Introduction
Here within lies the assessment of a hypothetical portfolio of assets held by an
investment trust. The portfolio contains long equity positions in stocks listed on the
London Stock Exchange and a cash position earning annually, the fixed interest rate of
1.0%. The present values of the cash holdings from the inception date were calculated
using the formula in the Appendix section. The equities contained within the portfolio are
of the following public limited companies:
1. Greggs PLC
2. Tate & Lyle PLC
3. Persimmon PLC
4. Barratt Developments PLC
5. Bellway PLC
6. Unilever PLC
7. Associated British Foods PLC
8. Royal Mail PLC
9. Taylor Wimpey PLC
10. Imperial Brands PLC
A public limited company in one with a minimum share capital of £50,000 traded on an
exchange by the law of the United Kingdom. According to the United Kingdom
Companies Act 2006 c. 46 Part 1 a limited public company is “limited by shares or
limited by guarantee”. This means that shareholders do not have legal responsibility for
the debts of the business, however, it is the business that bares the risks of all its debts.
Being share (equities) it infers that units of ownership of the business are sold on the
stock exchange. Because they are sold publicly and traded daily, each equity’s share
price is subject to public sentiment as well as economic factors which we shall discuss
(Knuppe, 2023).
Methodology
Portfolio Fund and Risk Assessment
It should be noted that, although the portfolio’s inception date is the 1
January 2021, returns of assets traded on the London Stock Exchange
cannot be assessed from the aforementioned date but only from the 4
January 2021, because the London Stock Exchange is closed on New
Year’s
Day
and
only
opens
after
the
holiday.
Visit
https://www.londonstockexchange.com/equities-trading/business-days for
more information. Furthermore, the writer makes certain assumptions about
the previous portfolio manager’s decision. For instance that the portfolio
manager bought at or near to the open price on the 4th of January 2021.
Please make notice of the Table 6 in the Appendix Section, it describes the
portfolio by weightings and includes the Purchasing Price and Selling Price
on the 29th of September 2023. The selling Price is assumed to be Closing
Price of the asset on the 29th September.
The Portfolio’s Performance
Assessing the performance of the portfolio was a difficult task for various
reason. Firstly, because the performance measures have to be related.
Secondly, because, the chosen metrics of evaluation have to be within the
competencies of the new portfolio manager as it is essential for him to
effectively communicate his findings. Lastly, the aim is to measure the
portfolio without over complicating an easily communicated concept with
too much mathematical and financial jargon.
The chosen metrics of evaluation were, the portfolios’ return, the portfolio’s
Sharpe Ratio, the portfolio’s beta and the Treynor ratio. The Sharpe Ratio
was chosen because it is an industry standard measure for determining the
return in excess of the risk-free return and relative to the volatility of the
portfolio measured as the standard deviation of the portfolio (Hardy &
Pagdin, 2018). Formulas for calculating these measures are in the
Appendix section.
Portfolio Rate of Return
The aim of the portfolio is to achieve capital growth and maximize long term
capital returns. Over the investment period the fund has underperformed.
Firstly, the fund made losses and the annualized daily return on the whole
portfolio is -0.58%, calculated as the portfolio weighted return. This is done
basically, by obtaining the weightings of each asset within the portfolio
multiplying them respectively by their annualized daily returns based on
their adjusted closing prices. It should also be noted that the portfolio
began with a value of £559514800.00 and on the final date was worth
£461596800.00, a loss of £97918000.
Secondly, the Sharpe ratio calculated for the portfolio over the investment
horizon is -0.28276%. This makes the fund an inferior investment option to
other investment vehicles and unworthy for a rational investor as other
funds that track the same performance benchmark have manage to have
higher and positive risk-adjusted returns. Such funds include CT FTSE
All-Share Tracker Fund 2 Acc GB0033138131 and ETF with a Sharpe ratio
of +0.91, the Vanguard FTSE U.K. All Share Index Unit Trust GBP Acc with
a Sharpe ratio of +0.81 and SPDR® FTSE UK All Share UCITS ETF Acc
ETF with a Sharpe ratio of +0.79. The mentioned Sharpe ratio were
obtained from the Financial Times and snapshots of this information is
included in the Appendix.
