Not financial advice. This article reproduces a quantitative model for educational purposes only — it is not investment advice. Past performance does not predict future results. Consult a licensed adviser before investing.
Three-Framework Quantitative Assessment · Data Period: Jun 2025 – Jun 2026 · 12 JSE Stocks · 259 Trading Days · Report Date: June 2026
Executive Summary
This report presents a comprehensive quantitative analysis of 12 JSE-listed stocks using three complementary academic frameworks applied to daily closing price data spanning June 2025 to June 2026. The analysis integrates reverse stress testing, forward-looking stressed capital estimation, and behavioural portfolio theory to produce actionable investment signals for a R5,000 budget.
| Best Performer | Worst Performer | Binding SVaR | Top Recommendation |
|---|---|---|---|
| GFI +133.7% | NPN −126.8% | −0.92% (W.Asia) | Gold Fields (GFI) |
Key Finding: Gold Fields (GFI) scores 11/12 across all three frameworks — highest past-year return (+133.7%), lowest volatility in its return group (12.1% p.a.), and the most positive sentiment state of the entire dataset (X = +5.16). With R5,000 an investor can purchase 8 shares at R596.27, for a total outlay of R4,770.
Key Warning: Naspers (NPN) is the clearest avoid — it recorded an annualised return of −126.8%, carries the deepest stress scenario exposure under an AI bubble shock (−45%), and is in deep loss sentiment territory (X = −0.67).
Methodology
The analysis combines three academic frameworks, each addressing a distinct dimension of investment risk:
Paper 1 — Parametric Reverse Stress Testing (RST)
Source: Sparviero & Viola (2025). Develops a conditional framework in which a multivariate stress scenario coherent with the empirical dependence structure is reconstructed from a single exogenous shock on one asset.
- Shocked asset: Naspers (NPN), cumulative −30% over 63 trading days
- Daily equivalent shock: \( s = (1 - 0.30)^{1/63} - 1 = -0.565\% \) per day
- 1,000 Gaussian simulations drawn from the conditional distribution of non-shocked assets
- Modal scenario vector \( \bar{r}^{\,s} \) identifies the most likely joint response across all 12 stocks
Paper 2 — GPR-HS Stressed SVaR
Source: Vadrevu (2026). Extends a Hybrid Gaussian Process Regression – Historical Simulation framework to forward-looking macro stress scenarios using SACS covariance stabilisation and the ANI volatility initialisation strategy.
- Three scenarios: West Asia War, Climate Risk, AI Bubble / Regulation
- 252-day stressed paths with deterministic drift plus historical stochastic residuals
- SACS covariance matrix averaged from two historical crisis sub-periods
- SVaR and SES computed under Student-t distribution \( (\nu = 5) \) at 99% confidence
Paper 3 — Asymmetric Nonlinear Extrapolation
Source: Yan et al. (2026). Extends the Atmaz (2022) return extrapolation framework to incorporate saturation (Weber–Fechner law) and loss aversion in a CRRA investor’s optimal portfolio choice. The asymmetric extrapolation function is:
- Sentiment \( X \) computed as a 63-day normalised EWMA of past returns
- Portfolio decomposed into myopic demand \( \omega^*_M \), variance hedge \( \omega^*_V \), and sentiment hedge \( \omega^*_S \)
- Welfare cost measures loss relative to the rational benchmark; saturation benefit = 0.87 pp vs the linear rule
Data Overview — Jun 2025 to Jun 2026
All computations use 259 trading days of daily log returns \( r_t = \ln(P_t / P_{t-1}) \) derived from closing prices sourced from JSE.co.za and supplementary data providers. Prices are denominated in ZAR.
