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Methodology: how Paperball calculates

In short

Paperball calculates on historical daily prices and includes what changes the return of a real portfolio: costs, distributions, taxes and rebalancing. This page describes the rules in the calculation engine and every change to the calculation that has changed saved results.

The calculation rules

  • Costs: order fees (fixed, percentage, with minimum and maximum), front-end loads per security and an annual custody fee as a percentage of the portfolio value. A fund's ongoing charges (TER) are already included in its price and are not deducted again.
  • Distributions: every distribution counts on the ex-dividend date — the first trading day on which the security trades without entitlement to the distribution — and is taxed. What happens to it next is your choice: by default it is ignored and shown as a metric of its own. When reinvested, the backtest uses it to buy units of the same security, without an order fee and without a front-end load. With a savings or withdrawal plan it can instead pay the withdrawals from the cash account or go into the liquidity buffer.
  • Taxes: the tax mechanics of 8 countries. The calculation covers taxes on capital gains and distributions and, depending on the country, also a partial exemption for funds (Teilfreistellung), an annual tax-free allowance, a reduced rate after a holding period, wealth tax and, for Germany, the advance lump-sum tax on funds (Vorabpauschale). In addition, there are tax-advantaged accounts such as ISA, Roth IRA or ISK. The user sets the tax rates; non-binding example values per country can be adopted. The result is an approximation, not tax advice.
  • Rebalancing: on each rebalancing date, Paperball Finance sells every position whose value exceeds its target value by more than 1 % of the portfolio value, down to the target value. The proceeds go to the positions that are more than 1 % below their target value; if none is that far below, to all underweight positions – in each case in proportion to their distance from the target value. Sales incur an order fee and tax on the gain realised; purchases incur an order fee and a front-end load.
  • Savings plan: in the backtest, Paperball Finance invests one instalment on the first trading day of each savings period, optionally with an annual increase in the instalment and with separate savings plan fees. The savings phase can be followed by a withdrawal plan in the same run.
  • Withdrawal plan: a fixed amount adjusted for inflation or a percentage of the value, monthly to annually; with a fixed amount, optionally with an interest-bearing cash buffer.
  • Period: a backtest starts no earlier than the day from which all selected securities have continuous daily prices. The security whose price history begins latest determines the start; a shorter period can be selected. Paperball Finance does not extend short price histories using index or substitute series; users' own price files that reach further back can extend them. Monte Carlo on asset mixes calculates only within the period that is covered by all building blocks with a price history of their own.
  • Currency: Paperball Finance calculates backtests and Monte Carlo simulations in euros. Prices in other currencies are converted for each day at the reference rate of a central bank that was published on that day or most recently before it. Before 1999, Paperball Finance calculates via the Deutsche Mark: with the daily rates of a central bank, converted at the fixed rate of 1.95583 Deutsche Mark per euro; these are available from 1971 for the US dollar, pound sterling, Swiss franc, yen, Canadian dollar and the Scandinavian currencies. For the US dollar, the fixed parities of the Deutsche Mark apply from October 1949 to 1970: 4.20, from March 1961 4.00 and from October 1969 3.66 Deutsche Mark per dollar; during the float in autumn 1969, the last rate before it applies. There is no currency hedging: return, volatility and drawdown include exchange rate movements from the perspective of a euro investor. If an exchange rate is missing or the currency of a security is unknown, the backtest does not start. Results can be displayed in other currencies: the finished euro amounts are then converted at the exchange rate on the start date and on the end date; the metrics in per cent remain unchanged. The exception is the years before November 1949 in the building block US equities since 1871: from December 1924 to June 1941 it calculates via the Reichsmark (monthly averages of New York rates); before and after that there is no usable exchange rate, so its return counts in dollars there, without exchange rate movements. If US equities since 1871 and German equities since 1871 are in the same asset mix, both count in purchasing power in the months without an exchange rate between mark and dollar. The building blocks global equities since 1871 work the same way: until July 1914 via the fixed gold parity, from December 1924 to June 1941 via the Reichsmark (sterling via its dollar rate), and in the months without an exchange rate between mark and dollar each country in purchasing power. In an asset mix with German or global equities since 1871, US equities count via the gold value of the dollar (the dollar rate of sterling) until December 1878, because the dollar only returned to gold in 1879.
  • Metrics: annual return (CAGR), volatility, maximum drawdown and the longest time without a new high (in trading days), Sharpe, Sortino, Calmar, Sterling and Martin ratios, Ulcer index, plus beta and alpha against a selectable benchmark. The Sharpe ratio, the Sortino ratio and alpha are calculated against a risk-free rate that the user sets. The Community-Backtests calculate all portfolios to a uniform standard: without taxes and fees, with a risk-free rate of zero and starting no earlier than a uniform cut-off date. This makes the metrics of the Community-Backtests comparable with one another. The benchmarks include distributions (total return) and are converted into euros.