Thirdly, the fund’s calculated Treynor ratio is -4.33958 with a Beta of
1.134941162. This infers that the fund has terrible risk-adjusted returns and
is relatively riskier than the market. Making it an irrational choice as an
investment, that is, by being risky and producing no capital gains over the
investment period.
Finally, the fund’s aim is to earn a return above the FTSE ALL Share Index.
If that were so it would make it a better investment than a portfolio that
mimics precisely the stock portfolio of the index. The index over the
investment period earned an annualized interest of 4.764977% and the
fund’s return is way much less.
The Portfolio’s Risk
To determine the risk of the portfolio the chosen measures of volatility were
the portfolio’s standard deviation and the portfolios beta. The market risk or
systematic risk of an asset represents how the overall market impacts the
asset. It can be estimated by calculating what is called the Beta (β) of the
asset. The Beta of the asset is calculated by first obtaining the covariance
of the asset to the overall market and dividing it by the variance of the
overall market. It represents the relationship of the asset to the market in
proportion to the variability of the overall market’s returns. Formulas for
calculating the covariance and Beta are included in the Appendix. The most
desirably beta is one less than that of the market and the markets beta is
always 1. For the portfolio given, the overall market that the assets were
benchmarked to is the FTSE ALL Share Index. The portfolio’s beta is
calculated by summing the product of the individual asset’s betas and their
weightings withing the portfolio.
The stock portfolio was calculated using the multi asset standard deviation
formula for calculating the risk of a portfolio with multiple assets. This
formula is disclosed in the Appendix. It was determined that the entire
portfolio has an annualized standard deviation of approximately 17.506%.
This metric represents the volatility of the portfolio and hence it’s risk. It
should be noted that the portfolio of equities excluding the cash holding has
a standard deviation of 19.2565%, this is because the cash asset is not
correlated with the market and hence has no market risk. It should also be
noted that because of diversification the stock portfolio’s risk is less than
that of each asset’s individual standard deviation. This level of risk is way
above that of the FTSE ALL Share Index that has a standard deviation of
13.688%.
A contributor to the riskiness and underperformance of the fund is the
purchase and over weighting of risky assets that are correlated. This
defeats the purpose of diversification. It becomes just like buying one risky
asset that is as risky or more risky that the overall market. Assets like
Barratt Developments PLC, Persimmon PLC, Greggs and Bellway have the
highest betas in the portfolio as well as the highest correlations with one
another as seen in the correlation matrix in the Appendix. For example
Persimmon has a correlation of 0.867349 with Barratt Development and
Barrat Development PLC has a correlation with Bellway of 0.866067.
Recommendations
It should be noted that the portfolio underperformed with a loss of -0.58%.
This may be only due to the fact that the assets in the portfolio are not of
the necessary weightings for the achievement of minimized risk and a
desired target return above the expected return of the FTSE ALL Share
Index. In other words, the portfolio is not optimized with a target return
above that of the FTSE ALL Share index. The author of this report also
recommends that the portfolio should be rebalanced with a target return
above the risk free rate of the 10 year UK Gilts of 4.37%. Buying or selling
assets within the portfolio to achieve the weighting as suggested in the
table in the Appendix. The target return rate of 12% was chosen humbly,
firstly, as it is above the yield of the risk free rate and the expected return of
the market and also because the target is within one standard deviation of
the market’s daily annualized returns of 13.688%. It should be noted that
the systematic risk is not diversifiable.