Annualised Returns and Volatility
| Asset | Company | Price (ZAR) | Ann. Return | Ann. Vol | Sharpe | Score | Signal |
|---|---|---|---|---|---|---|---|
| NPN | Naspers | 856.62 | −126.8% | 26.7% | −4.75 | 1 | Avoid |
| AGL | Anglo American | 820.95 | +64.2% | 23.5% | 2.73 | 9 | Consider |
| FSR | FirstRand | 92.65 | +0.4% | 12.0% | 0.03 | 4 | Neutral |
| SBK | Standard Bank | 314.67 | +63.4% | 12.7% | 5.01 | 9 | Consider |
| MTN | MTN Group | 212.57 | +81.6% | 12.5% | 6.53 | 10 | Strong buy |
| SOL | Sasol | 215.15 | +34.0% | 40.2% | 0.85 | 6 | Neutral |
| SHP | Shoprite | 288.60 | −27.2% | 12.9% | −2.11 | 2 | Avoid |
| CPI | Capitec | 4325.28 | +43.7% | 14.0% | 3.12 | 8 | Consider |
| GFI | Gold Fields | 596.27 | +133.7% | 12.1% | 11.05 | 11 | Strong buy |
| SSW | Sibanye | 40.36 | +43.2% | 13.1% | 3.30 | 9 | Consider |
| SLM | Sanlam | 85.08 | −7.4% | 12.1% | −0.61 | 4 | Neutral |
| VOD | Vodacom | 149.77 | +43.0% | 12.4% | 3.47 | 8 | Consider |
Sharpe ratio approximated as \( (\text{Ann. Return} - 2\%) / \text{Ann. Vol} \). Score out of 12 based on the multi-framework composite ranking.
Annualised Return by Stock
Green = positive, red = negative (% p.a.)
Risk vs Return
Annualised volatility (x) against annualised return (y)
Composite Multi-Framework Score
Out of 12 — higher is stronger
Paper 1: Reverse Stress Testing Results
The parametric RST framework was applied using NPN as the shocked asset (−30% cumulative over 63 trading days, daily equivalent −0.565%). The conditional mean of each non-shocked asset given the NPN shock constitutes the modal stress scenario vector \( \bar{r}^{\,s} \). 1,000 simulations were run from the joint conditional Gaussian distribution.
Modal Scenario and Cumulative Return Quantiles
| Asset | Modal r̄ˢ | Base Q5 | Base Q50 | Base Q95 | Stress Q50 | Stress Q95 | Shift |
|---|---|---|---|---|---|---|---|
| NPN | −0.5646% | −22.9% | −0.6% | 21.8% | −35.6% | −35.6% | −35.0% |
| AGL | +0.0240% | −25.4% | −1.1% | 24.5% | −1.5% | 24.0% | +0.4% |
| FSR | +0.0340% | −12.7% | 3.3% | 19.3% | 2.8% | 18.8% | −0.5% |
| SBK | +0.2420% | −2.8% | 14.2% | 30.7% | 13.5% | 32.2% | −0.7% |
| MTN | +0.1190% | −12.8% | 10.3% | 33.4% | 5.8% | 29.9% | −4.5% |
| SOL | +0.0580% | −27.0% | 4.0% | 37.8% | 2.5% | 33.5% | −1.5% |
| SHP | +0.1580% | −6.4% | 8.2% | 22.7% | 10.1% | 23.5% | +1.9% |
| CPI | +0.0150% | −15.1% | 2.5% | 20.1% | 1.9% | 18.6% | −0.6% |
| GFI | −0.0680% | −25.5% | −1.8% | 24.0% | −2.8% | 21.8% | −1.0% |
| SSW | +0.1430% | −24.3% | 7.2% | 40.5% | 9.4% | 42.2% | +2.2% |
| SLM | +0.0720% | −10.7% | 4.0% | 19.2% | 3.8% | 18.9% | −0.2% |
| VOD | −0.0720% | −18.3% | −5.1% | 8.1% | −4.0% | 9.6% | +1.1% |
Stressed-Median Shift under NPN Shock
Change in cumulative-return median vs baseline (pp)
RST Key Findings
- NPN receives the full −35.6% cumulative stressed Q50, confirming it as the shock absorber
- Most stocks show near-zero correlation with NPN — median shifts are small (< 5%) confirming genuine diversification
- SSW and SHP show slight positive stressed medians — resilient or mildly benefiting in this scenario
- MTN shows the largest meaningful downward shift (−4.5%) — consistent with modest positive correlation to NPN
- GFI and VOD have near-zero to slightly negative modal responses — consistent with their defensive characteristics
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Paper 2: GPR-HS Stressed SVaR Results
The Hybrid GPR-HS framework with SACS covariance stabilisation was applied to three forward-looking macro scenarios. Each scenario prescribes 252-day cumulative shocks across all 12 assets. GPR volatility models are fitted with the ANI strategy, and portfolio SVaR/SES are computed at 99% confidence under a Student-t distribution with \( \nu = 5 \).
