2.4: Risk Measurement and Metrics(Exercises)
- Page ID
- 24475
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- The Texas Department of Insurance publishes data on all the
insurance claims closed during a given year. For the thirteen years
from 1990 to 2002 the following table lists the percentage of
medical malpractice claims closed in each year for which the injury
actually occurred in the same year.
Year % of injuries in the year that are closed in that year 1990 0.32 1991 1.33 1992 0.86 1993 0.54 1994 0.69 1995 0.74 1996 0.76 1997 1.39 1998 1.43 1999 0.55 2000 0.66 2001 0.72 2002 1.06 Calculate the average percentage of claims that close in the same year as the injury occurs.
- From the same Texas Department of Insurance data on closed
claims for medical malpractice liability insurance referred to in
Problem 1, we can estimate the number of claims in each year of
injury that will be closed in the next 16 years. We obtain the
following data. Here the estimated dollars per claim for each year
have been adjusted to 2007 dollars to account for inflation, so the
values are all compatible. Texas was said to have had a “medical
malpractice liability crisis” starting in about 1998 and continuing
until the legislature passed tort reforms effective in September
2003, which put caps on certain noneconomic damage awards. During
this period premiums increased greatly and doctors left high-risk
specialties such as emergency room service and delivering babies,
and left high-risk geographical areas as well causing shortages in
doctors in certain locations. The data from 1994 until 2001 is the
following:
Injury year Estimated # claims Estimated $ per claim 1994 1021 $415,326.26 1995 1087 $448,871.57 1996 1184 $477,333.66 1997 1291 $490,215.19 1998 1191 $516,696.63 1999 1098 $587,233.93 2000 1055 $536,983.82 2001 1110 $403,504.39 - Calculate the mean or average number of claims per year for medical malpractice insurance in Texas over the four-year period 1994–1997.
- Calculate the mean or average number of claims per year for medical malpractice insurance in Texas over the four-year period 1998–2001.
- Calculate the mean or average dollar value per claim per year for medical malpractice insurance in Texas over the four-year period 1994–1997 (in 2009 dollars).
- Calculate the mean or average dollar value per claim per year for medical malpractice insurance in Texas over the four-year period 1998–2001 (in 2009 dollars).
- Looking at your results from (a) to (e), do you think there is any evidence to support the conclusion that costs were rising for insurers, justifying the rise in premiums?
- Referring back to the Texas Department of Insurance data on
closed claims for medical malpractice liability insurance presented
in Problem 5, we wish to see if medical malpractice was more risky
to the insurer during the 1998–2001 period than it was in the
1994–1997 period. The data from 1994 until 2001 was:
Injury year Estimated # claims Estimated $ per claim 1994 1021 $415,326.26 1995 1087 $448,871.57 1996 1184 $477,333.66 1997 1291 $490,215.19 1998 1191 $516,696.63 1999 1098 $587,233.93 2000 1055 $536,983.82 2001 1110 $403,504.39 - Calculate the standard deviation in the estimated payment per claim for medical malpractice insurance in Texas over the four-year period 1994–1997.
- Calculate the standard deviation in the estimated payment per claim for medical malpractice insurance in Texas over the four-year period 1998–2001.
- Which time period was more risky (in terms of the standard deviation in payments per claim)?
- Using the results of (c) above, do you think the medical malpractice insurers raising rates during the period 1998–2001 was justified on the basis of assuming additional risk?