UNIVERSITY OF SOUTH AFRICA (UNISA)
Department of Finance, Risk Management and Banking
⋄
Investment Analysis
Assignment 02 — Semester 2, 2026
⋄
Module Code: INV4801
Module Name: Investment Analysis
Assignment No.: Assignment 02
Due Date: Wednesday, 19 August 2026, 20:00
Unique Number: 236775
Lecturer: Prof. Godfrey Marozva, CFA
Submitted in partial fulfilment of the requirements for Investment Analysis
at the University of South Africa.
,UNISA | INV4801 Assignment 02 — Investment Analysis
Question 1(a): The Re-parameterised GARCH(1,1) Model
Volatility in equity markets is rarely constant. Periods of geopolitical tension and commodity
price swings, such as those affecting the Johannesburg Stock Exchange Top 40 Index, tend
to cluster into calm and turbulent phases. The generalised autoregressive conditional het-
eroskedasticity, GARCH(1,1), model developed by Bollerslev (1986) as an extension of Engle’s
(1982) ARCH framework captures this behaviour by making today’s conditional variance a
function of yesterday’s squared surprise in returns and yesterday’s variance estimate. The re-
parameterised form used in practice replaces the constant term with a single intercept, so that
the model reads σt2 = γ + αu2t−1 + βσt−1
2 , where γ is the constant weight, α is the weight on
the previous period’s squared unexpected return, and β is the weight on the previous period’s
variance (Bollerslev, 1986; Alexander, 2008).
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 height-
ened 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 Models. 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.
1(a)(i) Computing the Conditional Variance for Today
Question
Using re-parameterized GARCH (1,1), compute the conditional variance for today. (6)
The portfolio manager has supplied the following inputs:
Two items in the case belong to a benchmarking exercise rather than to the variance recursion
itself. The stated long-run daily standard deviation of 1% is consistent with γ, since (0.01)2 =
0.0001 = γ, confirming that γ already embeds the long-run variance contribution. The com-
parable company’s estimated standard deviation of 1.2% is a cross-sectional reference point
for judging whether the Top 40 fund is more or less volatile than a peer; it does not enter
Page 1 of 15
, UNISA | INV4801 Assignment 02 — Investment Analysis
Table 1: GARCH(1,1) Input Parameters
Parameter Value
Constant term, γ 0.00010
ARCH weight, α −0.08
GARCH weight, β 0.35
2
Previous period variance, σt−1 0.0144
Return surprise, ut−1 7.77%
the time-series recursion, which depends only on the fund’s own lagged variance and lagged
squared surprise (Bollerslev, 1986). Including it in the calculation would be a conceptual error,
since GARCH conditions on a security’s own history, not on a peer’s risk level.
Squaring the return surprise gives u2t−1 = (0.0777)2 = 0.00603729. Substituting into the recur-
sion:
σt2 = γ + αu2t−1 + βσt−1
2
= 0.00010 + (−0.08)(0.00603729) + (0.35)(0.0144)
= 0.00010 − 0.00048298 + 0.00504
= 0.00465702
Key Distinction
Today’s conditional variance: σt2 ≈ 0.004657, or 0.4657%.
1(a)(ii) Computing the Conditional Standard Deviation for Today
Question
Compute the conditional standard deviation for today. (3)
Volatility is reported in standard deviation terms because variance is not directly comparable
to a return figure. Taking the square root of the conditional variance obtained above:
√
σt = 0.00465702
≈ 0.0682, or 6.82%
Page 2 of 15
Department of Finance, Risk Management and Banking
⋄
Investment Analysis
Assignment 02 — Semester 2, 2026
⋄
Module Code: INV4801
Module Name: Investment Analysis
Assignment No.: Assignment 02
Due Date: Wednesday, 19 August 2026, 20:00
Unique Number: 236775
Lecturer: Prof. Godfrey Marozva, CFA
Submitted in partial fulfilment of the requirements for Investment Analysis
at the University of South Africa.
,UNISA | INV4801 Assignment 02 — Investment Analysis
Question 1(a): The Re-parameterised GARCH(1,1) Model
Volatility in equity markets is rarely constant. Periods of geopolitical tension and commodity
price swings, such as those affecting the Johannesburg Stock Exchange Top 40 Index, tend
to cluster into calm and turbulent phases. The generalised autoregressive conditional het-
eroskedasticity, GARCH(1,1), model developed by Bollerslev (1986) as an extension of Engle’s
(1982) ARCH framework captures this behaviour by making today’s conditional variance a
function of yesterday’s squared surprise in returns and yesterday’s variance estimate. The re-
parameterised form used in practice replaces the constant term with a single intercept, so that
the model reads σt2 = γ + αu2t−1 + βσt−1
2 , where γ is the constant weight, α is the weight on
the previous period’s squared unexpected return, and β is the weight on the previous period’s
variance (Bollerslev, 1986; Alexander, 2008).
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 height-
ened 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 Models. 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.
1(a)(i) Computing the Conditional Variance for Today
Question
Using re-parameterized GARCH (1,1), compute the conditional variance for today. (6)
The portfolio manager has supplied the following inputs:
Two items in the case belong to a benchmarking exercise rather than to the variance recursion
itself. The stated long-run daily standard deviation of 1% is consistent with γ, since (0.01)2 =
0.0001 = γ, confirming that γ already embeds the long-run variance contribution. The com-
parable company’s estimated standard deviation of 1.2% is a cross-sectional reference point
for judging whether the Top 40 fund is more or less volatile than a peer; it does not enter
Page 1 of 15
, UNISA | INV4801 Assignment 02 — Investment Analysis
Table 1: GARCH(1,1) Input Parameters
Parameter Value
Constant term, γ 0.00010
ARCH weight, α −0.08
GARCH weight, β 0.35
2
Previous period variance, σt−1 0.0144
Return surprise, ut−1 7.77%
the time-series recursion, which depends only on the fund’s own lagged variance and lagged
squared surprise (Bollerslev, 1986). Including it in the calculation would be a conceptual error,
since GARCH conditions on a security’s own history, not on a peer’s risk level.
Squaring the return surprise gives u2t−1 = (0.0777)2 = 0.00603729. Substituting into the recur-
sion:
σt2 = γ + αu2t−1 + βσt−1
2
= 0.00010 + (−0.08)(0.00603729) + (0.35)(0.0144)
= 0.00010 − 0.00048298 + 0.00504
= 0.00465702
Key Distinction
Today’s conditional variance: σt2 ≈ 0.004657, or 0.4657%.
1(a)(ii) Computing the Conditional Standard Deviation for Today
Question
Compute the conditional standard deviation for today. (3)
Volatility is reported in standard deviation terms because variance is not directly comparable
to a return figure. Taking the square root of the conditional variance obtained above:
√
σt = 0.00465702
≈ 0.0682, or 6.82%
Page 2 of 15
