FINANCIAL ANALYSIS REPORT
OPTINOSE INC. (OPTN) & SILICON LABORATORIES INC.
(SLAB) STOCK PORTFOLIO
FINA304-RISK MANAGEMENT
BY SARAH O’HANLON
s.o-hanlon@nup.ac.cy
1
Contents
Summary ............................................................................................................................................ 3
Problem Introduction ........................................................................................................................ 3
Addressed and Unaddressed Decision Questions ................................................................................. 3
Data Overview and Relevance ........................................................................................................... 3
Model Assumptions and Impact of Errors ........................................................................................... 4
Model Critique and Validation ........................................................................................................... 4
Presentation of results ....................................................................................................................... 5
Discussion of Possible Improvement Options and Further Work ........................................................... 5
Modelling Strategy............................................................................................................................... 6
Decision Questions ........................................................................................................................... 6
Available Data ................................................................................................................................. 6
Methods of Addressing Decision Questions ........................................................................................ 6
Different Model assumptions ............................................................................................................. 6
Model selection ................................................................................................................................ 7
MODEL OVERVIEW.......................................................................................................................... 7
Model structure ................................................................................................................................ 7
Key Components .............................................................................................................................. 7
Results Analysis ............................................................................................................................. 10
Validation...................................................................................................................................... 17
References and Data Sets ................................................................................................................ 17
TECHNICAL APPENDICES ............................................................................................................. 18
Equation Derivations ...................................................................................................................... 18
Result Interpretation Guide.............................................................................................................. 19
2
SUMMARY
Problem Introduction
This report explores risk estimation using two simulation methods, Historical Simulation and VarianceCovariance Simulation, for a portfolio of two stocks, OPTN and SLAB, listed on the NASDAQ Stock
Exchange. Daily returns were used for one year, from 20 December 2023 to 20 December 2024. Using
historical simulation and variance-covariance methods, the objective is to evaluate Value at Risk (VaR) at a
99% probability for 1-day, 2-day, and 3-day horizons.
Following this, we shall discuss the risk-return of the stocks and the portfolio and reduce the portfolio's risk
exposure.
Addressed and Unaddressed Decision Questions
The decision questions addressed in this analysis primarily focus on assessing market risk for a two-stock
portfolio using Historical Simulation and Variance-Covariance methods. Specifically, they explore the
market risk calculation at a 99% confidence level over horizons of 1, 2, and 3 days.
Additionally, the analysis requires strategies for reducing the bank's risk exposure as a market risk manager
and comparing the two risk methodologies to identify any discrepancies in results and their underlying
causes.
The data does not include some aspects of risk analysis, such as changes in market dynamics or the
relationships between stocks. The simulations do not consider stress testing or scenario analysis. This would
provide a better understanding of portfolio behaviour during extreme market conditions. These limitations
underline the need for further analysis to enhance risk management assessments.
Data Overview and Relevance
of Model Choice
Our data was collected from Yahoo Finance.
The data consists of 252 daily historical stock prices for two companies from December 2023 to December
2024, with returns calculated based on closing prices.
OPTN – Optinose Inc
SLAB – Silicon Laboratories Inc.
Since the trade periods are assumed to be identical, the mean, variance, and correlation do not change over
time, thus forecasting and modelling is easier but may not accurately represent market realities.
3
The analysis includes two risk methods: historical simulation and variance-covariance simulation. Historical
simulation depends solely on empirical data, capturing observed market behaviour. However, calculations are
more complex, and historical-based calculations do not necessarily fit future scenarios. On the other hand,
variance-covariance simulation assumes the data follows a normal distribution, which simplifies the
calculations but underestimates the risk for non-normally distributed data, such as extreme movements in the
market or “fat tails”.
Investors and risk managers can evaluate the estimated maximum possible loss (Value at Risk) with the help
of the model outputs. By enhancing strategic decision-making, these insights help to reduce market risk and
achieve effective risk management.
Model Assumptions and Impact of Errors
We assume that future risk reflects historical patterns for historical simulation. However, in the event of any
change in market dynamics, the results may be invalid as they cannot predict risks outside the sample
period/data set.