Appendix
Optimized Portfolio Weight Allocation
Tables
Table 1: Portfolio Returns
Portfolio
Weightings(w)
0.065879831
0.085900793
Annualized Daily
Returns(Ri)
0.211997189
0.044911973
Persimmon PLC
Barratt Developments PLC
Bellway PLC
0.202965642
0.195544778
0.189417918
-0.174968391
-0.016252083
0.004569973
w * Ri
0.013966339
0.003857974
-0.03551257
2
-0.00317801
0.000865635
Unilever PLC
Associated British Foods
PLC
Royal Mail PLC
Taylor Wimpey PLC
Imperial Brands PLC
Cash (GB£)
0.119221761
0.021016133
0.00250558
0.020339459
0.004534824
0.054707426
0.083350675
0.001072379
0.024618982
0.006376213
0.008197843
0.131427412
0.01
0.000500737
2.8915E-05
0.000448483
0.010954564
1.07238E-05
Asset
Greggs PLC
Tate & Lyle PLC
Σ(𝑤 * 𝑅𝑖)
Portfolio Return
Table 2: Correlation Matrix of the Assets
-0.005551631
(w *Ri) %
-0.55516314
5
Table 3: Variance Covariance Matrix (Excluding Cash)
Table 4: Variance Covariance Matrix (Inclusive of Cash)
Table 5: Portfolio Beta
Asset
Greggs PLC
Tate & Lyle PLC
Persimmon PLC
Barratt Developments PLC
Bellway PLC
Unilever PLC
Associated British Foods
PLC
Royal Mail PLC
Taylor Wimpey PLC
Imperial Brands PLC
Cash (GB£)
Portfolio Weightings
Beta B
0.065879831
1.324179615
0.085900793
0.711306411
0.202965642
1.330261072
0.195544778
0.189417918
1.324366992
1.289225291
0.119221761
0.395848052
0.020339459
1.135886253
0.004534824
0.054707426
1.130898878
1.422600834
0.083350675
0.001072379
0.72198174
0
Wi * B
0.08723672
9
0.06110178
5
0.26999729
2
0.25897304
9
0.24420237
0.04719370
2
0.02310331
2
0.00512842
7
0.07782683
0.06017766
6
0
Portfolio
Beta
1.13494116
2
Table 6: The Portfolio’s Assets Their Weightings, Market Values at the Inception Date
and Final Date
Asset
EPIC
Size of
holding
Beginnin
g Market
Price(£)
Beginnin
g Market
Value(£)
Ending
Price
(£)
Ending
Market
Value(£)
Weightin
g (%)
GRG
20000
1843.00
36860000.
00
685.50
13710000.0
0
6.587851
Name
Greggs
PLC
Tate & Lyle
PLC
Persimmo
n PLC
Barratt
Developm
ents PLC
Bellway
PLC
Unilever
PLC
Associated
British
Foods
PLC
Taylor
Wimpey
PLC
Royal Mail
PLC
Internation
al
Distributio
n Services
PLC
Imperial
Brands
PLC
TAT
E
PSN
BDE
V
BWY
ULV
R
ABF
TW
801.03
48061800.0
0
801.03
48061800.0
0
8.589907
2839.00
113560000.
00
2839.0
0
113560000.
00
20.29616
683.80
109408000.
00
441.10
70576000.0
0
19.55409
3028.00
105980000.
00
2284.0
0
79940000.0
0
18.94141
4447.00
66705000.0
0
4062.0
60930000.0
0
11.92193
66
2276.00
11380000.0
0
2068.0
0
10340000.0
0
169.15
15223500.0
0
117.30
10557000.0
0
2.72084
340.10
5101500.00
3912000.00
0.911772
50010000.0
0
8.3349
60000
40000
160000
35000
15000
2.0339
05
5000
90000
260.80
RMG
(IDS)
15000
1554.50
IMB
30000
46635000.0
0
1667.00
Cash
(GB£)
£588764.
10
1.0%PA
TOTAL
Portfolio Beta
(B)
1.134941
Treynor Ratio
Portfolio Return
%(Rx)
-0.55516
(Rx-Rf)/B
-4.33958
600000.00
611342.47
559514800.
00
461596800.
