UNIVERSITY OF SOUTH AFRICA
College of Economic and Management Sciences
Department of Finance, Risk Management and Banking
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
INV4801: Investment Analysis
Assignment 02 — 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
INV4801
Module Code:
Investment Analysis
Module Name:
GARCH Volatility, International Market
Assignment Topic:
Integration and Performance Attribution
Assignment 02
Assignment Number:
236775
Unique Number:
19 August 2026
Due Date:
50
Total Marks:
Submitted in partial fulfilment of the requirements for Investment Analysis — UNISA 2026
, UNISA | INV4801 Investment Volatility, Market Integration & Performance Attribution
Question 1(a): GARCH(1,1) Volatility Modelling
Question: A portfolio manager at a Johannesburg-based investment firm
is tasked with managing a fund heavily exposed to the South African Top
40 Index. Following a period of heightened market uncertainty due to
geopolitical tensions and fluctuating commodity prices, the firm decides to
model daily equity return volatility more accurately using a Time-Varying
Volatility-ARCH Model. The portfolio manager gathered the following daily
information: α = −0.08, γ = 0.00010, and β = 0.35. Given these parameters,
the daily standard deviation is 1%. Suppose the previous period estimated
variance was 0.0144, the comparable company estimated standard deviation
was 0.012 and the current period return is 7.77% above the expected value.
Time-varying volatility models allow a portfolio manager to update a conditional variance es-
timate as new return information arrives, rather than assuming volatility is constant through
time. The Generalised Autoregressive Conditional Heteroskedasticity model of order (1,1), de-
veloped from Bollerslev’s extension of Engle’s ARCH framework, is the workhorse specification
used by risk managers for this purpose (Bollerslev, 1986).
1.1 The Re-parameterised GARCH(1,1) Equation
2 ,
The standard GARCH(1,1) conditional variance equation is written as σt2 = ω + αu2t−1 + βσt−1
where ω is a constant, α is the weight on the most recent squared return surprise, and β is
the weight on the previous period’s conditional variance (O’Connell, 2026a). Because ω on
its own is difficult to interpret, the model is commonly re-parameterised by replacing ω with
γVL , where VL is the long-run (unconditional) average variance and γ is the weight the model
assigns to that long-run level (Hull, 2018). This gives the re-parameterised form used below:
σt2 = γVL + αu2t−1 + βσt−1
2
(1)
For the process to be covariance stationary and for the conditional variance to remain econom-
ically meaningful, the weights are required to satisfy γ + α + β = 1, with α ≥ 0 and β ≥ 0
(Hull, 2018).
Page 2 of 15
College of Economic and Management Sciences
Department of Finance, Risk Management and Banking
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
INV4801: Investment Analysis
Assignment 02 — 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
INV4801
Module Code:
Investment Analysis
Module Name:
GARCH Volatility, International Market
Assignment Topic:
Integration and Performance Attribution
Assignment 02
Assignment Number:
236775
Unique Number:
19 August 2026
Due Date:
50
Total Marks:
Submitted in partial fulfilment of the requirements for Investment Analysis — UNISA 2026
, UNISA | INV4801 Investment Volatility, Market Integration & Performance Attribution
Question 1(a): GARCH(1,1) Volatility Modelling
Question: A portfolio manager at a Johannesburg-based investment firm
is tasked with managing a fund heavily exposed to the South African Top
40 Index. Following a period of heightened market uncertainty due to
geopolitical tensions and fluctuating commodity prices, the firm decides to
model daily equity return volatility more accurately using a Time-Varying
Volatility-ARCH Model. The portfolio manager gathered the following daily
information: α = −0.08, γ = 0.00010, and β = 0.35. Given these parameters,
the daily standard deviation is 1%. Suppose the previous period estimated
variance was 0.0144, the comparable company estimated standard deviation
was 0.012 and the current period return is 7.77% above the expected value.
Time-varying volatility models allow a portfolio manager to update a conditional variance es-
timate as new return information arrives, rather than assuming volatility is constant through
time. The Generalised Autoregressive Conditional Heteroskedasticity model of order (1,1), de-
veloped from Bollerslev’s extension of Engle’s ARCH framework, is the workhorse specification
used by risk managers for this purpose (Bollerslev, 1986).
1.1 The Re-parameterised GARCH(1,1) Equation
2 ,
The standard GARCH(1,1) conditional variance equation is written as σt2 = ω + αu2t−1 + βσt−1
where ω is a constant, α is the weight on the most recent squared return surprise, and β is
the weight on the previous period’s conditional variance (O’Connell, 2026a). Because ω on
its own is difficult to interpret, the model is commonly re-parameterised by replacing ω with
γVL , where VL is the long-run (unconditional) average variance and γ is the weight the model
assigns to that long-run level (Hull, 2018). This gives the re-parameterised form used below:
σt2 = γVL + αu2t−1 + βσt−1
2
(1)
For the process to be covariance stationary and for the conditional variance to remain econom-
ically meaningful, the weights are required to satisfy γ + α + β = 1, with α ≥ 0 and β ≥ 0
(Hull, 2018).
Page 2 of 15
