Calculation of Mean, Standard Deviation and Efficient Portfolio

This report provides a detailed analysis of the calculation of arithmetic mean, geometric mean, and standard deviation for different asset classes. It also discusses the construction of an efficient portfolio and the risk-return characteristics of each asset.

Investment management

Contents
EXECUTIVE SUMMARY.......................................................................................................................3
MAIN BODY.............................................................................................................................................4
1. Calculation of the Arithmetic Mean (AM), Geometric Mean (GM) and Standard Deviation (σ) of
returns of each of the five asset classes....................................................................................................4
2. Construction of an efficient portfolio.............................................................................................11
3. Discussion on Modern Portfolio Theory (MPT) models................................................................13
CONCLUSION........................................................................................................................................16
REFERENCES........................................................................................................................................17

EXECUTIVE SUMMARY
The report abstracts about different aspects regarding to analysis of given assets. In first
and second question, information about calculation of different kinds of mean, standard deviation
is included in order to assess risk-return features. In the end part of report information related to
various types of modern portfolio theories is mentioned along with impact of diversification is
also summarized.

MAIN BODY
1. Calculation of the Arithmetic Mean (AM), Geometric Mean (GM) and Standard
Deviation (σ) of returns of each of the five asset classes.
Arithmetic mean (μ)- In the context of statistics, term arithmetic mean can be defined as
an average of given data set (Qian, Yang and Chu, 2019). In relation to given information
about return of various five assets calculation of arithmetic mean is done below in such
manner:
Arithmetic mean (μ)= Sum of data/ number of data
Australian
Shares
Australian
Bonds
US Shares US Fed
Funds
Brent Oil
Sum of data 104.90 77.5 153.55 33.04 199.60
Number of data 20 20 20 20 20
Calculation 104.90/20 77.5/20 153.55/20 33.04/20 199.60/20
Arithmetic mean 5.25 3.85 7.68 1.65 9.98
Geometric Mean - This can be understood as a form of mean which typically shows
central tendency of given data set by help of making product of values (Lu, Ma and
Zhang, 2020). In relation to given data set of five assets, calculation of geometric mean is
done below in such manner:
Formula of geometric mean: n√x1.x2.x3……x n
Spread sheet formula: =PRODUCT (A1:A6) ^ (1/COUNT (A1:A6))
Spread sheet formula: =PRODUCT (A1:A6) ^ (1/COUNT (A1:A6))
Australian
Shares
Australian
Bonds
US Shares US Fed
Funds
Brent Oil
Geometric Mean 9.42 3.32 0.69 14.63

Standard deviation: Standard deviation= √ (variance)
Variance: [∑ (x – μ) 2 / N]
Australian Shares’ standard deviation:
Australian Shares (x- μ) (x- μ)²
μ = 5.25
2.8 -2.45 6.00
6.7 1.45 2.10
-12.1 -17.35 301.02
9.7 4.45 19.80
22.8 17.55 308.00
17.6 12.35 152.52
19 13.75 189.06
11.8 6.55 42.90
-41.3 -46.55 2166.90
30.8 25.55 652.80
-2.6 -7.85 61.62
-14.5 -19.75 390.06
14.6 9.35 87.42
15.1 9.85 97.02
1.1 -4.15 17.22
-2.1 -7.35 54.02
7 1.75 3.06
7 1.75 3.06
-6.9 -12.15 147.62
18.4 13.15 172.92
[∑ (x –
μ)2= 4875.17

Variance: 4875.17/20
= 243.76
Standard deviation: √ (243.76)
= 15.61
Australian Bonds:
Australian Bonds (x- μ) (x- μ)²
μ=3.85
6.3 2.45 6.00
4.3 0.45 0.20
4.8 0.95 0.90
5.3 1.45 2.10
5.3 1.45 2.10
5.5 1.65 2.72
6.3 2.45 6.00
6.8 2.95 8.70
4.3 0.45 0.20
3.8 -0.05 0.00
4.8 0.95 0.90
4.3 0.45 0.20
3 -0.85 0.72
2.5 -1.35 1.82
2.5 -1.35 1.82
2 -1.85 3.42
1.5 -2.35 5.52
1.5 -2.35 5.52
1.5 -2.35 5.52
0.75 -3.1 9.61
[∑ (x – μ) 2=