00
10 yr UK Gilt yield %(Rf)
4.37
0.107236
This extract from the Financial Times of the Portfolio includes taxes and transaction
charges.https://www.londonstockexchange.com/raise-finance/equity/how-list-equity/calc
ulating-fees
Table 7: The Portfolio’s Assets Volatilities
Asset
Greggs
PLC
Tate &
Lyle
PLC
Persim
mon
PLC
Barratt
Develo
pments
PLC
Bellway
PLC
Unileve
r PLC
Standard
Deviation of
Returns
(DAILY)
0.0199205183
0427920000
Daily
Standard
Deviation
of
Returns%
1.99
0.0131294594
1252150000
1.31
0.0216508711
2597730000
2.17
0.0195680275
8679150000
1.96
0.0195752225
3983000000
1.96
0.0120668541
2755670000
1.21
Annualized
Standard
Deviation
of Returns
(%)
31.497105
01
Annualized Daily
Returns
0.211997189254
25100000
20.759498
1
0.044911973182
90410000
4.49120
0.3423303
3
34.233033
04
-0.17496839063
-17.49684
0.3093976
82
0.3095114
45
0.1907937
16
30.939768
25
30.951144
47
19.079371
62
-0.0162521
0.004569973153
09859000
0.021016132551
56800000
-1.62521
Annualized
Standard
Deviation
of Returns
0.3149710
5
0.2075949
81
Annualized
Daily
Returns (%)
21.19972
0.45700
2.10161
Associa
ted
British
Foods
PLC
Royal
Mail
PLC
Taylor
Wimpe
y PLC
Imperial
Brands
PLC
Cash
(GB£)
0.0167221022
9840210000
0.2643996
53
1.67
26.439965
26
0.024618982323
55780000
2.46190
36.042930
89
0.006376212793
47344000
0.63762
0.81978
0.0227955510
2983200000
2.28
0.3604293
09
0.0200705163
3102380000
2.01
0.3173427
27
31.734272
71
0.008197842548
73967000
1.29
0.2046265
12
20.462651
2
100.00
1
100
0.131427412212
76800000
0.010000000000
00000000
0.0129417169
5004250000
1.0000000000
0000000000
Table 8: Calculated Betas
Stock
Beta
Greggs PLC
1.32417961
5
Tate & Lyle
PLC
0.71130641
1
Persimmon
PLC
1.33026107
2
Barratt
Development
s PLC
1.32436699
2
Bellway PLC
1.28922529
1
Unilever PLC
0.39584805
2
13.14274
1.00000
Associated
British Foods
PLC
1.13588625
3
Royal Mail
PLC
1.13089887
8
Taylor
Wimpey PLC
1.42260083
4
Imperial
Brands PLC
0.72198174
Table 9:CAPM Returns
Stock
Greggs PLC
Tate & Lyle
PLC
Persimmon
PLC
Barratt
Development
s PLC
CAPM
Re
0.04893
0.04650
9
0.04895
4
0.04893
1
Bellway PLC
0.04879
2
Unilever PLC
0.04526
4
Associated
British Foods
PLC
0.04818
6
Royal Mail
PLC
Taylor
Wimpey PLC
Imperial
Brands PLC
0.04816
7
0.04931
9
0.04655
2
Table 10
Risk Identified
Systematic Risk
Non Systematic
Risk
Liquidity Risk
Inflation Risk
Interest rate risk
Description
The fund's portfolio has a beta above 1 and therefore a high market risk.
Each firm has its own internal workings. Humans make mistakes and may make
wrong decisions with resources affecting a firm’s financial health and ability to
own and acquire assets. Leading to lower net asset values and lower revenues
and profits.
Some stocks may not be highly regarded by the market though they may have
intrinsic value.
If inflation depletes the businesses purchasing power. Leading to lower profit
margins for the business. Making the businesses a less worthy investment.
If interest rates rise, fixed income instruments become more worthwhile
investments.
