loan-invest-sim

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loan-invest-sim

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Is it worth borrowing at 2% to buy global ETFs? Reproducible Monte Carlo study + browser simulator (Traditional Chinese)

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# loan-invest-sim

[繁體中文](README.zh-TW.md) · **English**

**Is it worth borrowing at 2% to buy global ETFs?** A reproducible Monte Carlo study using 32 years of real market data, in real (inflation-adjusted) TWD, after tax and FX.

Taiwanese physicians can borrow unsecured at rates well below the long-run return of a global equity portfolio. This repo works out what actually happens if you do — including what the losing 10.6%–22.4% of outcomes look like, which most versions of this argument leave out.

**Interactive simulator:** https://invest.drpwchen.com — run the simulation with your own loan terms, entirely in your browser, nothing uploaded.

**Comparison Lab (v0.5.0):** the same page now answers three more questions on a multi-asset engine — sweep the stock/bond mix from 0:100 to 100:0, compare four rebalancing strategies (monthly/quarterly/annual/5% band, lump-sum and DCA), and see concentration vs. diversification: a single Taiwanese stock (drawn point-in-time from all 673 stocks listed on 2004-02-11, *including 4 that later delisted*, terminal return −30% per Shumway 1997) vs. 0050 vs. S&P 500 vs. global equity vs. global 80:20.

**Full report (Traditional Chinese):** [docs/REPORT.md](docs/REPORT.md)

**Write-ups (Traditional Chinese):**
- [The 90% win rate, and the other half of the story](https://drpwchen.com/posts/loan-invest-sim/) — full distribution, method, limits, and how many rounds of correction this model went through
- [Get the order right before you borrow](https://drpwchen.com/posts/before-you-borrow/) — the four gates to clear first, plus win rate as the loan rate goes from 2.5% to 6%
- [Personal finance basics, six parts](https://drpwchen.com/posts/personal-finance-1-getting-started/) — asset allocation, insurance, budgeting (2019)

---

## Headline findings

**Every figure below is real TWD purchasing power** — net of inflation, of a simplified dividend withholding tax, and of TWD/USD FX. Nominal figures are labelled as such wherever they appear.

Data: 1994-06-02 to 2026-08-05 (7,912 trading days). 10,000 simulated paths, stationary block bootstrap, seed 42.

**A 30-year backtest of three stock/bond mixes (annual rebalancing):**

| Mix | Real CAGR | Nominal CAGR | Real volatility | Real max drawdown |
|---|---|---|---|---|
| 90:10 | 6.07% | 7.49% | 17.69% | -51.41% |
| 80:20 | 5.76% | 7.18% | 15.95% | -45.54% |
| 60:40 | 5.06% | 6.46% | 12.92% | -33.24% |

**Loan lump-sum vs. dollar-cost averaging** (NT$5M loan, 20-year evaluation horizon):

| Scenario | Win rate for the loan | Loan final value P50 | DCA final value P50 |
|---|---|---|---|
| 7y @ 2.15% | 79.2% | 16,273,270 | 13,848,331 |
| 7y @ 2.47% | 77.6% | 16,273,270 | 14,001,654 |
| 10y @ 2.15% | 82.3% | 16,273,270 | 12,988,943 |
| 20y @ 2.15% | 89.4% | 16,273,270 | 10,735,985 |
| 20y @ 2.47% | 87.8% | 16,273,270 | 11,058,779 |

*How the two arms are made comparable:* the DCA arm contributes the loan's monthly payment quarterly, so that **nominal total contributions equal nominal total repayment**. Because the simulated paths are real returns while a loan payment is a fixed nominal amount, each contribution is then deflated by its own timing at π = 0.1098%/month (1.326%/yr, the historical DGBAS CPI average). Both sides therefore lose purchasing power over time, treated identically. Versions before v0.3.0 fed the fixed nominal payment straight into real paths, which silently assumed the DCA investor's contributions rose with inflation while the borrower's did not — see the [changelog](CHANGELOG.md).