  • Monte Carlo: two calculation methods are available. “Drawn anew” (default): from the historical monthly returns, Paperball Finance draws pieces of consecutive months at random and with replacement and joins them into 10,000 possible paths – a historical block bootstrap. The default is pieces of 12 months; the piece length can be set from 1 to 60 months or made random, 12 months long on average (stationary bootstrap after Politis and Romano). Within a piece, crisis and recovery stay together; no normal distribution is assumed. Optionally, the return can be set cautiously: every monthly return is scaled down by the same factor until the average return matches a more cautious value – suggested from long equity and bond series, compared after inflation, or chosen freely; it is never raised. “Real start years”: without randomness, Paperball Finance plays the plan through in real order from every January of the price history from which the whole term still fits in – a historical simulation with rolling windows, using real prices only and only with at least 20 start years. Both methods only assemble what occurred in the price history; they cannot generate market phases outside the data. No calculation is run with a price history shorter than 5 years, and with one shorter than 10 years the interface shows a warning. The calculation covers savings and withdrawal phases, optionally with tax under the same settings as in the backtest, approximated per path. As the sustainable amount, Paperball Finance shows the highest withdrawal that would have lasted until the end of the term in at least 90 % of the paths.
  • Inflation: Where the price history starts in February 1871 or later, Paperball Finance takes inflation from the same months as the returns, in the currency in which the returns count: official German consumer prices from November 1949, the cost-of-living index of the German Reich from December 1924 to June 1941, and US consumer prices in the other months before that – without a euro or Reichsmark exchange rate, returns count in dollars there. If it starts earlier, a fixed rate applies. If an asset mix contains German or global equities since 1871, the German cost-of-living index applies until October 1949; from November 1918 to January 1924 this building block counts in purchasing power, without inflation. The building blocks global equities since 1871 count in purchasing power, without inflation, in the months without an exchange rate between mark and dollar.
  • Building blocks: instead of a saved portfolio, a Monte Carlo simulation can calculate an asset mix of building blocks. Building blocks are series for equities, bonds and gold, the money market rate, a fixed interest rate and, if desired, the securities the user has saved. The Bund building blocks are not traded funds: Paperball Finance calculates them from the yields that a central bank has published monthly since 1972, as a zero-coupon bond with a constant remaining maturity of 2, 7 or 10 years that is bought anew every month. The building block for US Treasuries is calculated in the same way: from the yields on ten-year US Treasury bonds that have been published daily since 1962, as a bond with a semi-annual coupon and a constant remaining maturity. Two building blocks go back further than any fund: German equities since 1954 from an academic series (all shares in the top Frankfurt segment) and US equities since 1926 from an academic data collection (the entire US stock market); from 2014 and 1993 respectively, they are continued with traded ETFs. The building block for German government bonds since 1956 is calculated in the same way: from the yield on ten-year German government bonds published monthly by an international organisation, as a bond with an annual coupon; for the years before 1972 it is probably based on the average yield on German bonds outstanding, an average across all remaining maturities. The longest series is US equities since 1871: until 1926 estimated from two research data collections (monthly averages of a broad index and month-end prices of individual stocks), then the same series as US equities since 1926; before November 1949 it calculates via the Reichsmark from December 1924 to June 1941 and otherwise in dollars, without a currency effect, because there is no usable exchange rate. Gold goes back to 1960: until 2009 the gold price in dollars from the commodity prices of an international organisation (monthly averages), converted into euros, from 2010 Xetra-Gold. German equities since 1871 go back just as far: until 1953 an index of German blue-chip stocks with dividends from a research study (monthly averages), then the same series as German equities since 1954; from August 1914 to October 1918 the stock exchange was closed, and from November 1918 to January 1924 the return counts in purchasing power. Global equities since 1871 are estimated, in five versions: like the MSCI World (until 2007 70 % US, then the iShares MSCI World), with a fixed 30 or 50 % US or US only (from 2008 from the SPDR S&P 500 ETF and the iShares MSCI EAFE), and historically weighted by the market value of the countries at the time (from 2008 the iShares MSCI World). Until 2008 they are made up of US, British, German, French, Italian and Japanese equities. The British equities come from a central bank's collection of historical economic data, without dividends, using the dividend yield of US equities instead; for France, Italy and Japan there are only annual figures from a research data collection, spread over twelve months using the pattern of the British months. The selection for new asset mixes offers the longest series per asset class – the ten-year German government bond as one series since 1956, until August 1972 from the international organisation's yield, then from the yield curve –; country series and individual index ETFs keep working in saved asset mixes and as benchmarks in the backtest.
  • Data: Paperball Finance obtains prices, distributions and master data of the securities from a licensed market data provider. The benchmarks are ETF series built from these prices in which the distributions are reinvested (total return). Sector and country breakdowns of funds come from the holdings lists of the fund companies, supplemented by AI research via web search. Exchange rates and money market rates are reference values of central banks, bond yields come from central banks and an international organisation, and the long equity series from academic research. Users can upload their own prices as CSV, Excel or ODS files.

