Management application the group project report

After collecting the data, our group calculated the retums of these assets, analyzed the correlation coefficient from various perspectives, and computed the portfolio mean and variance f

THE STATE BANK MINISTRY OF EDUCATION

HOCHIMINH UNIVERSITY OF BANKING

:

HOCHIMINH UNIVERSITY OF BANKING

MANAGEMENT APPLICATION THE GROUP PROJECT REPORT

Major / Specialization: Insurance — Finance — Banking

Trinh Trac Quynh - 110102210081 Khưu Nguyễn Khai Tri - 110102210036 Tran Khanh Linh - 110102210066

Võ Nguyễn Quý Tiên - 110102210031 Nguyễn Trần Hải Anh - 120603210006 Trần Thị Hồng Nhung - 120603210129

Hỗ Tiến Dương - 110320210038 Nguyễn Như Phát - 120603210132

Lê Trần Minh Thư - 120603210161

Lecturer: Ha Binh Minh

Ho Chi Minh City, November - 2024

Table of Contents

1H PORTEOLIO MEANS AND VARIANCES OE N=2 ASSETS che 5

1V PORFORLIO MEANS AND VARIANCES OF N=4 ASSETS Hi 7

L OVERVIEW

In this chapter, we delve into the fundamental principles of portfolio calculations, beginning with a straightforward example involving two assets, they are stocks: VINAMILK (VNM) and VINGROUP (VIC) We gathered historical data on their closing prices for the three most recent months from 6/8/2024 to 5/11/2024 After collecting the data, our group calculated the retums of these assets, analyzed the correlation coefficient from various perspectives, and computed the portfolio mean and variance for this pair of assets Subsequently, we experimented with the case of N assets (GAS and VCB); everything becomes convenient when using matrix notation and exploiting Excel's matrix handling capabilities

PetroVietnam Gas Corporation (GAS, PV Gas) is a Vietnam National Oil and Gas Group member unit PV Gas mainly operates in gathering, transporting, storing, processing, exporting, importing, and trading gas and gas products Its stock code is GAS Vietcombank

is the largest joint-stock commercial bank in Vietnam, with nearly 23,000 employees (2023 December), and offers a wide range of products and services, from deposit to lending activities with a network of branches and transaction offices spanning 50 of the 63 provinces across the country, the stock code is VCB Vinamilk, formally the Vietnam Dairy Products Joint Stock Company, is the largest dairy company in Vietnam, and the stock code is VNM The largest conglomerate in Vietnam, Vingroup Joint Stock Company, concentrates on

technology, industrial, real estate development, retail, and healthcare services The largest

conglomerate in Vietnam, Vingroup Joint Stock Company, concentrates on technology,

industrial, real estate development, retail, and healthcare services, and the stock code is VIC

In this section, we compute the return statistics for VINAMILK (stock symbol VNM) and VINGROUP (VIC) Here is the price and the return data

ILI Price and returns for VNM and VIC

These data give the closing price of two stocks from 6" August 2024 to 5" November 2024

and our group calculated the return based on the formula:

t

Hine

At the top section of the Excel sheet, we focus on calculating the return statistics for each

stock The monthly return reflects the percentage return an investor would earn by buying the stock at the end of a specific month (t-1) and selling it at the end of the next month

Figure 1 Historical data and Price and Returns For VNM and VIC Stock

We assume that historical performance provides valuable insights into future return patterns Based on this, the average of past data is treated as an estimate of the expected monthly retum for each stock Similarly, the historical data is used to infer the variance of potential

future returns We used the AVERAGE, VAR.S, and STDEV.S functions to perform these

calculations in Excel

The results are presented in the table below:

PRICE AND RETURNS FOR VINAMILK AND VINGROUP

IN 3 MONTHS FROM 6/8/2024- 5/11/2024

Figure 2 Price and Return

Firstly, the monthly mean retum of the two stocks represents the percentage profit or loss an

investor would achieve The monthly return for VNM is -0.166%, and for VIC, it is 0.000%,

calculated using the AVERAGE function for each stock's historical data Secondly, the monthly variance was computed using the VAR.S function, and from these results, the monthly standard deviation for each stock was determined