The loan wins 77.6%–89.4% of the time. **The other 10.6%–22.4% is the part that matters.** In the worst 5% of paths the NT$5M ends up at 5,533,691 in real terms — after you have already repaid between 5,390,153 and 6,341,311 in nominal principal and interest.

And along the way: the median path draws down **-37.9%** at some point, the 25th percentile **-45.5%**, the 5th percentile **-58.4%**. The loan payment is due regardless.

**Probability of being underwater** (year-end real assets below the real value of the outstanding loan balance) is concentrated in the first three years, and is worse for longer loans because principal amortises more slowly:

| Scenario | Year 1 | Year 3 | Year 5 | Year 10 |
|---|---|---|---|---|
| 7y @ 2.15% | 6.9% | 0.3% | 0.0% | 0.0% |
| 10y @ 2.15% | 11.5% | 2.2% | 0.2% | 0.0% |
| 20y @ 2.15% | 19.9% | 8.5% | 3.7% | 0.3% |

Note that the v0.3.0 correction raised win rates but left this table, the loan-side final values, and the drawdowns **completely unchanged** — it improved how often you win, not what losing looks like. (The v0.3.2 independent-review corrections did move both sides, slightly against the loan; see the AI collaboration statement.)

Full tables, including the stock/bond mix comparison and block-length sensitivity, are in [results/summary.md](results/summary.md).

---

## AI collaboration statement

This project went through three rounds of correction. All followed the same principle: establish that the framework is internally consistent, then let the numbers land where they land.

**Modelling — Gemini 3.1 Pro.** The original analysis (`finance_test.ipynb`) covered data download and stitching, tax and FX handling, deflation by Taiwan's official CPI, the historical backtest, the Monte Carlo comparison, and the rebalancing-frequency test. The architecture and the data-handling ideas (stitching daily *returns* rather than prices; using real DGBAS CPI rather than a flat inflation assumption) were sound.

**Round 1 — Claude review (v0.1.0).** A code and methodology review identified four defects:

| # | Defect | Consequence |
|---|---|---|
| B1 | Part 5 was labelled "real" but never divided by the CPI multiplier | The same 90:10 portfolio reported 6.29% in one table and 7.57% in another |
| B2 | Part 2 referenced a variable defined only in Part 5 | The notebook could not be run top to bottom |
| M1 | The Monte Carlo engine used `scipy.stats.t.fit()`, whose `loc` is a robust estimate close to the median and overstates drift on left-skewed returns | Simulated 20-year median annualised return of 13.5% against a historical real CAGR of 6.29% — **roughly double** |
| M2 | The 60:40 mix fitted a Student-t with df = 2.06, where variance nearly diverges | 60:40 final values collapsed to about a third of 90:10 — a pure statistical artifact |

The parametric engine was replaced with a **stationary block bootstrap** (Politis-Romano) that resamples real historical daily returns, plus machine-checkable gates that fail the run rather than being loosened.

**Round 2 — user correction (v0.3.0).** A subtler asymmetry survived round 1, and the repo owner caught it: the simulated paths are *real* returns, but a loan payment is a *fixed nominal* amount whose purchasing power erodes every month. Feeding that fixed nominal payment straight into a real path silently assumed the DCA investor's contributions were indexed to inflation while the borrower's were not — favouring DCA and understating the loan's win rate. Contributions are now deflated by their own timing, and the fair-comparison framing was restated as *nominal contributions = nominal repayment*.

**Round 3 — independent re-verification (v0.3.2).** After v0.3.1 went public, a fresh-context reviewer (an AI agent with no access to the development history) re-implemented the backtest and cash-flow math from the spec and the cached data, and reconciled against this pipeline. Every headline number reproduced to full precision — but two residual biases surfaced, **both favouring the loan side**: ① the simulation used 252 trading days per year while the merged multi-market calendar actually has 245.9, so every "20-year" path silently drew 20.5 years of historical drift (+3% terminal wealth); ② the underwater probability compared real assets against the *nominal* outstanding balance, overstating early-year odds by 10–35% relative (the browser engine had it right all along). After the fix the sanity-gate residual shrank from 0.164pp to 0.006pp.

