Glossary: terms starting with P

Varying data, periods or parameters until a result looks significant. In backtests this often happens unconsciously, when only the successful variants are pursued further.

More: Backtest overfitting: when the past looks too good

The probability of seeing a result at least as extreme if in truth there is no effect. A small p-value argues against chance, but it is not the probability that the strategy works.

Loan that individual investors make to consumers or businesses through an online platform, usually in small portions. The return is the interest rate minus defaults; on top come risks from loan originators and the platform.

Source: Iyer et al. (2016); Regulation (EU) 2020/1503, Art. 21 (2020)

Distance between two prices in pairs trading, such as ln A − β × ln B or the difference of the normalized prices. If it reverts to its mean, the position makes money; if it keeps widening, the position loses. Not to be confused with the bid-ask spread.

Source: Vidyamurthy (2004)

More: Pairs trading and statistical arbitrage: idea, methods, research · Methodology: Long/short and statistical arbitrage: methodology

Trading two similar securities against each other: if their prices drift unusually far apart, the relatively expensive one is sold short and the relatively cheap one is bought; the trade is closed when the gap narrows again. The position is largely market neutral.

Source: Gatev et al. (2006); Vidyamurthy (2004)

More: Pairs trading and statistical arbitrage: idea, methods, research · Methodology: Long/short and statistical arbitrage: methodology

Value at risk from the mean and standard deviation, assuming a normal distribution, e.g. the mean minus 1.645 standard deviations at the 95% level. Underestimates losses when tails are fat.

Source: Jorion (2007)

More: Value at risk and conditional VaR: how much you could lose in a bad case

The base index from which an index variant is derived, e.g. a global index from which an ESG or climate variant is built. The variant's deviation is measured with active share and tracking error.

More: ESG ETFs: index rules, ratings, returns and the limits of sustainable funds

EU Paris-aligned benchmark: a climate index with a greenhouse gas intensity at least 50% below that of the investment universe, which must then fall by at least 7% a year, plus exclusions of coal (from 1% of revenue), oil (from 10%), gas (from 50%) and power generation with more than 100 g CO₂e per kWh (from 50%).

Example: After ten years with a 7% reduction a year: 0.5 × 0.93¹⁰ = 24.2% of the universe's initial intensity.

Source: EU Climate Benchmarks Regulation (2020)

More: ESG ETFs: index rules, ratings, returns and the limits of sustainable funds

Hedge fund fee model in which salaries, bonuses, data, technology and offices are charged directly to the fund instead of being paid from a fixed management fee, usually on top of a performance fee. Typical of multi-strategy funds.

Source: Bloomberg (2025)

A result depends not only on the final price but on the entire price path, as with leveraged ETFs, CPPI or Asian options. With daily reset leverage L, volatility without a trend costs roughly the factor exp(−L × (L − 1) / 2 × σ² × T).

Example: Leverage 2, index unchanged after one year, volatility 20%: exp(−0.04) = 0.961, i.e. −3.92%.

Source: Avellaneda and Zhang (2010)

More: Leverage in a portfolio: return, risk, costs and margin calls

A sequence of iterated integrals of a path that summarizes the course of several time series, including the order of their movements. Linear functions of the signature can approximate continuous, even path-dependent functions of the path; in portfolio theory they serve as building blocks of signature portfolios or as a substitute for correlation matrices.

Source: Allan et al. (2023); Cuchiero and Möller (2025); Gregnanin et al. (2024)

More: Beyond Markowitz: estimation error, noise and true diversification · Methodology: Dimensions methodology: how Portfolio Lab counts independent risks

Pension financing in which the contributions of those in work pay current pensions directly, without any capital being saved. This is how the statutory pension insurance works in Germany. Its implicit return follows growth in the total wage bill, not the capital market.

Source: Samuelson (1958); Aaron (1966)

Interest that is not paid in cash but added to the loan; the debt grows. With a PIK toggle, the borrower may choose. According to FSB (2026), PIK is used in about 12% of private credit loans; a rising share can point to payment difficulties.

Example: 8% PIK on $100 million over three years: 100 × 1.08³ = $125.97 million of debt at the end.