The variance-covariance model assumes the returns follow a normal distribution. In reality, returns can
follow different distributions. Fatter tails mean a higher likelihood of significant losses (underestimated risk).
Model Critique and Validation
When evaluating risk using actual data, the historical simulation method is useful because it captures true
historical patterns, including significant market occurrences from the past. Its dependence on historical data,
which isn't always current or applicable, is a drawback. Its capacity to forecast severe threats that have not
materialised within the observed sample period is limited.
Variance-Covariance: While this method is computationally efficient and scalable, suitable for large
portfolios, it simplifies risk assessment by assuming a normal distribution of returns. This assumption can be
a weakness when returns deviate from normality, such as in financial crises or volatile periods, which
typically have a skewness about them, overlooking critical risks and underestimating the potential for
extreme events.
Models are sensitive to changes in the input data such as in the event of spikes in volatility or non-stationarity
in returns.
In summary, Historical Simulation effectively captures past market behaviour but is constrained by its
reliance on historical data, which may not accurately predict future scenarios. On the other hand, Variance-
4
Covariance is computationally efficient at the cost of oversimplifying risk by assuming returns follow a
normal distribution, underestimating extreme market events.
Therefore, the model choice should depend on the current conditions in which the portfolio operates, striking
a balance between accuracy and efficiency. A suitable model must be detailed enough to account for extreme
risk scenarios while avoiding excessive complexity or resource demands that increase costs and processing
time.
Presentation of results
Model
Metric
Historical Simulation
Variance-Covariance
1 Day VaR
2 Day VaR
3 Day VaR
Percentile (99%)
-7.26%
-
-
Maximum Loss (USD)
$14,522
$20,537
$25,125
Percentile (99%)
-7.55%
-10.68%
-13.08%
Maximum Loss (USD)
$15,121
$21,385
$26,191
The above results show that the two models provide comparable estimates for 1-day, 2-day, and 3-day VaR.
However, there are slight differences. This is seen from a higher 99th percentile in the covariance-variance
model than that of the historical simulation's 99th percentile. The higher maximum losses in the variancecovariance model may result from the sensitivity in its normal distribution assumption.
At the same time, the small one-year sample of data used shows its limitations in calculating more accurate
results.
Discussion of Possible Improvement Options and Further Work
The bank's risk exposure can be decreased by putting the following strategies into practice:
Diversification: Add assets with low or negative correlations. A loss in one asset is often offset by gains in
another when they are negatively connected since generally, they move in opposite directions. Thus, the
impact of large losses from any asset or group of highly correlated assets is reduced, which strengthens the
portfolio's resistance to market fluctuations.
Hedging: During weak market conditions, investing in derivatives like options or futures helps balance
possible losses in underlying investments. This will restrict excessive losses and offer downside protection.
Dynamic Asset Allocation: When faced with times of extreme volatility, adjusting assets according to market
conditions might reduce risk. It helps to maintain portfolio stability during such periods and is often
employed to adjust risk exposure. For example, if a recession is believed to be coming, the bank may reduce
exposure to equities and increase assets like bonds or cash.
5
Incorporate stress-testing and scenario analysis: Since VaR does not capture extreme events so accurately,
they may accompany VaR with regular stress testing and scenario analysis to evaluate the portfolio's
performance under extreme market conditions, vulnerabilities can be identified, and strategies can be
adjusted based on such findings, allowing proactive measures to be undertaken. In addition, to combine
historical simulation with parametric models, for capturing better non normal returns. it is also useful to
identify specific historical events that may have occurred during the period of the sample data to discuss their
relevance in future scenarios.
Improved data utilisation: Since banks use VaR to ensure they are within acceptable risk limits, historical
data over a more extended period, such as five years, should be used to provide more precise expected results
closer to actual results.
Use distributions with larger tails for a possible better fit of the data.
Utilise alternate data or other datasets (macroeconomic variables like GDP, inflation, and interest rates) to
increase the precision of risk assessment and predictive modelling. This supplementary information might
help detect any hazards that conventional financial data might miss.