**Net effect of the three rounds:** median final values halved (33,417,127 → 16,273,270) and the 60:40 collapse disappeared — round 1. Win rates rose by roughly 5–7 percentage points — round 2. Loan P50 down another ~3%, the 20y win rate 90.0% → 89.4%, underwater odds revised down — round 3. Round 2 favoured the loan; round 3 went against it. Each correction was made because the framework was inconsistent, not because of which way it moved the answer.

The one-line summary for all three: **the direction was right, the magnitude needed to be made honest.**

---

## Reproduce

Everything — the CPI fetch from Taiwan's DGBAS, the price download from Yahoo Finance, all five parts, and every output file — runs from one entry point:

```bash
uv run --with yfinance,pandas,numpy,scipy python analysis/run_all.py
```

Flags: `--refresh-prices`, `--refresh-cpi` to bypass the local cache in `data/`.

Outputs:

- `results/results.json` — machine-readable, every table plus run metadata
- `results/summary.md` — human-readable, all tables including before/after comparisons
- `web_data/monthly_returns.json` — monthly real and nominal return series for the browser simulator

The run is deterministic (`seed = 42`, with per-task seeds derived by hash so results don't depend on execution order). It aborts with `FAILED` if the CPI spot-check, the Part 2 / Part 5 consistency gate, or the bootstrap sanity gates don't pass.

---

## Fork it and change the assumptions

Disagree with any of this? Every parameter is a constant at the top of `analysis/run_all.py`:

| What to change | Where |
|---|---|
| Stock/bond mixes and their component weights | `PORTFOLIOS` |
| Number of simulated paths | `N_SIMS` |
| Bootstrap block length (main and sensitivity) | `BLOCK_MAIN`, `BLOCK_SENS` |
| Random seed | `SEED` |
| Tickers and the pre-ETF backfill funds | `TICKERS_ETF`, `TICKERS_FUND` |
| Taiwan equity ticker and its first month | `TW_TICKER`, `TW_FIRST_MONTH` |
| Winsorization limits and the FX jump filter | `LIMIT_EQUITY`, `LIMIT_BOND`, `FX_JUMP_LIMIT` |
| Dividend tax drag | `TAX_DRAG_ANNUAL` |
| Data start date | `START_DATE` |

The Part 3 scenario matrix (loan term × interest rate) and the Part 4 parameters (loan size, term, rate) are defined inline in `main()` — search for `scenarios = [` and `loan4, rate4, yrs4`.

To compare engines, `simulate()` accepts `engine="student_t"`, which reproduces the original parametric approach. It is kept only for the before/after comparison and should not be used as a basis for decisions.

---

## Method summary

- **Universe:** VTI, VGK, VPL, VWO (equity, equal-weighted), VGIT, BWX (bonds, half each); pre-ETF history backfilled with the corresponding mutual funds (VTSMX, VEURX, VPACX, VEIEX, VFITX) by splicing *daily returns*, not prices.
- **Taiwan equity option (v0.2.0; chain re-based in v0.6.0):** the web simulator can swap part of the equity sleeve into 0050 (Yuanta Taiwan Top 50, dividends included). The Taiwan leg now starts **1997-08** and deliberately includes the 1997 Asian crisis and the 2000 dot-com bust — a 2003-07 start (the previous behaviour) sampled only the bull years and flattered Taiwan. Three splices: 1997-08..2002-12 is an **approximation** (^TWII price index + the TWSE Fact Book market dividend yield / 12; no total-return index existed yet — the same construction reproduces the real TR index on 2003-2008 with monthly correlation 0.9953 and mean |annual gap| 1.3pp, systematically slightly optimistic and worst in crash years), 2003-01..2009-01 is the TWSE total-return index, 2009-02 onward is 0050 itself. Both pre-2009 segments track the whole market (0050 only listed 2003-06) and carry no ETF expense ratio. 0050 is TWD-denominated: no FX conversion and no 30% US dividend withholding drag. Nothing before 1997-08 exists in any free source, so the chain honestly stops there.
- **Real TWD:** portfolio value in USD → multiplied by TWD/USD → divided by a daily CPI multiplier interpolated (pchip) from DGBAS monthly 消費者物價指數 總指數.
- **Monte Carlo:** stationary block bootstrap of each portfolio's historical real daily TWD returns; geometric block lengths with an expected block of 21 trading days; 10,000 paths. This preserves volatility clustering and fat tails while pinning drift to history. A simulated year is the panel's measured 245.9 trading days (the merged multi-market calendar has fewer common trading days than a single exchange's 252). The Monte Carlo standard error on win rates is about ±0.4pp.
- **Fair comparison:** the DCA arm invests the loan's monthly payment quarterly for the loan term, so nominal contributions equal nominal repayment; each contribution is then deflated by its own timing at π = 0.1098%/month, since the paths are real returns and the payment is a fixed nominal amount. Both arms are evaluated on the same random paths at year 20.