Price Charts and Returns Charts
Greggs PLC
Stationarity Test
Tate & Lyle PLC
Persimmon PLC
Barratt Developments PLC
Bellway PLC
Unilever PLC
Associated British Foods
Royal Mail PLC now International Distribution Services PLC
Taylor Wimpey PLC
Imperial Brands PLC
The Funds Portfolio vs FTSE ALL Share Index
Betas Obtained From Financial Times
Formulas
Formula for Present Value of Cash Due to 1.0% Interest
𝑃𝑉(𝑇) = 600000/𝑒
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑒𝑙𝑎𝑝𝑠𝑒𝑑 𝑠𝑖𝑛𝑐𝑒 𝑖𝑛𝑐𝑒𝑝𝑡𝑖𝑜𝑛 𝑑𝑎𝑡𝑒
365
Where e = Euler’s number
Returns
𝑟𝑖 =
𝑝𝑡+1−𝐷𝑖𝑣𝑡+1
𝑝𝑡
–1
𝑝𝑡+1 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑟𝑖𝑐𝑒
𝑝𝑡 = 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑑𝑎𝑦𝑠 𝑝𝑟𝑖𝑐𝑒
𝐷𝑖𝑣𝑡+1 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝑟𝑖 = 𝑑𝑎𝑖𝑙𝑦 𝑟𝑒𝑡𝑢𝑟𝑛
Expected Return (Arithmetic Mean)
𝐸[𝑟𝑥] =
𝑛
1
∑ 𝑟𝑖
𝑛
𝑛=𝑖
Annualized Daily Returns
(1 + 𝑟𝑖 )
250
− 1
Volatility of Stock as Standard Deviation of Daily Returns
σ=
Variance
𝑛
1
∑
𝑛
𝑛=𝑖
(𝑟𝑖 − 𝐸[𝑟𝑥])2
2
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = σ * σ = σ
Formula for Stock Beta
𝑐𝑜𝑣(𝑟 ,𝑟 )
𝑎 𝑚
β𝑖 = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠
𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡
Where 𝑟𝑎 = 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑜𝑐𝑘 𝑎𝑛𝑑
𝑟𝑚 = 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡
Portfolio Beta
∞
β𝑝 = ∑ (𝑤𝑖 * β𝑖)
𝑛=1
𝑤ℎ𝑒𝑟𝑒 𝑤𝑖 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 𝑎𝑛𝑑 β𝑖 𝑏𝑒𝑡𝑎 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘
Correlation of the Assets
ρ𝑖,𝑘 =
𝑐𝑜𝑣(𝑖,𝑗)
σ𝑗*σ𝑘
Formula for Calculating Portfolio Risk
𝑛
σ𝑝 =
2
2
𝑁−1
∑ 𝑤𝑖 σ𝑖 + ∑
𝑖=1
𝑁
∑ 𝑤𝑖𝑤𝑗𝑐𝑜𝑣𝑖,𝑗
𝑖=1 𝑗=𝑖+1
Formula for Sharpe Ratio
𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =
(𝑅𝑖− 𝑅𝑓)
σ
𝑊ℎ𝑒𝑟𝑒 𝑅𝑖 𝑖𝑠 𝑡ℎ𝑒 𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑑 𝑎𝑛𝑑 𝑅𝑓 𝑖𝑠 𝑡ℎ𝑒 𝑅𝑖𝑠𝑘 𝐹𝑟𝑒𝑒 𝑅𝑎𝑡𝑒.
The risk free rate chose in the analysis is that of the yield of the 10 year UK gilt of
4.37%.
Treynor ratio
𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑅𝑎𝑡𝑖𝑜 =
(𝑅𝑖− 𝑅𝑓)
β𝑝
𝑊ℎ𝑒𝑟𝑒 𝑅𝑖 𝑖𝑠 𝑡ℎ𝑒 𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑑 𝑎𝑛𝑑 𝑅𝑓 𝑖𝑠 𝑡ℎ𝑒 𝑅𝑖𝑠𝑘 𝐹𝑟𝑒𝑒 𝑅𝑎𝑡𝑒.
'
β𝑝 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑠 𝑏𝑒𝑡𝑎.
Capital Asset Pricing Model
[ ]
[ ]
𝐸 𝑟𝑒 = 𝑟𝑓 + β(𝐸 𝑟𝑚 − 𝑟𝑓)
Funds That Track the FTSE ALL Share Index
By visually inspecting the line above it can observe that over the past 5 years the fund
was correlated with the market.
References
Figure
1.
Companies
Act
2006,
Part
https://www.legislation.gov.uk/ukpga/2006/46/section/4
1
Section
4
Mondello, E., (2023). Applied Fundamentals in Finance. Springer
Pagdin, I., Hardy, M., (2018). Investment and Portfolio Management. Kogan
Page Limited
Retrieved November 8, 2023, from
https://www.mollie.com/growth/what-is-a-public-limited-company
Retrieved November 8, 2023, from
https://www.legislation.gov.uk/ukpga/2006/46/pdfs/ukpga_20060046_en.pdf
Soros, G., (2003). The Alchemy of Finance. John Wiley & Sons
The Financial Times (2023, November 8) https://www.ft.com/.
World Health Organization (2020, May 30). Youth advocate in Kenya’s
tobacco
control
drive.
https://www.legislation.gov.uk/ukpga/2006/46/pdfs/ukpga_20060046_en.pdf