64.02
Variance: 64.02/20
= 3.201
Standard deviation: √ (3.201)
= 1.79
US Shares’ standard deviation:
US Shares (x- μ) (x- μ)²
μ=7.68
-9.1 -16.78 281.57
-11.89 -19.57 382.98
-22.1 -29.78 886.85
28.68 21 441.00
10.88 3.2 10.24
4.91 -2.77 7.67
15.79 8.11 65.77
5.49 -2.19 4.80
-37 -44.68 1996.30
26.46 18.78 352.69
15.06 7.38 54.46
2.11 -5.57 31.02
16 8.32 69.22
32.39 24.71 610.58
13.69 6.01 36.12
1.38 -6.3 39.69
11.96 4.28 18.32
21.83 14.15 200.22
-4.48 -12.16 147.87

31.49 23.81 566.92
[∑ (x – μ)2=
6204.30
Variance: 6404.30/20
= 320.21
Standard deviation: √ (320.21)
= 17.89
US Fed Funds’ standard deviation:
US Fed Funds Rate (x- μ) (x- μ)²
μ=1.65
6.4 4.75 22.56
1.82 0.17 0.03
1.24 -0.41 0.17
0.98 -0.67 0.45
2.16 0.51 0.26
4.16 2.51 6.30
5.24 3.59 12.89
4.24 2.59 6.71
0.16 -1.49 2.22
0.12 -1.53 2.34
0.18 -1.47 2.16
0.07 -1.58 2.50
0.16 -1.49 2.22
0.09 -1.56 2.43
0.12 -1.53 2.34
0.24 -1.41 1.99
0.54 -1.11 1.23

1.3 -0.35 0.12
2.27 0.62 0.38
1.55 -0.1 0.01
[∑ (x – μ) 2=
69.31
Variance: 69.31/20
= 3.46
Standard deviation: √ (3.46)
= 1.86
Brent Oil (USD)’ standard deviation:
Brent Oil (USD) (x- μ) (x- μ)²
μ=9.98
-5.7 -15.68 245.86
-17.4 -27.38 749.66
49.6 39.62 1569.74
-0.1 -10.08 101.61
36.6 26.62 708.62
43.2 33.22 1103.57
-4.1 -14.08 198.25
60.2 50.22 2522.05
-63.1 -73.08 5340.69
81.4 71.42 5100.82
17.9 7.92 62.73
12.8 2.82 7.95
-0.3 -10.28 105.68
-2.7 -12.68 160.78
-48.8 -58.78 3455.09

-33.9 -43.88 1925.45
51.5 41.52 1723.91
21.2 11.22 125.89
-24.1 -34.08 1161.45
25.4 15.42 237.78
[∑ (x – μ) 2=
26607.57
Variance: 26607.57/20
= 1330.38
Standard deviation: √ (1330.38)
= 36.47
Overall standard deviation of each asset:
Australian
Shares
Australian
Bonds
US Shares US Fed
Funds
Brent Oil
Standard
deviation
15.61 1.79 17.89 1.86 36.47
Risk return characteristics of each asset:
Australian Shares- In relation to this assets, it can be find out that value of mean is of
5.25 and standard deviation is of 15.61. This shows that value of standard deviation is too
higher which means this investment is highly risky and can produce high return.
Australian Bonds- In relation to this assets, it can be find out that value of mean is of 3.85
and standard deviation is of 1.79. This shows that value of standard deviation is lower as
compared to mean which states this investment is less risky and can produce less return.
US Shares- In relation to this assets, it can be find out that value of mean is of 768 and
standard deviation is of 17.89. This shows that value of standard deviation is higher as