Scenario Shock Calibration
| Asset | Hist. Vol | West Asia War | Climate Risk | AI Bubble | Rationale |
|---|---|---|---|---|---|
| NPN | 26.7% | −25% | −20% | −45% | Direct tech/Tencent exposure in AI scenario |
| AGL | 23.5% | −30% | −35% | −10% | Commodity supply chain; physical climate risk |
| FSR | 12.0% | −20% | −15% | −15% | Banking credit stress; fintech repricing |
| SBK | 12.7% | −20% | −15% | −15% | Banking credit stress; fintech repricing |
| MTN | 12.5% | −30% | −20% | −20% | EM currency + energy; telecom AI repricing |
| SOL | 40.2% | +10% | −25% | −10% | Energy price spike beneficiary in war scenario |
| SHP | 12.9% | −15% | −20% | −10% | Consumer input costs; food supply disruption |
| CPI | 14.0% | −20% | −15% | −25% | Consumer stress; digital banking repricing |
| GFI | 12.1% | +15% | −15% | +10% | Gold safe-haven in war; hedge in AI bubble |
| SSW | 13.1% | +5% | −20% | +5% | Commodity hedge in geopolitical scenarios |
| SLM | 12.1% | −20% | −20% | −20% | AUM losses across all scenarios |
| VOD | 12.4% | −15% | −10% | −15% | Telecom infrastructure costs |
Terminal SVaR and SES (Day 252)
| Scenario | SVaR (99%) | SES (99%) | |SES|>|SVaR| | Binding? |
|---|---|---|---|---|
| West Asia War | −0.9234% | −1.2218% | Yes | No |
| Climate Risk | −0.8950% | −1.1842% | Yes | No |
| AI Bubble / Regulation | −0.8838% | −1.1695% | Yes | Yes |
Stressed VaR vs Expected Shortfall by Scenario
Terminal day-252 values at 99% confidence (%)
SVaR Key Findings
- The AI Bubble scenario is the binding capital constraint — SVaR −0.8838%, SES −1.1695%
- All three scenarios maintain |SES| > |SVaR| — coherent risk measure property preserved throughout
- SVaR spread across scenarios is 9.6 basis points — indicating stable, consistent capital estimation
- GFI and SSW receive positive shocks under West Asia War and AI Bubble — acting as natural portfolio hedges
- SOL (Sasol) benefits under West Asia War (+10%) — energy price spike beneficiary but highest volatility stock
Paper 3: Asymmetric Nonlinear Extrapolation Results
The asymmetric nonlinear extrapolation framework was applied using parameters \( \theta = 0.15 \) (extrapolation degree), \( \lambda = 3 \) (saturation rate), \( \beta = 0.5 \) (loss aversion asymmetry), and \( \gamma = 3 \) (CRRA risk aversion). Sentiment \( X \) is a 63-day normalised EWMA of past returns, scaled to unit variance.
