Exam (elaborations) INV4801 Assignment 2 (COMPLETE ANSWERS) 2025 (165590) -
DUE 29 August 2025 • Course • Investments: Portfolio Management (INV4801) •
Institution • University Of South Africa (Unisa) • Book • Portfolio Management
a) Volatility Dynamics in South African Equity Markets 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 Models. The portfolio manager gathered the
following daily information: α = 0.07, γ = 0.000015, and β = 0.91. Given these
parameters, the daily standard deviation is 1%. Suppose the previous period
estimated variance was 0.0012 and the current period return is 4.5% above the
expected value. (i) Compute the conditional variance for today. (5) (ii) Compute the
conditional standard deviation for today. (2) (iii) What will happen to the variance if
the current return is in line with expectation? (2)
We use the (standard) GARCH/ARCH-style variance equation implied by the
parameters:
𝜎𝑡2
𝛼 + 𝛾 𝜀𝑡−12 + 𝛽 𝜎𝑡−12σt2
=α+γε t−1 2
+βσ t−1 2
(you were given 𝛼
0.07 , 𝛾
0.000015 , 𝛽
0.91 , 𝜎 𝑡 − 1 2
0.0012 α=0.07, γ=0.000015, β=0.91, σ t−1 2
,=0.0012 and the surprise 𝜀
4.5 %
0.045 ε=4.5%=0.045.)
(i) Conditional variance for today
Compute the squared shock:
𝜀2
0.045 2
0.002025 ε 2 =0.045 2 =0.002025
Multiply by 𝛾 γ:
𝛾𝜀2
0.000015 × 0.002025
0.000000030375 γε 2 =0.000015×0.002025=0.000000030375
Multiply previous variance by 𝛽 β:
𝛽𝜎𝑡−12
0.91 × 0.0012
0.001092 βσ t−1 2
=0.91×0.0012=0.001092
Add all terms:
, 𝜎𝑡2
𝛼+𝛾𝜀2+𝛽𝜎𝑡−12
0.07 + 0.000000030375 + 0.001092
0.071092030375 σ t 2
=α+γε 2 +βσ t−1 2
=0.07+0.000000030375+0.001092=0.071092030375
Answer (i): 𝜎 𝑡 2 ≈ 0.07109203 σ t 2
≈0.07109203
(variance in the same units as given)
(ii) Conditional standard deviation for today
Take the square root:
𝜎𝑡
0.071092030375 ≈ 0.26663 σ t
= 0.071092030375
≈0.26663
Expressed as a percentage:
0.26663 ≈ 26.663 % 0.26663≈26.663%
Answer (ii): 𝜎 𝑡 ≈ 0.2666 (i.e. 26.66 % ) σ t
≈0.2666 (i.e. 26.66%)
(iii) What if the current return is in line with expectation?
DUE 29 August 2025 • Course • Investments: Portfolio Management (INV4801) •
Institution • University Of South Africa (Unisa) • Book • Portfolio Management
a) Volatility Dynamics in South African Equity Markets 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 Models. The portfolio manager gathered the
following daily information: α = 0.07, γ = 0.000015, and β = 0.91. Given these
parameters, the daily standard deviation is 1%. Suppose the previous period
estimated variance was 0.0012 and the current period return is 4.5% above the
expected value. (i) Compute the conditional variance for today. (5) (ii) Compute the
conditional standard deviation for today. (2) (iii) What will happen to the variance if
the current return is in line with expectation? (2)
We use the (standard) GARCH/ARCH-style variance equation implied by the
parameters:
𝜎𝑡2
𝛼 + 𝛾 𝜀𝑡−12 + 𝛽 𝜎𝑡−12σt2
=α+γε t−1 2
+βσ t−1 2
(you were given 𝛼
0.07 , 𝛾
0.000015 , 𝛽
0.91 , 𝜎 𝑡 − 1 2
0.0012 α=0.07, γ=0.000015, β=0.91, σ t−1 2
,=0.0012 and the surprise 𝜀
4.5 %
0.045 ε=4.5%=0.045.)
(i) Conditional variance for today
Compute the squared shock:
𝜀2
0.045 2
0.002025 ε 2 =0.045 2 =0.002025
Multiply by 𝛾 γ:
𝛾𝜀2
0.000015 × 0.002025
0.000000030375 γε 2 =0.000015×0.002025=0.000000030375
Multiply previous variance by 𝛽 β:
𝛽𝜎𝑡−12
0.91 × 0.0012
0.001092 βσ t−1 2
=0.91×0.0012=0.001092
Add all terms:
, 𝜎𝑡2
𝛼+𝛾𝜀2+𝛽𝜎𝑡−12
0.07 + 0.000000030375 + 0.001092
0.071092030375 σ t 2
=α+γε 2 +βσ t−1 2
=0.07+0.000000030375+0.001092=0.071092030375
Answer (i): 𝜎 𝑡 2 ≈ 0.07109203 σ t 2
≈0.07109203
(variance in the same units as given)
(ii) Conditional standard deviation for today
Take the square root:
𝜎𝑡
0.071092030375 ≈ 0.26663 σ t
= 0.071092030375
≈0.26663
Expressed as a percentage:
0.26663 ≈ 26.663 % 0.26663≈26.663%
Answer (ii): 𝜎 𝑡 ≈ 0.2666 (i.e. 26.66 % ) σ t
≈0.2666 (i.e. 26.66%)
(iii) What if the current return is in line with expectation?