Source: FSB (2026); IMF (2024)

More: Private debt in practice: direct lending, unitranche, mezzanine, loan structure and fund vehicles

Timing of payments in financial mathematics: in advance at the start of a period, in arrears at the end. A payment in advance earns interest for one period longer; the future value and the present value of an annuity are therefore higher by the factor (1 + i).

Example: $100 a year for ten years at 5%: future value in arrears $1,257.79, in advance $1,320.68.

More: Compound interest: how returns on returns make wealth grow · Methodology: Methodology of the free calculators: FIRE, savings plan, withdrawal and leveraged ETF

Share of profit that a company pays out as dividends. A ratio close to or above 100% leaves little room: if profits fall, the dividend has to be cut or paid out of the company's substance.

Example: Earnings per share $4, dividend $3: payout ratio 75%.

More: Dividend strategies and dividend ETFs under scrutiny: irrelevance, factors, risks

Colloquially, a share with a very low price and a small market capitalization, often traded over the counter or in lightly regulated market segments, with little trading volume and few disclosure requirements.

Source: Leuz et al. (2017)

More: Pump and dump and penny stocks: how the scam works

The claim to a future pension accrued so far, e.g. from Social Security, a civil service pension or an employer pension plan. Its present value is a large, bond-like item in total wealth.

More: Your whole wealth: total wealth from portfolio, home, human capital and car · Methodology: ALM methodology: how Portfolio Lab measures assets against liabilities

Share of observations up to a given value. The 90th percentile is exceeded in only 10% of cases; a percentile rank of 85% means: higher than on 85% of days.

More: VIX: how to read the stock market's fear gauge

The share of historical values that lie below the current value. A VIX with a percentile rank of 90 is higher than on 90% of the days in the period under review.

Methodology: Volatility dashboard: methodology

Breaking down the return difference between portfolio and benchmark into its causes, e.g. allocation, selection and interaction according to Brinson, or factor contributions from a regression.

More: Performance attribution: Brinson decomposition, factors and linking over time

How much the variant that was best in the backtest falls off on unused data: in CSCV, the regression of the winner's out-of-sample metric on its in-sample metric across all splits. A flat or falling line means that a good in-sample value reveals little or nothing about out-of-sample performance.

Source: Bailey et al. (2016)

More: Probability of backtest overfitting: does the best variant deliver what the backtest promises?

A performance-related fee for the manager, usually a share of the profit above a hurdle or above the high-water mark. Classic for hedge funds and CTAs is 20% alongside a 2% management fee, “2 and 20” for short.

Example: Gross return 8%, 2% management fee, 20% performance fee: 8 − 2 − 0.2 × 6 = 4.8% net.

Source: Bhardwaj et al. (2014)

More: CTAs and managed futures: industry, position sizing, costs and crisis protection

An allocation proposed by Harry Browne, with one quarter each in equities, long-term government bonds, gold and money market. Each component is meant to shine in a different economic environment.

More: All Weather portfolio: ready for every economic climate?

Futures contract without an expiry, mainly on crypto assets. To keep its price close to spot, buyers and sellers regularly pay each other a funding rate; unlike a classic future, it does not have to converge to the spot price.

Source: He et al. (2022)

The highest annual withdrawal, as a percentage of starting wealth and adjusted for inflation each year, at which wealth at the end is at least as large in real terms as at the start. It is lower than the safe withdrawal rate, which is allowed to use up the capital.

More: Safe withdrawal rate: how good is the 4% rule? · Methodology: Backtest methodology: data, process, rebalancing and costs

Distortion of average returns because a rare, very bad event does not occur in the sample at all, or occurs too rarely. Named after the Mexican peso; regarded as one explanation for the high carry returns.

Source: Burnside et al. (2011)

More: Carry trade: earning interest with currencies and bonds

Covered bond issued by a bank that is additionally secured by a pool of mortgage or public-sector loans. Pfandbriefe are regarded as very safe and yield slightly more than government bonds.

Methodology: Risk factors in the look-through: methodology

The ETF actually buys the securities in the index, either all of them (full replication) or a representative selection (sampling). The counterpart to synthetic replication via swaps.