What the calculation does not do

  • A backtest shows how an allocation would have developed in the past. That is not a reliable indicator of future results and not a recommendation to buy or sell anything.
  • The tax calculation reproduces a country's tax mechanics, with rates that you set yourself. It is an approximation and not tax advice.
  • Monte Carlo only combines what occurred in the price history. It cannot create market phases outside the data; the paths describe possible developments, not a forecast.

Changes to the calculation

If the calculation changes in a way that would give saved results different figures, Paperball raises the calculation version and recalculates saved portfolios. Current calculation version: 23.

Ulcer Index and Sortino ratio: the Ulcer Index is now given in percentage points, like the maximum drawdown; before, it was 100 times too small. The Sortino ratio averages the squared losses over all days instead of only the losing days, which makes it roughly one and a half times as high.
Retirement savings account (Altersvorsorgedepot) in the backtest: a year's state grants are added as a payment at the end of June of the following year. Contributions above the subsidy ceiling are taxed on payout only on their share of gains.
Sustainable withdrawal: the displayed amount, rounded down to whole euros, is itself checked once more and therefore lasts over the whole path.
Distributions: instead of one switch there are four options — reinvest, offset against withdrawals, put into the liquidity buffer or ignore.
Withdrawal: even with high minimum or fixed fees, the calculation sells exactly enough for the withdrawal to arrive after fees and taxes.
Rebalancing: if no position is far enough below its target value, the sale proceeds go to all underweight positions. Previously they remained unused in this case.
Withdrawal: results show what each withdrawal was paid from — distributions, sales or the liquidity buffer.
Sterling ratio: the denominator is the average annual maximum drawdown plus ten percentage points, as in its definition.
Distribution yield: averaged only over complete calendar years; a partial year no longer counts.
Distributions: if an ex-date falls into a price gap of another security, the distribution is booked on the next common trading day instead of being dropped. Saved portfolios calculate their spread across sectors and countries with the same rule as the backtest.
Advance lump-sum tax (Vorabpauschale): in the backtest, funds bear the tax on the base return every year; on sale it reduces the gain.
Withdrawal: results state the sustainable withdrawal, the largest fixed amount that would have lasted over the whole path. Tax on the liquidity buffer's interest is paid only from money that is available.
Savings plan in the backtest, withdrawals optionally taken first from overweight positions, reinvestment of distributions as a choice and an annual custody fee.
Withdrawal plan and liquidity buffer in the calculation engine. Saved portfolios calculate with the tax of their backtest.
Rebalancing: the threshold from which a portfolio is rebalanced depends only on the share of the portfolio value — a small portfolio is rebalanced like a large one.

Frequently asked questions about the calculation

Does Paperball calculate with the prices of the funds or of their indices?

With the prices of the securities themselves, day by day. The backtest does not extend short price histories with index or substitute series; your own price files that reach further back can extend them. Only in Monte Carlo can a short price history optionally be supplemented with estimated crisis years — labelled as estimated.

What happens to saved results when the calculation changes?

Paperball recalculates saved portfolios as soon as their inputs or the calculation version change; this is checked daily. The result list of a backtest stays as it was calculated until it is recalculated.

Is a result a recommendation?

No. Paperball shows how allocations would have developed in the past and what range Monte Carlo produces. It does not recommend what anyone should buy or sell.

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Last updated: 25 Oct 2026