Full specification: [docs/METHODOLOGY_SPEC.md](docs/METHODOLOGY_SPEC.md). Browser simulator specification: [docs/WEB_SPEC.md](docs/WEB_SPEC.md).

---

## Known limitations

- **BWX is backfilled with VFITX** (US treasuries) before its inception — a US-bond proxy standing in for international bonds, which understates FX exposure in that period.
- **The tax drag applies 30% withholding to bonds too.** US treasury interest is actually exempt for foreign holders, so the model is conservative here.
- **Winsorization smooths the 1997 Asian financial crisis TWD move**, understating extreme FX shocks.
- **The CPI deflator is held flat** after the last published month (2026-06-01).
- **Inflation is a single fixed rate** (π = 0.1098%/month) with no uncertainty of its own. A high-inflation regime would erode both the DCA contributions and the borrower's real payment burden, in opposite directions; that interaction is not modelled.
- **The bootstrap can only resample what happened.** 1994–2026 includes the Asian financial crisis, the dot-com bust, the global financial crisis and the pandemic. It does not include a Taiwan Strait conflict, a structural TWD devaluation, or a reversal of globalisation.
- **The model assumes you never miss a payment and never sell.** Most real-world failures of this strategy come from those assumptions breaking, and the model says nothing about them.
- **"Underwater" is not a margin call.** A personal loan has no forced-liquidation mechanism — as long as you make the monthly payment, nobody can make you sell at the bottom. The underwater probability measures **forced-exit risk**: if you had to wind up in that year, selling everything would still not clear the remaining principal.
- **Mixes that include Taiwan equity are still an extrapolation.** Since v0.6.0 the Taiwan sample includes the Asian crisis and the dot-com bust (full-chain real CAGR ~8.3%, versus ~13% on the old bull-only 2003+ window), so the flattery is much reduced — but resampling Taiwan's own past three decades as "the future" still assumes Taiwan keeps outperforming, and the post-2003 bull run still dominates the sample. The simulator surfaces this warning when the slider moves and suggests revising the expected return down by hand. The pre-2009 series is an index proxy and carries no dividend tax drag, both of which flatter it further.
- **No transaction costs, FX spreads, brokerage fees, loan origination fees, or prepayment penalties.** The loan rate is assumed fixed; Taiwanese unsecured loans are typically index-linked.

---

## Disclaimer

This is an educational historical simulation. **It is not investment, financial, legal, or tax advice**, and it is not an offer or solicitation for any financial product. Past performance does not indicate future results. **A loan must be repaid whether or not the investment loses money.** Consult a qualified professional and accept your own risk.

Full text, English and Traditional Chinese: [DISCLAIMER.md](DISCLAIMER.md).

---

## License

MIT — see [LICENSE](LICENSE).

Data sources: Yahoo Finance (historical prices and FX), Directorate-General of Budget, Accounting and Statistics, Executive Yuan, Taiwan (CPI). Neither endorses this project.

Author: [drpwchen](https://drpwchen.com)

Find this useful? [Buy me a drink](https://drpwchen.com/support/) 🧋

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