Monitoring and rebalancing: Monitor market movements regularly and rebalance the portfolio by adjusting
asset weights, ensuring they align with the bank's risk level and investment objectives.
MODELLING STRATEGY
Decision Questions
Quantify portfolio market risk over 1, 2, and 3-day horizons with a 99% probability and compare the two
modelling methods to assess reliability.
Available Data
Historical daily prices for two stocks (Optinose Inc. and Silicon Laboratories Inc.) over 1 year (252
observations).
Methods of Addressing Decision Questions
The historical simulation calculates VaR using empirical percentiles of past returns for 1, 2, and 3 days.
Variance-Covariance uses portfolio standard deviation, asset correlations, and the square root of periods to
calculate risk using statistical assumptions about the normality of returns.
Different Model assumptions
Independence and identical distribution of returns.
For historical simulations, observed returns are representative of future scenarios.
6
For variance-covariance simulation, returns are considered to follow normal distributions with constant
variance over time.
Model selection
Both models were chosen for relevance and are two of four modes for market risk calculation: Historical
simulation for empirical accuracy and variance-covariance for simplicity and speed, offering an efficient
alternative.
MODEL OVERVIEW
Model structure
Both models estimate VaR by assessing the probability distribution of daily returns.
The Historical Simulation uses percentile-based historical and empirical estimations, while VarianceCovariance relies on mean and standard deviation statistical assumptions.
Key Components
Data: Arithmetic returns are calculated from daily price changes. OPTN's standard deviation is 5.16%, and
SLAB's is 2.95%, with a combined portfolio standard deviation of 3.24%. The Portfolio correlation is
22.34%. Each stock is 50% weighted. The Z score at 99% is 2.33.
Combined
Portfolio
OPTN
SLAB
Percentage/weights of Portfolio Invested
50%
50%
100%
Portfolio Standard Deviation
5.16%
2.95%
3.24%
Portfolio Correlation
22.34%
Investment Amount
$ 100,000.00 $ 100,000.00 $ 200,000.00
Mathematics:
Calculations of daily returns on stocks:
As a first step, we use the formula
𝑃𝑡 −𝑃𝑡−1
𝑃𝑡−1
Calculation for the 99th percentile:
As a first step, we create a synthetic portfolio by combining the returns of the two stocks and calculating their
weighted returns based on the amount invested in each stock.
Since we invested $100.000 in each stock, the weight of each is 0.5 (50%)
$100.000
$200.000
7
We then multiply each stock's weight by every daily return for one year and add them together.
(0.5 *𝑑𝑎𝑖𝑙𝑦 𝑟𝑒𝑡𝑢𝑟𝑛𝑆𝑡𝑜𝑐𝑘 𝐴𝑛 ) + (0.5 *𝑑𝑎𝑖𝑙𝑦 𝑟𝑒𝑡𝑢𝑟𝑛𝑆𝑡𝑜𝑐𝑘 𝐵𝑛 )
Where 𝑛= days 1-252
Once we have the data for the synthetic portfolio, we use the excel formula on the synthetic portfolio:
PERCENTILE.EXC (array, 0.01)
The calculated 99% percentile is then multiplied by the total investment amount for the 1-day horizon:
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑙𝑜𝑠𝑠$ = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 ∗ $200.000
To determine the loss for the 2-day and 3-day horizon:
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑙𝑜𝑠𝑠𝑑𝑎𝑦 1 ∗ √𝑛
When 𝑛 = 2, 𝑛 = 3
Calculation of Standard Deviation of Stocks:
Once the daily returns are found for each of the two stocks, we can find the stock's standard deviation
(volatility) for the 1-year period using the Excel formula STDEV.S for the whole column of daily returns.
We use the sample standard deviation formula as we are using sample data for one year only.
