# Miami University Probabilistic Risk Measures Questions

Important General Instructions for Reporting Numerical Answers:

– Do not round intermediate calculations.

– Unless otherwise instructed, solve problems in the given units. IE: if given units are in \$K, complete your computations in these units and, as applicable, report your answer in these units (without writing “\$” or “K”).

– Do not report any numerical answer as a percent. IE: for example, write

– Report negative numbers with a leading minus sign, like this (for example):

-23.451

Not like this: (23.451).

– Note that Canvas removes trailing insignificant figures. If you type, for example, 41.350, Canvas will remove the last decimal place and record your answer as 41.35 This is fine because 41.35 = 41.350.

– Assume time is measured in years unless otherwise stated.

Consider this information for questions 1 — 4:

Here is your boss’ unfinished analysis of three zero coupon bonds, as shown in the above table. Your boss assumes CF1 of each bond is Bernoulli-distributed.

Q1:

What is the expected value return of bond 1? Compute your answer to four decimal places and do not represent your answer as a percent.

Q2:

What is the probability of the expected worst-case CF1 for bond 2? Compute your answer to four decimal places and do not represent your number as a percent.

Q3:

What is the expected worst-case CF1 for bond 3? Compute your answer to 1 decimal place.

Q4:

Using the awesome power of probability and statistics, which is the best way to characterize bond 3?

A) We believe that the bond’s price (CFo) and estimated return indicate that it is a Treasury bond.
C) We expect bonds of this type to provide a CF1 of \$100 about 99% of the time. However, we believe that, about 1% of the time this class of bonds to do worse. So, on average for this type of investment we expect a CF1 of about \$99.85, and we further expect that on average, investing in bonds like this will produce a 5.1% return.
B) Our view is that, because the bond has a low expected-value return, it is a poor investment prospect.
E) Because past performance does not guarantee future results, it is imprudent to say anything about the potential performance of this bond.
D) We believe that the market price (CFo) of this bond (and others like it) is too high. We therefore recommend not purchasing this type of bond until market conditions become more favorable.

Use this info for the next three questions.

Consider these historical annual returns for the SPSM small-cap ETF. (Google it as needed).

 Date Return 12/31/99 -22.180% 12/31/00 0.270% 12/31/01 4.190% 12/31/02 38.310% 12/31/03 16.520% 12/31/04 25.600% 12/31/05 27.330% 12/31/06 31.530% 12/31/07 21.430% 12/31/08 10.500% 12/31/09 -0.790% 12/31/10 14.120% 12/31/11 -16.340% 12/31/12 9.810% 12/31/13 14.830% 12/31/14 -2.430% 12/31/15 -2.670% 12/31/16 -38.250% 12/31/17 12.500% 12/31/18 32.650%

Q5:

What is the average return? Compute your answer to three decimal places.

Q6:

What is the standard deviation of the returns? (Hint, use sample stdev formula not the population formula. Recall this from your Stats class or google it). Compute your answer to three decimal places.

Q7：

Assume you’ve made a relative-frequency distribution graph of the above returns, which you believe enables this data to be approximated with (modeled by) a Normal Density function. To use this Normal Density function to model future SPSM returns, which big assumption is most important?

B) SPSM’s future returns, over the long haul, will look like its historical returns.
D) Inflation we be contained during the forecast window.
A) There will be no periods of recession in the forecasting window.
C) World peace will not be declared anytime soon.

Q8:

Given a set of annual historical returns that are well approximated by the normal disribution, with expected value 10% and standard deviation 20%. Which statement(s) are true?

A) The VaR return will be achieved about 10% of the time.
B) The VaR return will be achieved about 5% of the time.
E) A and D
F) B and D
D) On average, over the long haul, the VaR return or something worse will occur about once in 10 years.
C) On average, over the long haul, the VaR return or something worse will occur about once in 20 years.

Consider this normal distribution graph and the information below for the next two questions:

You have analyzed 30 years of historical, annual returns of Vanguard’s VIOO ETF. VIOO aims to follow the performance of the S&P 600 Small-Cap stock index. (The S&P 500 is a large-cap index). You found the average return = 10% and the standard deviation of returns = 20%. You plan to model future VIOO returns (for better or worse) as normally distributed, based on your historical analysis.

Q9:

What is the probability that, in any given future year, the VIOO return will be 30% or better? Round your answer to three decimal places.

Q10:

What is the probability that, in any given future year, the VIOO return will be -10% or worse? Round your answer to three decimal places.