compared to mean which states this investment is highly risky and can produce more
return.
US Fed Funds- From above calculated values, this can be assessed that mean is of 1.65
and SD is of 1.86. there is not a huge gap between these values. Therefore, this asset will
be generating average value of return at moderate risk level.
Brent Oil- In regards to it, the value of mean is of 9.98 and SD is of 36.47 which shows
that this asset will be highly risky and return generating due to huge amount of gap
between these values.
2. Construction of an efficient portfolio.
Portfolio variance 5.787343 5.694239 5.588566 5.755681 5.525636
Portfolio SD 2.40569 2.386261 2.364015 2.3991 2.350667
Portfolio mean 0.005245 0.038526 0.076775 0.01652 0.099801
Variance covariance matrix
Australian
Shares
Australian
Bonds
US
Shares
US Fed
Funds
US Fed
Funds
Australian
Shares 243.7585 1.120138 209.4695 7.40211 322.1304
Australian
Bonds 1.120138 3.200869 -8.81274 1.964945 12.32105
US Shares 209.4695 -8.81274 310.2151 -5.09292 223.39735
US Fed Funds 7.40211 1.964945 -5.09292 3.465736 9.27954
US Fed Funds 322.1304 12.32105 223.3974 9.27954 1330.3786

2.34 2.35 2.36 2.37 2.38 2.39 2.4 2.41
0
0.02
0.04
0.06
0.08
0.1
0.12
0.00524518387399
008
0.03852599811655
46
0.07677499783534
09
0.01652000341003
24
0.09980099528692
04
Efficiecnt frontier
Portfolio SD
Portfolio mean
Portfolio return: w T u= SUMPRODUCT (w, u)
Portfolio variance: w T∑ w= MMULT(TRANSPOSE(w), MMULT (∑, w)
Correlation matrix
Australia
n Shares
Australia
n Bonds
US
Shares
US Fed
Funds
US Fed
Funds
Australian Shares 1 0.040101
0.76174
5 0.25467
0.56567
1
Australian Bonds 0.040101 1
-
0.27967
0.58995
5
0.18881
1
US Shares 0.761745 -0.27967 1
-
0.15532
0.34774
3
US Fed Funds 0.25467 0.589955
-
0.15532 1 0.13666
US Fed Funds 0.565671 0.188811
0.34774
3 0.13666 1

Discussion: In accordance of above measured value of different kinds of aspects of
related to given data of five assets, it can be find out that assets one has standard
deviation of 2.40 which is higher among all given assets. It indicates that under this
assets, there are more risk along with possibility of return is lower. It is so because this
asset has lower value of mean. In terms of mean value, it can be find out that asset five
has higher value which is 0.99 which is a positive indication of return. If investors will
make invest in such asset than there is highly possibility of generating return with a good
pace in upcoming time period. The assets four has moderate level of risk and return
which shows that it will be viable for those investors who want to make a lower
investment with expectation of generating minimum return.
3. Discussion on Modern Portfolio Theory (MPT) models.
Modern portfolio theory is a commonly used investment paradigm developed to help
shareholders reduce market risk while optimizing fund returns (Bodie, Kane & Marcus,
2018). It is an investment philosophy focused on the idea that the stocks are more
accurate and effective than the stakeholders. Investors can select the investments in their
portfolios using modern portfolio theory. In particular, modern portfolio theory promotes
a buy-and - hold approach with periodic readjustment. Asset allocation theories seems
like one of those technologies that failed to achieve their full potential
Practical capacity. Amid the beauty of the initial mean-variance model, statistical
rationality and there seems to be some innate appeal and the abundance of computational
capacity proof that MPT in the market climate is not being sufficiently used.
Markowitz mean-variance model of asset allocation-
Markowitz shows how to reduce the mix of asset groups or shares in a portfolio. Portfolio
risk at the anticipated return level or increase the projected return at the risk stage,
providing the fundamental basis for diversification (Santacruz, L 2016). Provided a
portfolio composed of n properties with wi as the fraction of the overall value of the
portfolio retained.