The standard deviation dollar amount can be found by multiplying the investment amount in each stock by
the standard deviation of the daily returns of each stock:
$100.000 ∗ 𝜎𝛢,𝛣
Calculation of Standard Deviation of stocks in dollar amount:
𝑃0 * 𝜎𝑝𝑒𝑟 𝑑𝑎𝑦
Calculation of the Daily Earnings at Risk (DEaR) the 1-day VaR for each stock:
𝑀𝑎𝑥 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑜𝑠𝑠 𝑎𝑡 99% 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑃0 * 𝜎𝑝𝑒𝑟 𝑑𝑎𝑦 * 𝑍
Where:
𝑃0 = investment amount ($100.000)
Z= 2.33
Calculation of the VaR for 2- and 3-day horizons for each stock:
𝑉𝑎𝑅 = 𝐷𝐸𝑎𝑅 ∗ √𝑛
Where 𝑛 is the number of horizons
8
Calculation of the Portfolio Correlation:
To calculate the Portfolio standard deviation, we use the Excel function:
CORREL (array1, array2)
The arrays are the columns with the calculated daily returns for each stock, which provides the correlation
coefficient between asset returns.
A value between -1< ρ < 1.
Where:
-1 shows a strong inverse relationship between stocks, moving in opposite directions.
0 indicates no relationship between stocks.
+1 shows a strong positive correlation between stocks, moving in the same direction.
Calculation of Standard Deviation of Portfolio & in dollar value:
Using the correlation above, we can now use the formula:
2
2
𝜎𝛼,𝛽,$ 𝑝𝑒𝑟 𝑑𝑎𝑦 = √𝜎𝛼,$
+ 𝜎𝛽,$
+ 2 ∗ 𝛲𝛼.𝛽 ∗ 𝜎𝛼,$ ∗ 𝜎𝛽,$
And
𝜎𝛼,𝛽 = √(𝑤𝛼2 𝜎𝛼2 ) + (𝑤𝑏2 𝜎𝑏2 ) + 2(𝛲𝛼.𝑏 𝑤𝛼 𝑤𝑏 𝜎𝛼 𝜎𝛽 )
Where:
𝑤𝛼 = weight of asset A
𝑤𝑏 = Weight of asset B
𝜎𝛼 = standard deviation of asset A
𝜎𝛽 = standard deviation of asset B
𝛲𝛼.𝑏 = correlation of assets A & B
Calculations for the Portfolio VaR (1 day, 2-day, 3-day horizon):
𝒛 ∗ 𝝈𝝆,$ ∗ √𝒏
Where:
Z = 2.33
𝜎𝜌,$ = the standard deviation of the portfolio in $
𝑛 = the horizons
9
Calculations for the 99th percentile for 1-day, 2-day and 3-day horizon:
1-day: portfolio standard deviation * Z (2.33)
2-day: [portfolio standard deviation * Z (2.33)] * √𝟐
3-day: [portfolio standard deviation * Z (2.33)] * √𝟑
Results Analysis
Figure Ошибка! Источник ссылки не найден. below shows the daily closing prices of OPTN stock over
the specified one-year period, which shows significant volatility in the asset price, peaking at approximately
$1,80 in March 2024, followed by a steady decrease over the year. By December 2024, the price stabilises at
approximately $0.40, showing a substantial loss in value. This decline could be due to adverse events
impacting the company. Such fluctuations indicate high short-term risk. Thus, effective risk management
strategies, such as estimating Value at Risk to navigate volatile market conditions, are important.
Asset Price ($)
Asset Price of OPTN 20th December 2023- 20th
December 2024
2
1,8
1,6
1,4
1,2
1
0,8
0,6
0,4
0,2
Months
Figure 1
As we can see from Figure Ошибка! Источник ссылки не найден. below, many of the daily returns for
OPTN are concentrated around the range of -1.62% to 1.27%, which shows that the returns were primarily
small, with the more extreme return values tapering off as the returns move away from the centre. Significant
negative returns occur but with a relatively low frequency, implying that significant losses are rare. Whilst
positive greater returns also seem to be rare, they have a slightly higher frequency in comparison to the
negative returns. The range of returns for this stock over the one year is from -21.88% to 18.63%, which
illustrates significant volatility over this period.