Optimal Portfolio Decomposition
The optimal portfolio weight decomposes into three components:
- Myopic demand: \( \omega^*_M = \dfrac{\mu + \theta f(X) - r}{\gamma v} \) — the sentiment-distorted mean-variance optimum
- Variance hedge: \( \omega^*_V = \dfrac{\rho\,\sigma_v}{\gamma}\,\Psi_v \) — reduces the long position under negative return–variance correlation
- Sentiment hedge: \( \omega^*_S = \dfrac{\alpha}{\gamma}\,\Psi_X \) — hedges the anticipated mean-reversion of sentiment
Asset Sentiment States and Welfare Costs
| Asset | μ ann. | σ ann. | Sentiment X | f(X) | ω* total | Welfare loss | State |
|---|---|---|---|---|---|---|---|
| NPN | −126.8% | 26.7% | −0.667 | −1.911 | −4.23 | 11.5% | Loss |
| AGL | 64.2% | 23.5% | +1.016 | +0.998 | +4.13 | 5.9% | Gain |
| FSR | 0.4% | 12.0% | −0.103 | −0.496 | −2.50 | 7.4% | Loss |
| SBK | 63.4% | 12.7% | +2.774 | +1.000 | +5.00 | 27.2% | Gain |
| MTN | 81.6% | 12.5% | +3.098 | +1.000 | +5.00 | 28.6% | Gain |
| SOL | 34.0% | 40.2% | +1.415 | +1.000 | +0.57 | 1.4% | Gain |
| SHP | −27.2% | 12.9% | −0.427 | −1.653 | −4.99 | 61.0% | Loss |
| CPI | 43.7% | 14.0% | +1.083 | +0.999 | +5.00 | 10.5% | Gain |
| GFI | 133.7% | 12.1% | +5.158 | +1.000 | +5.00 | 21.2% | Gain |
| SSW | 43.2% | 13.1% | +1.681 | +1.000 | +5.00 | 21.2% | Gain |
| SLM | −7.4% | 12.1% | +0.049 | +0.208 | −1.56 | 1.2% | Neutral |
| VOD | 43.0% | 12.4% | +1.862 | +1.000 | +5.00 | 25.2% | Gain |
Behavioural Sentiment State (X) by Stock
Positive = gain territory, negative = loss territory
Behavioural Welfare Cost by Stock
Certainty-equivalent loss vs rational benchmark (%)
Welfare Sensitivity (from paper — Table 5)
| θ (extrap. degree) | 0.05 | 0.10 | 0.15 (baseline) | 0.20 | 0.25 |
|---|---|---|---|---|---|
| Linear benchmark (%) | 0.33 | 1.13 | 2.22 | 3.50 | 4.90 |
| Asymmetric (this paper) | 0.23 | 0.74 | 1.34 | 1.87 | 2.23 |
| Saturation benefit (pp) | 0.11 | 0.39 | 0.87 | 1.63 | 2.68 |
Welfare Cost vs Extrapolation Degree
Linear rule vs asymmetric saturating rule (%)
Extrapolation Key Findings
- 9 of 12 stocks are in gain sentiment territory — consistent with the strong momentum environment of the past year
- SBK, MTN, GFI, SSW all at maximum f(X) = 1.0 — extrapolation has fully saturated — protection from further overreaction
- SHP has the highest welfare cost at 61.0% — investors are most likely making expensive behavioural mistakes here
- SOL has the lowest welfare cost at 1.4% — near-neutral sentiment, rational behaviour approximately preserved
- The asymmetric extrapolator saves 0.87 percentage points of CEW vs the linear rule at baseline θ
Investment Recommendations
The following recommendations are derived from the composite multi-framework score, which integrates past-year returns, RST stress resilience, SVaR scenario exposure, and behavioural sentiment state.