Equation 1: 𝑟𝑝 = ∑ 𝑤𝑖𝑟𝑖 𝑛 𝑖=1 Equation 2: 𝜎𝑝 2 = ∑ ∑ 𝑤𝑖𝑤𝑗𝜎𝑖.
Modern portfolio theory suggests that the risk and return features of an investment must
not be interpreted in isolation and it should be measured by how the portfolio impacts the
risk - return of the total portfolio. MPT shows a shareholder should create a portfolio of
various properties to optimize the rewards for a given degree of risk.
Limitations and practical difficulties of MPT model:
 Planned returns are usually provided out.
 Risk is calculated by estimated return variances.
 There is a way to reliably calculate the estimated yields and differences of
investment properties, and covariance matrix between them.
 The desired utility of the investor is maximized over a single investment period.
 The shareholder has an expected utility that is quaternion.
Multi period model: MPI model is a pure new investment method that avoids usage,
which defines a shareholder with a purpose of resource utility. The capital utility of a
shareholder w, with n periods on the horizon, is described as:
If returns are believed to be time frame-to-period independent and non - normality, so
applies for a very large class of marginal utility maximization. Convergence means that
the usage for each cycle of the following static, moronic, judgment law is ideal.
Suitable model that is preferred:
In the above part two models have been discussed. Though, there are some other models
too that are used in accordance of nature of investment portfolio (McKay, Shaoiro&
Thomas, 2018). From above mentioned two portfolios, Markowitz mean-variance model

of asset allocation is used mostly by investors in order analyze efficiency of different
kinds of investments. Though, there are some issues under this model but it is suitable for
larger sized assets as well as for assessing risk and return of more than two portfolios.
Diversification failure:
One of the most troubling challenges in investment management is that when
shareholders most need, diversification appears to vanish. We believe that many
shareholders still do not completely understand the effect, especially on risk to loss, of
severe fluctuations on portfolio performance. We look at what determines stock-to-credit,
inventory-to-hedge funds, inventory-to-private cash, inventory-to-risk considerations, and
inventory-to-bond correlations in tail events. We implement a technique of transfer
learning to increase the reliability of measurements of tail association. Lastly, we are
addressing consequences for multi-asset finance.
Impact:
There are many positives of diversifying here, although there might even be certain
downsides (Page & Panariello, 2018). Managing a complex portfolio, particularly if
investors have several holdings and investments, can be quite tedious. Second, it will
place a dent at the rim. Not all asset classes operate the same, but it can be pricey to
purchase and sell — from trading costs to interest charged. And investors can end up
restricting what they show back with because higher risk coincides with huge returns.
CONCLUSION
On the basis of above project report it can be concluded that before making an
investment, this is essential to make proper analysis of various kinds of aspects including
risk and return. In the report different types of five assets are analyzed by help of efficient
frontier. In addition to this, vital range of portfolio theories are useful in order to assess
efficiency of investments.

REFERENCES
Books and journal:
Bodie, Z, Kane, A & Marcus, AJ 2018, Investments , 11th edn, McGraw-Hill Education, New
York (ISBN: 9781259277177).
McKay, S, Shaoiro, R & Thomas, R 2018 , ‘What free lunch? The costs of overdiversification’,
Financial Analysts Journal , vol. 74, no. 1, pp. 44-57
Page, S & Panariello, RA 2018, ‘When diversification fails’, Financial Analysts Journal , vol. 74,
no. 3, pp. 19-32.
Santacruz, L 2016, ‘Asset allocation theory and practice in Australian investment management’,
The Journal of Wealth Management, vol. 19, no. 2, pp. 47-67.
DOI:10.3905/jwm.2016.19.2.047.
Qian, W.M., Yang, Y.Y., Zhang, H.W. and Chu, Y.M., 2019. Optimal two-parameter geometric
and arithmetic mean bounds for the Sándor–Yang mean. Journal of Inequalities and
Applications, 2019(1), pp.1-12.
Lu, X.M., Ma, Z. and Zhang, C., 2020. Generalized-mean Cramér-Rao bounds for
multiparameter quantum metrology. Physical Review A, 101(2), p.022303.