CLA 2 Paper & PPT - Advanced Statistical Concepts and Business Analytics

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Probabilities of Different Classes

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The return of the future regarding the various classes normally distributed and having a known population mean and the standard deviation is well indicated in the table below.

	class of investment	Standard Deviation	return means
1	Fixed annuities	8.310	0.540
2	Equivalent cash	7.730	0.810
3	Treasury Bond of US	8.800	5.980
4	investment Grade of U.S	9.330	7.920
5	Non-US Government Bonds	10.950	10.470
6	Domestic large cap stock	11.710	15.300
7	international Equities	14.020	17.160
8	Domestic Midcap Stocks	13.640	18.190
9	Domestic small -cap stocks	14.930	21.820

1. Finding the probability of the return that is less than Zero

	class of investment	Standard Deviation	mean	Z=	P (Z<=Z1)
1	Fixed annuities	8.310	0.540	-15.390	0.0000
2	Equivalent cash	7.730	0.810	-9.540	0.0000
3	Treasury Bond of US	8.800	5.980	-1.470	0.0706
4	investment Grade of U.S	9.330	7.920	-1.180	0.1194
5	Bonds of Non-US Government	10.950	10.470	-1.000	0.1587
6	stock of Domestic large cap	11.710	15.30	-0.770	0.2220
7	The International Equities	14.020	17.160	-0.820	0.2070
8	Stocks of Domestic Midcap	13.640	18.190	-0.750	0.2267
9	Stocks of Domestic small cap	14.930	21.820	-0.680	0.2483

It is observed that the probability involving occurring of losses is more in the securities with high returns. This is usually a common phenomenon that is kept in the investment market. If there is a high return, it usually arrives with a higher risk, i.e., the standard deviation. The probability of losses occurring is much dependent to a standard deviation. When the standard deviation is high, there is a high probability of failure. This is well justified with the likelihood that we found.

2. a) Probability to a return greater than 5% for the securities

	class of investment	Standard Deviation	return mean	Z=	P(Z<=Z sec)	P(Z>Z1) =1-Pr(Z>Z1)
1	Fixed annuities	8.310	0.540	-6.130	0.000	1.000
2	Equivalent cash	7.730	0.810	-3.370	0.0004	0.9996
3	Treasury Bond of US	8.800	5.980	-0.640	0.2626	0.7324
4	investment Grade to U.S	9.330	7.920	-0.550	0.2928	0.7088
5	Non-US Government Bond	10.950	10.470	-0.570	0.2849	0.7157
6	Domestic large- cap stock	11.710	15.30	-0.440	0.3305	0.6695
7	international Equities	14.020	17.160	-0.530	0.2996	0.7004
8	Domestic Midcap Stocks	13.640	18.190	-0.470	0.3174	0.6826
9	Domestic small cap stocks	14.930	21.820	-0.460	0.3245	0.6755

b)The probability that the returns are more significant than the 10%

	class of investment	Standard Deviation	return mean	Z=	P(Z<=Z1)	P(Z>Z1) =1-Pr(Z>Z1)
1	Fixed annuities	8.310	0.540	3.130	0.9991	0.0009
2	Equivalent cash	7.730	0.810	2.800	0.9975	0.0025
3	Treasury Bond of US	8.800	5.980	0.200	0.5795	0.4206
4	investment Grade of U.S	9.330	7.920	0.080	0.5337	0.4663
5	Non-US Government Bonds	10.950	10.470	-0.090	0.4690	0.5359
6	Domestic large cap stock	11.710	15.30	-0.110	0.4355	0.5398
7	international Equities	14.020	17.160	-0.230	0.4074	0.5926
8	Domestic Midcap Stocks	13.640	18.190	-0.200	-0.4207	0.5793
9	Domestic small cap stocks	14.930	21.820	-0.230	0.4106	0.5910

c) The probability that the returns are more significant than the 20%

	class of investment	Standard Deviation	return mean	Z=	P (Z<=Z sec)	P (Z>Z1) =1-Pr(Z>Z1)
1	Fixed annuities	8.310	0.540	21.650	1.0000	0.0000
2	Equivalent cash	7.730	0.810	15.150	1.0000	0.0000
3	Treasury Bond of US	8.800	5.980	1.870	0.9693	0.0307
4	investment Grade of U.S	9.330	7.920	1.350	0.9115	0.0885
5	Non-US Government Bonds	10.950	10.470	0.860	0.8051	0.1949
6	Domestic large cap stock	11.710	15.30	0.540	0.7054	0.2946
7	international Equities	14.020	17.160	0.350	0.6368	0.3632
8	Domestic Midcap Stocks	13.640	18.190	0.350	0.6368	0.3632
9	Domestic small cap stocks	14.930	21.820	0.230	0.5910	0.4090

d)The probability that the returns are more significant than the 50% of each the security

	class of investment	Standard Deviation	return mean	Z=	P(Z<=Z sec)	P(Z>Z1) =1-Pr(Z>Z1)
1	Fixed annuities	8.310	0.540	77.200	1.0000	0.0000
2	Equivalent cash	7.730	0.810	52.190	1.0000	0.0000
3	Treasury Bond of US	8.800	5.980	6.890	1.0000	0.0000
4	investment Grade of U.S	9.330	7.920	5.140	1.000	0.0000
5	Non-US Government Bonds	10.950	10.470	3.730	0.9999	0.0001
6	Domestic large cap stock	11.710	15.30	2.500	0.9938	0.0062
7	international Equities	14.020	17.160	2.100	0.9821	0.0179
8	Domestic Midcap Stocks	13.640	18.190	2.000	0.9772	0.0228
9	Domestic small cap stocks	14.930	21.820	1.610	0.9452	0.0548

3. Essentially zero and greater than 50%

i. Cash equivalent

ii. Fixed annuities

iii. U.S investment-grade corporate bonds.

iv. U.S treasury Bond

Greater than 20% &5 %

i. Domestic midcap stock

ii. International equities

iii. Domestic small-cap stocks

More than 50% and 5%

i. Stocks of Domestic small-cap

Classes with probability loss is zero

i. Fixed annuities

ii. Cash equivalents.

5. The class probability of loss are more significant than 1%are:

i. U.S Treasury bond,

ii. Domestic small-cap stocks

iii. Domestic midcap stocks

iv. International equities

v. Domestic large-cap stock

vi. Non-U.S government bond

vii. U.S. investment-grade corporate bonds.

Class who’s probability of loss is more than 10%:

i. Domestic large-cap stocks,

ii. Non-U.S government bond,

iii. Domestic midcap stocks

iv. Domestic small-cap stocks

v. U.S investment-grade corporate bonds,

vi. International equities,

Class with probability loss is more significant than 20%,

i. Domestic large-cap stocks

ii. Domestic midcap stocks

iii. Domestic small-cap stocks

iv. International equities

Reference

Bhatt, (2018). New matrix-based methods for the analytic evaluation of the multivariate cumulative normal distribution function. Transportation Research Part B: Methodological, 103