10
Frequency
Historical Daily Returns for
OPTN
December 2023-December 2024
80
70
60
50
40
30
20
10
0
Frequency
Daily Returns
Figure 2
The daily closing prices of SLAB shares over a one-year period are displayed in Figure 3 below. Although
there are occasional fluctuations, the annual trend of stock prices is comparatively steady, ranging from $90
to $160. Late January and early March see the highest stock prices, which then progressively decline and
level off during the summer. In November, a price recovery signalled increased market activity. SLAB's
performance is less volatile than that of OPTN's stock, making it a safer investment. Such trends demonstrate
the importance of monitoring asset correlation and variance to manage market risk and diversify portfolios.
Asset Price ($)
Asset Price of SLAB 20th December 2023 - 20th
December 2024
160
150
140
130
120
110
100
90
80
Months
Figure 3
11
Therefore, as shown in Figure 4 below, most of the daily returns are concentrated around the range of -0.89%
and 2.2%, which shows that the returns were moderately positive. The distribution seems symmetric, with
frequencies lessening as returns deviate from the centre. Significant negative returns are rare, with the
frequency decreasing as the loss becomes more extreme. While greater positive returns seem infrequent, there
are more small positive returns than negative ones. Looking at the full range of daily returns, we can see that
this stock has a lower volatility than OPTN. As a result, the returns suggest consistent small gains over the
period.
Frequency
Historical Daily Returns for
SLAB
December 2023- December 2024
70
60
50
40
30
20
10
0
Frequency
Daily Returns
Figure 4
The table in Figure 5 and chart in Figure 6 below provides a comparative analysis of value-at-risk (VaR)
outcomes using two different models: historical simulation and variance-covariance simulation. The results
are assessed across 1-day, 2-day, and 3-day horizons, focusing on maximum losses and percentiles.
The portfolio's VaR in both models at 2—and 3-day horizons is expected to be larger than the 1-day VaR
since longer horizons allow for more market movements, which result in a higher probability of larger
profit/loss.
The historical simulation model estimates a maximum loss for one day of $14,522, while the variancecovariance model estimates a slightly higher maximum loss of $15,121. The percentile metrics show a loss of
-7.26% for the historical simulation compared to -7.55% for the variance-covariance model.
For the 2-day and 3-day VaR, the variance-covariance model estimates higher maximum losses than the
historical simulation model.
12
The variance-covariance model forecasts higher losses, probably as it is based on the normal distribution
assumption, which does not effectively compensate for extreme occurrences or non-linear correlations in the
data. In contrast, historical simulation captures actual market behaviour, possibly resulting in more accurate
results. On the other hand, since the historical simulation model relies on historical patterns, this may have
underestimated future risks if the market conditions do not follow those of the historical data. From our data,
since we saw significant volatility/large drop in the asset price of OPTN, our historical data already factored
in this event, whereas our variance-covariance approach, being limited by normality, underestimated this
loss.
The difference in both model outcomes is important, as overestimating risk may lead to conservative/riskaverse decisions by management. However, underestimating risk by basing future results on past
performance may leave the company/financial institution open to significant losses that cannot be recovered.
Model
Historical Simulation
Variance-Covariance
Metric
1 Day VaR
2 Day VaR
3 Day VaR
Percentile (99%)
-7.26%
-
-
Maximum Loss (USD)
$14,522
$20,537
$25,125
Percentile (99%)
-7.55%
-10.68%
-13.08%
Maximum Loss (USD)
$15,121
$21,385
$26,191
Figure 5
Maximum Loss Comparison across models
Maximum loss ($)
30 000
$25 125 $26 191
25 000
20 000
15 000
$20 537 $21 385
$14 522 $15 121
10 000
5 000
0
1-Day
2-Day
3-Day
Time Horizon
Historical Simulation
Variance-Covariance
Figure 6
13
VaR Plot (99% confidence level) for portfolio
(historical return simulation)
Frequency/probability
70
60
---- 99% VaR threshold (-7.26%)
VaR Zone
50
Frequency
40
30
20
10
0
portfolio returns
Figure 7
The above graph in Figure 7 illustrates the distribution of portfolio returns and highlights the 99% Value and
Risk (VaR) threshold at -7.26%. It shows that most portfolio returns are concentrated around the centre, with
a peak frequency near a small positive return (around 0.67%). The VaR zone, indicated in red, represents the
worst 1% of portfolio returns. At the 99% confidence level, the portfolio is expected to receive losses
exceeding -7.26% only 1% of the time. The results highlight the risk level associated with the portfolio and
provide a benchmark to evaluate potential downside exposure. It underlines the importance of monitoring and
mitigating extreme losses, especially during volatile market conditions.