R5,000 Investment Guide
| Rank | Stock | Price | Shares | Cost | Change | Score | Key reason |
|---|---|---|---|---|---|---|---|
| 1 | Gold Fields (GFI) | R596.27 | 8 | R4,770 | R230 | 11/12 | Highest return + lowest vol + AI hedge |
| 2 | MTN Group (MTN) | R212.57 | 23 | R4,889 | R111 | 10/12 | Strong momentum + low vol + resilient |
| 3 | Standard Bank (SBK) | R314.67 | 15 | R4,720 | R280 | 9/12 | Consistent performer + banking strength |
| 4 | Anglo American (AGL) | R820.95 | 6 | R4,926 | R74 | 9/12 | Commodity cycle recovery + high return |
Stocks to Avoid
| Asset | Company | Ann. Return | Score | SVaR Shock (AI) | Reason to avoid |
|---|---|---|---|---|---|
| NPN | Naspers | −126.8% | 1/12 | −45% | Worst performer; deep loss state; max AI shock |
| SHP | Shoprite | −27.2% | 2/12 | −10% | Highest behavioural distortion (61% welfare cost) |
Gold Fields (GFI) — Best Pick
- Past year return: +133.7% annualised at 12.1% volatility — best risk-adjusted performer
- RST: slightly negative modal response (−0.068%/day) confirms defensive characteristics vs NPN
- SVaR: +15% under West Asia War and +10% under AI Bubble — genuine portfolio hedge
- Sentiment: X = +5.16, deepest gain state in the dataset; f(X) fully saturated at 1.0
- Welfare cost: 21.2% — acceptable given the return; saturation prevents further overreaction
- R5,000 allocation: 8 shares at R596.27 = R4,770.16, R229.84 returned as change
MTN Group (MTN) — Second Pick
- Past year return: +81.6% at 12.5% volatility — second-best return with comparable low vol
- RST: small downward shift (−4.5%) under NPN shock, consistent with modest positive correlation
- SVaR: moderate shocks across all scenarios (−20% to −30%), no extreme vulnerability
- Sentiment: X = +3.10, strong positive momentum; welfare cost 28.6% elevated but return justifies
- R5,000 allocation: 23 shares at R212.57 = R4,889.11, R110.89 returned as change
Naspers (NPN) — Clear Avoid
- Past year return: −126.8% annualised — worst performer by a large margin
- RST: the shock anchor; receives full −35.6% cumulative loss in stressed scenario
- SVaR: deepest AI bubble shock at −45% — no other stock is more exposed
- Sentiment: X = −0.667, in loss state; welfare cost 11.5% from behavioural over-extrapolation of losses
- Do not invest R5,000 here under any of the three frameworks
Limitations and Risk Disclosures
Model Limitations
- The parametric RST assumes joint Gaussianity of returns — fat tails and regime changes may cause underestimation of tail risk
- SACS covariance is averaged across two historical sub-periods — does not capture dynamic correlation spikes within a stress episode
- GPR sentiment EWMA uses a 63-day window — may not capture very short-term momentum reversals
- Portfolio weights are clipped to ±5 — extreme market states may produce larger optimal positions in practice
- Welfare costs assume rational benchmark with θ = 0 — real investors may have persistent biases not captured here
- SOL (Sasol) has a 40.2% annualised volatility — more than three times the average; position sizing requires extra care
Data Notes
- 259 trading days from Jun 12 2025 to Jun 9 2026
- Prices sourced from JSE.co.za; bridged observations use GBM interpolation where noted in source data
- Log returns computed as \( r_t = \ln(P_t / P_{t-1}) \); all volatilities and returns are annualised using 252 trading days
- The prior analysis (Jun 2024 – Jun 2025) showed materially different results — GFI and SOL were in loss states; VOD and SBK were in gain states
This report is produced by a quantitative model for informational and educational purposes only. It does not constitute financial advice. Past performance is not indicative of future results. All investments carry risk. Consult a licensed financial adviser before making investment decisions.
References
- Sparviero, M. and Viola, L. (2025). Reverse Stress Testing for Multivariate Scenarios: A Conditional Framework for Stressed Time Series. arXiv:2606.09274v1.
- Vadrevu, U. (2026). Forward-Looking Stress Testing Under Macro Scenarios: Stable SVaR Estimation Using a Hybrid GPR-HS Framework with SACS. arXiv:2606.07575v1.
- Yan, D., Ye, W., Zong, Z. and Chen, W. (2026). Asymmetric Nonlinear Return Extrapolation and Optimal Portfolio Choice under Stochastic Volatility. arXiv:2606.10805v1.
- Atmaz, A. (2022). Stock return extrapolation, option prices, and variance risk premium. The Review of Financial Studies 35(3), 1348–1393.
- Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2), 327–343.
- Basel Committee on Banking Supervision (2019). Minimum capital requirements for market risk. Bank for International Settlements.