Using the monthly figures as a basis, the annual values were calculated by multiplying the monthly figures by 12 months

IL2 Covariance and Correlation

Next, our group wants to calculate the covariance of the returns and present it in the table

below: The column Return minus average contains the return of VNM, VIC, and the

Product First is the return minus the average of VNM and VIC, which we calculated by using the stock return of each transaction minus the monthly mean of each stock Secondly, the Product column contains the multiple of returns minus the average of VNM and VIC

COMPUTING COVARIANCE AND CORRELATION

Figure 3 Covariance and Correlation of VNM and VIC Stocks

Next, our group defined the Covariance and Correlation as follows:

Covariance

-D5:D67)

2.3 The different view of the Correlation Coefficient

Another way to look at the correlation coefficient is to graph the VNM and VIC retums on the same axes and then use the Excel Trendline facility to regress the returns of VNM on those of VIC so our group has the graph below:

Graphing VIC (y-axis) against VNM (x-axis)

y = 0.1438x + 0.0002 R* = 0.0132

Figure 4 Trendlines of VNM and VICStocks

II PORTFOLIO MEANS AND VARIANCES OF N=2 ASSETS

During this part, we calculate the portfolio of VIC and VNM mean and variance based on the

last 3 months This portfolio contains 50% of each share, 50% of VIC, and 50% of VNM

Therefore, before calculating the returns, we need to add this information first

3.1 Means and Standard Deviation

CALCULATING THE MEANS AND STANDARD DEVIATIONS OF PORTFOLIO

Portfolio of VNM| 0.5|

Portfolio of VIC_ | 0.5|

To ensure accuracy for future use, we will calculate B3 by taking | minus B2 to calculate the exact percentage After we determine the portion of our portfolio, what will be the mean and variance of this portfolio? It is worth doing the brute force calculations at least once in Excel:

Figure 5 Means and Standard Deviation of a Portfolio

The mean portfolio return is exactly the average of the mean returns of the two assets:

It is calculated by multiplying the portfolio of VNM by VNM’s retum and then plus

with portfolio of VIC by VIC’s return Or we can write in the formula:

Expected portfolio return = E(rp) = 0.5E(rvnw) + 0.5E(rvic) 3.2 Porforlio Returns

In general, the mean return of the portfolio is the weighted average return of the component

stocks

Mean return, variance return, standard deviation, and covariance are calculated by using data

from each table for each stock, with the functions or Average, VAR.S, and STDEV.S Then

the data from the table also be applied to calculate the portfolio mean return, portfolio return variance, and portfolio retum standard deviation For these data, there are two ways to measure which are demonstrated in the figure above

3.3 Efficient frontier of the portfolio

A frequently performed exercise is to plot the means and standard deviations for

various portfolio proportions x To do this we built a table using Excel’s Data|What-If|

DataTable:

Portfolio mean return

-0.08%|=AVERAGE(D6:D68) -0.08%]=B2*G6+B3*H6

Portfolio return variance

0.0000896994/=V AR.S(D6:D68) 0.0000§96994|=B2^2*G7+B3ˆ2*H7+2*B2*B3*G9

Portfolio return standard đeviation

0.0094710|=SQRT(G16) 0.0094710|=STDEV.S(D6:D68)

0.0094710=STDEV.S(D6D68§)

Portfolio E(r,) and Ø, 0.10%

°

> °

sf 005% “

sy eo

E 0.00% °

& — o.00p00% 0.50000% 100000%, © 1.S0000% 2 00000% 2.50000%

c -0.05% ° zs Value) Axis

s x Horizontal (Value) Axis

3 Se

Ễ - °

E -010% °

2 °

£ °

§ 015%

® 0.20% °

° ° 0.25% °

Portfolio standard deviation p

a7 n 02a

Figure 6 Efficient frontier of the portfolio (N=2)

IV PORFORLIO MEANS AND VARIANCES OF N=4 ASSETS

4.1 Formula

This section calculates the mean and variance of a specific number of assets, which is different from the previous section The expected return of the portfolio whose proportions are given by X is the weighted average of the expected returns of the individual assets:

E(Œ.)E > x,Elri

which, in matrix notation, can be written as E(ry}= È_ x,Efr,) = xTE(r) = E(r)'x

i=1

The portfolio’s variance is given by:

N N N Var(r)= È› (x¿¿ï)f Var(r,)¿+2 2) Ð; x,.x,Cov(x,,x,)

i=1 i=1 j=i+l

Each asset’s variance appears once, multiplied by the square of the asset's proportion in

the portfolio; the covariance of each pair of assets appears once, multiplied by twice the

product of the individual assets’ proportions Another way of writing the variance is to

use the notation: Var(ri)=%i , COV(Iij)= Øụ

4.2 Covariance Between Two Portfolios

In Excel formulas, this is the array function MMult(MMult(x,S),Transpose(y))

Portfolio Calculations Using Matrices—An Example: Four risky assets have the

following expected returns and variance-covariance matrix:

A FOUR-ASSET PORTFOLIO PROBLEM

Mean Variance-covariance, S returns

E(r) 0.000070 0.000007 0.000014 0.000016 0.0970%| {=TRANSPOSE(N = 4'1S3:V3)} 0.000007 0.000273 0.000051 -0.000001 -0.1522%

0.000014 0.000051 0.000125 0.000018 -0.1592%

0.000016 -0.000001 0.000018 0.000202 0.0098%

We are calculating the two portfolios’ means, variances, and covariance We use the

function MMult for the matrix multiplications and the array function Transpose

Portfolio x and y statistics: Mean, variance, covariance, correlation

Covariance (xy) 0.0000345 {=MMULT(MMULT(@8:E3,A3:D6), TRANSPOSE(B9:E9))}

0.0000463|{=MMULT(MMULT(9:E9.A3:D6),TRANSE

We can now calculate the standard deviation and return of combinations of portfolios x

and y We have calculated the means, variances, and the covariance of the returns of

the two portfolios in the table below:

Proportion of x 03

Portfolio vaianes Portioko standard deviation 0, 00000551ƑP18018134(1818/20E13-01181/.818/814 0.743067} =30RTB20)

Table of returns (using Data Table)

Figure 7: Efficient frontier of the portfolio (N=4)

V ENVELOPE PORTFOLIO

An envelope portfolio is a portfolio of risky assets that gives the lowest variance of return of all portfolios having the same expected return An efficient portfolio is a portfolio that gives the highest expected return of all portfolios having the same variance Mathematically, we may define an envelope portfolio as follows: For a given return « = E(",), an efficient portfolio p = [X,, X, , Xy | is one that solves

To show that the concepts of envelope and efficient portfolio are non-trivial, we show that the two portfolios whose combinations are graphed in the previous examples are not either envelope or efficient This is easy to see if we extend the data table to include numbers for the individual stocks: We can see that stock VNM and stock VCB fall outside the frontier created by combinations of portfolios x and y Thus x and y cannot be efficient portfolios

0.10% NN .°

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= `

E °

5 °

a © 0.10% °

3 £ "6 a

2

a 0.20% <i e

x "

£ 0.30% °%

=

S °

0.40%

0.0% 0.5% 1.0% 1.3% 2.0% — 2.5% 3.0 5% 4.0%

ma-e£^f^ =e¬=2^rd deviation

Figure 7 Envelope Portfolio

VI CONCLUSION

In conclusion, all four stocks are worth investing in However, we have to consider the proportion before investing in the portfolio Excel can also be a handy tool for portfolio calculations and offers a range of benefits It provides a flexible and accessible platform that can handle a variety of financial calculations Excel’s built-in functions and formulas make

calculating portfolio returns, variances, standard deviations, and correlations easy Moreover,

Excel’s data visualization tools can help understand a portfolio's risk and return trade-off and

assist in making informed investment decisions However, while Excel is a powerful tool, it

is important to remember that the accuracy of the calculations depends on the quality of the input data Therefore, investors should ensure they use reliable and up-to-date data when using Excel for portfolio calculations

VIL REFERENCES

Benninga, S (2014) FINANCIAL MODELING (4 ed.) The MIT Press