Basic model without taxes:
UPM = -kRf + u[E(Rm) - Rf]
k = funds generating coefficient
u = underwriting beta
20
Example
k
u
Rf
E(Rm)
=
=
=
=
.5
.2
8%
16%
UPM = -0.5(8%) + 0.2[16% - 8%] = -2.4%
21
Tax Version of Insurance CAPM
UPM = - kRf(1 - TA)/(1 - T) + u[E(Rm) - Rf] +
(S/P)Rf[TA/(1 - T)]
TA = Tax rate on investment income
T = Tax rate on underwriting income
S/P = Surplus to premium ratio
22
Example
Same values as before
TA =
16.8%
T
30%
=
S/P =
1.0
UPM = -1.23%
23
Class Problem
Our stockholder-owned non-standard auto insurer decides to
use the tax version of the Capital Asset Pricing Model to
establish a benchmark underwriting profit margin. The funds
generating coefficient is 2.0 ($48 million in investable assets
on $24 million annual premiums). The underwriting beta is
0.25. The company’s tax rate is 34 percent on underwriting,
but only half of its investment income is taxable. The riskfree rate is 6.5 percent and the market risk premium is 9
percent.
Calculate the appropriate underwriting profit margin based on
a surplus value of $8 million and of $12 million.
24
Problems Applying the CAPM To
Insurance
• Market risk cannot be measured for a line of
business
• Bankruptcy risk is ignored
• Insurance risk that is not systematic with investment
risk is ignored
• Writing coverage at a CAPM-determined price does
not add value to an insurer
• CAPM may not even value stock returns
appropriately
25
Discounted Cash Flow Models
• Myers-Cohn uses policyholder perspective
• NCCI uses company perspective
• Myers-Cohn uses NPV
• NCCI uses IRR
26
General Risk-Adjusted Discount
Technique Formula
PV (P) = PV (L) + PV (E) + PV (TUW) + PV (TII)
where:
PV
P
L
E
TUW
TII
=
=
=
=
=
=
present value operator
premiums
losses and loss adjustment expense
underwriting expenses
taxes on underwriting profit or loss
taxes on investment income
27
Example 1
•
•
•
•
•
•
One-year policy
Premiums paid on the effective date of policy
Expenses = $20 paid on effective date of policy
Expected losses = $80 paid at the end of the year
Surplus allocated to policy = $50
Taxes on underwriting gain or loss paid at the end of
the year
• Taxes on investment income paid at the end of the year
• Tax rate of 34 percent applies to underwriting and
investments
28
• All cash flows discounted at 7 percent
Example 1
P = 80 + 20 + (P - 20 - 80) (.34) + (50 + P - 20) (.07) (.34)
1.07
1.07
1.07
P = 74.766 + 20 + .318 P - 6.355 - 25.421 + .667 + .022 P
.66 P = 63.657
P = 96.45
29
Summary of Nominal and Discounted
Values -- Example 1
Nominal Values
Discounted Values
Losses
80
74.77
Expenses
20
20
Taxes on Underwriting
-1.21
-1.13
Taxes on Investments
3.01
2.81
Total
101.80
96.45*
*Premium = Sum of the Discounted Values of Losses, Expenses and Taxes
30
Example 2
Risky cash flows discounted at 4 percent
(P-20)(.34)
80 (.34) (50+P-20)(.07)(.34)
80
P=
+ 20 +
- 1.04 +
1.07
1.07
1.04
P = 76.923 + 20 + .318 P - 6.355 - 26.154 + .667 + .022 P
P = 65.081 / 0.66 = $98.61
31
Example 3
Losses paid expected to be $40 at the end of the first
year and $40 at the end of the second year and the IRS
discount rate is 8%
40
40
40
(P - 20) (.34) (40 + 1.08 ) (.34)
P=
+
+ 20 +
2
1.04 (1.04)
(1.07)
1.04
40
(40 - 1.08 ) (.34)
(1.04)2
+
+
(50 + P - 20) (.07) (.34)
1.07
(50(.5) + P - 20 - 40) (.07) (.34)
(1.07)2
P = $98.45
32
Example 4
Expenses of $10 paid two years before the policy is
effective and $10 when the policy is effective
2)-10)(.34)
(P-(10(1.07)
40
40
2 +10 +
P=
+
+
10
(1.07)
(1.07)
1.04 (1.04)2
-
40
(40 + 1.08 ) (.34)
1.04
-
40
(40 - 1.08 ) (.34)
(1.04)2
(50
+
P
20)
(.07)
(.34)
(50(.5)
+
P
20
40)(.07)(.34)
+
+
1.07
(1.07)2
P = $100.00
33
Example 5
Premiums paid (not just received by the insurer) one
month after policy is effective
P
40 + 40 + 10 (1.07)2 +10 + (P-10 (1.07)2-10)(.34)
=
(1.07)1/12
1.04
(1.04)2
(1.07)
-
40
(40 + 1.08 ) (.34)
1.04
-
+
(50 + P - 20) (.07) (.34)
1.07
+
(40 -
40
)
(.34)
1.08
(1.04)2
(50(.5) + P - 20 - 40)(.07)(.34)
(1.07)2
P = $100.95
34
Additional Adjustments Possible
• Some expenses proportional to premiums
• Other risk-adjusted discount rates for
prepaid expenses and premium lags
• Different tax rates on underwriting and
investment income
• Mid-period cash flows
• Future renewals on book of business
35
Problems With Risk-Adjusted
Discount Technique
• Determining risk-adjusted discount rate
• Surplus allocation
– How much?
– How long?
36