On further analysis using Crystal Ball, we can see from the comparison charts below, Figures 8 and 9, that
the historical data for both stocks does not fit a normal distribution as shown by the Anderson-Darling
ranking. In both cases, the logistic distribution is the best-fitting distribution since it has the lowest sampling
errors at 1.9298 for the OPTN stock and 0.3269 for the SLAB stock compared to the other distributions
ranked. The logistic distribution is a better fit since it can model fatter tails compared to the normal
distribution used in variance-covariance simulation.
14
Figure 8
Figure 9
As a result, the historical daily returns do not fit the normal distribution in the variance-covariance
simulation, which would otherwise lead to less accurate results and a weaker risk assessment, as the model
underestimates the likelihood and impact of extreme market events.
Moreover, using the same Crystal Ball analysis on the portfolio returns (see Figure 9 below), the portfolio
better fits a logistic distribution with a sampling error of 0.6334.
15
Figure 10
Figure 11, below, is the Crystal Ball report confirming our above distribution findings. The logistic
assumptions for OPTN and SLAB are consistent with the statistical characteristics identified in our analysis.
For SLAB, the distribution is centred with a mean of 0 and a scale parameter of 0.02, illustrating relatively
low volatility. These assumptions reflect the stability and expected outcomes for an asset under typical
market conditions; the OPTN distribution has a mean of -0.01 and a higher scale parameter of 0.03,
illustrating slightly lower average returns and increased variability, which may be attributed to companyspecific challenges such as operation inefficiencies, financial difficulties or exposure to higher market risks.
As part of our online research, OPTN is a pharmaceutical company, and many recent articles discuss possible
investor concerns due to the drop in asset price, which is now below the NASDAQ requirement.
Therefore, given the historical data, these logistic distributions should be selected to accurately model returns
and serve as the basis for risk and return simulations.
16
Figure 11
Validation
Historical data validation involves comparing the model's output to historical data used as data inputs to
ensure its integrity. We have ensured that the historical data used in our simulation is accurate and complete.
The variance-covariance simulation is validated via statistical testing. Using the Anderson-Darling ranking,
we can establish whether the returns in our dataset follow a normal distribution to determine the best possible
fit.
Cross-verification is undertaken by comparing the VaR to historical drawdowns to see if any significant
discrepancies indicate our models do not fully capture the market risk.
References and Data Sets
Historical dataset: daily returns of 1 year (Yahoo Finance)
Statistical references:
Standard Z score for 99% probability
Standard deviation
Correlation Coefficient
Columbia Business School, Paul Glasserman (1999)
17
Market Risk VaR: Historical Simulation Approach, Chapter 9, NUP Moodle upload
Introduction to Simulation, Chapter 1, (Risk Analysis, Evans), NUP Moodle upload
Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91
Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political
Economy, 81(3), 637-654
Ibbotson, R. G., Chen, P., & Xie, J. (2011). The ABCs of Asset Allocation. Financial Analysts Journal,
67(1), 44-57
https://www.centralbank.cy/en/licensing-supervision/eu-wide-stress-testing
Jorion, P. (2011). Financial Risk Manager Handbook. 6th edition. John Wiley & Sons
Patil, R., & Kamath, V. (2023). 'An Empirical Study On Investment Decision In Volatile Market With
Special Reference To Investment In It Sector". (“‘An Empirical Study On Investment Decision”) Journal of
Namibian Studies: History Politics Culture. https://doi.org/10.59670/jns.v33i.4445
Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson Education
Jondeau, E., Poon, S.-H., & Rockinger, M. (2007). Financial Modeling Under Non-Gaussian Distributions.
Springer.
Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2014). Modern Portfolio Theory and
Investment Analysis (9th ed.). Wiley.
Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset
(3rd ed.). (“Damodaran: Investment Valuation: Tools and Techniques for ... - Wiley”) Wiley.
TECHNICAL APPENDICES
Equation Derivations
Portfolio standard deviation
This combines individual asset volatility, weights, and the correlation between the two assets. This formula is
taken from the variance-covariance matrix for portfolio risk, where each term represents a risk factor. This
formula considers the diversification effect by which the correlation between assets mitigates overall
portfolio risk.
Scaling Value at Risk (VaR)
Used to calculate the VaR for multi-day horizons.
The square root rule assumes that daily returns are independent and identically distributed. This
approximation is widely used in risk management but relies on the independent and identically distributed
assumption.
Synthetic Portfolio
18
To calculate the 99th percentile loss, a synthetic portfolio was created by combining the daily returns of the
two stocks, weighted equally based on the investment amounts. This estimates the behaviour of the combined
investment over time. The percentile Excel formula is applied.
Portfolio VaR using correlation
The Portfolio VaR under the variance-covariance model uses a formula that combines the portfolio standard
deviation in dollar amount, the Z score for the confidence level (2.33 for 99%), and the horizon adjustment
for 1-day, 2-day, and 3-day horizons.
Assumptions:
1. Returns are identical and independently distributed.
2. VaR variance-covariance assumes normal distributions
3. Static weights fixed at 0.5 each.
Limitations:
1. Single-year data of 252 trading days may not represent long-term trends
2. Excel functions may differ from other statistical/simulation software.
Result Interpretation Guide
Value at Risk (VaR) at a 99% confidence level for 1-day, 2-day, and 3-day horizons represents the maximum
possible loss in value of an asset or portfolio at a particular confidence level and over a specified time
horizon.
The historical method takes historical return data for the assets, such as the daily returns over the last year.
These returns are sorted from smallest to largest. To obtain the 99% confidence level, the return
corresponding to the 1st percentile (lowest 1%) is identified. The VaR is the value corresponding to this
return, meaning the amount the asset could lose with 99% confidence over the specified horizon.
The steps in the variance-covariance method take the same daily returns over the last year and use statistical
formulae such as the standard deviation to estimate the volatility. It also calculates the assets' covariance and,
using the statistical Z score, can estimate potential losses.
To compare VaR calculations at different horizons, such as the 2-day and 3-day horizon, we multiply the 1day horizon by the square root of time.
The statistical outputs and graphical representations provide insights into the portfolio's behaviour, including
the distribution of returns and potential losses.
The VaR results are the final values representing the risk at each horizon.
The histograms of the returns of the assets and the portfolio help visualise the distribution of returns,
identifying the frequency and severity of the extreme losses and gains. This helps assess the robustness of the
VaR estimates.
19
The VaR plot shows the distribution of potential losses. The area to the left of the VaR threshold (at 99%)
represents the worst 1% of losses).
Suppose the VaR calculated using the Variance-Covariance method exceeds the VaR of the historical
drawdown simulation. In that case, this suggests that the normality assumption in the variance-covariance
method does not fully capture the risk of extreme events. In contrast, any extreme loss or drop in asset price
over the analysis period would already be incorporated from the historical dataset.
Historical simulation reflects more on the actual market conditions and may not consider changes in market
behaviour or extreme tail events outside the sample period. For variance-covariance, assuming a normal
distribution may not capture fat tails or extreme market events as accurately as the historical simulation.
In conclusion, the VaR estimates from both simulations provide a confidence interval for prospective losses,
albeit based on different assumptions. The choice between both simulation methods should be based on the
nature of the asset, the time horizon, and the available historical data.
20