More: ETF basics: costs, replication and distributions

Data as they actually stood at each earlier point in time, without later corrections, revisions or reclassifications. Only with such data does a backtest avoid look-ahead bias, especially with alternative data, whose history providers can adjust retrospectively.

More: Alternative data: job ads, Reddit, X, satellites and card payments

The strategic asset allocation set down in writing: target weights per asset class, each with an index. Together they form the benchmark against which tactical deviations and security selection have to be measured.

Source: Brinson et al. (1986); GIC (2013)

More: Strategic and tactical asset allocation and the total portfolio approach

Separating market return (beta) from selection skill (alpha): a manager's alpha is transferred to a different market using futures or swaps. A forerunner of return stacking; it requires leverage and financing.

Source: Kung and Pohlman (2004)

More: Leveraged ETFs and return stacking: how they work, long holding periods, costs and research

A measure of diversification that links it to the non-normality of portfolio returns and is defined as a ratio of homogeneous risk measures of the same degree. That is why leverage does not change the measure. Maximizing it is a non-convex optimization problem.

Source: Barkhagen et al. (2023)

More: Beyond Markowitz: estimation error, noise and true diversification · Methodology: Dimensions methodology: how Portfolio Lab counts independent risks

Strategies that limit a portfolio's downside while leaving part of the upside open, e.g. CPPI or OBPI. They sell when prices fall and buy when they rise; they came under criticism in the 1987 crash.

Source: Leland (1980); Rubinstein and Leland (1981); Leland (1988)

More: CPPI: portfolio insurance with cushion, floor and multiplier

Extremely skewed distribution of proceeds in venture capital: a few holdings deliver almost all of a fund's return, while most deliver little or nothing. According to Hall and Woodward (2010), almost three quarters of the founders of venture-backed companies received nothing at exit.

Example: Fund with 20 holdings: one 25× hit delivers 100 of $172 million in proceeds, i.e. 58.14%.

Source: Hall and Woodward (2010); Kerr et al. (2014)

More: Venture capital: funding rounds, valuation, the power law and what research knows about returns

Value of a start-up before the money from a new financing round. The post-money valuation is pre-money plus the investment; the new investor's stake is the investment divided by post-money. An option pool created before the round is included in the pre-money and dilutes only the existing shareholders.

Example: $4 million pre-money plus a $1 million investment give $5 million post-money; the investor holds 1 / 5 = 20%.

More: Venture capital: funding rounds, valuation, the power law and what research knows about returns

A market in which contracts on future events are traded, e.g. election results, interest rate decisions or sports results. The price of a contract that pays 1 if the event occurs is regarded as an estimate of its probability, distorted by fees, interest and betting behavior.

Source: Wolfers and Zitzewitz (2004); Berg et al. (2008)

More: Prediction markets as bets on events: what they can and cannot do

In Bayesian statistics, the distribution of future returns that already incorporates the uncertainty about means and covariances. It is wider than the distribution with fixed, estimated parameters.

Source: Harvey et al. (2008)

More: Michaud resampled optimization: more stable portfolio weights through simulation

The percentage by which acquiring the underlying via a warrant is more expensive than buying it directly. For a call: (strike + price / ratio − spot) / spot. It equals the rise the underlying needs by expiry for you to break even.

Example: Spot $100, strike $110, price $0.53, ratio 0.1: (110 + 5.30 − 100) / 100 = 15.30%.

The deviation of an ETF's market price from its net asset value. Small in liquid markets; in periods of stress, such as for bond ETFs in March 2020, significant discounts can arise.

More: ETF basics: costs, replication and distributions

Today's value of a future payment, discounted at an interest rate. At 3%, $1,000 due in ten years is worth about $744 today.

More: Asset-liability management: aligning your wealth with future spending

Present value of a series of equal payments of $1 each: it tells you how much capital is needed today to make n payments at an interest rate i per period. In arrears (1 − (1 + i)^−n) / i, in advance (1 + i) times that. Capital divided by the factor gives the sustainable withdrawal.

Example: 360 months at 5% p.a., i.e. 0.4074% a month, in advance: factor 189.43; from $300,000, this supports $1,583.71 a month.

More: Safe withdrawal rate: how good is the 4% rule? · Methodology: Methodology of the free calculators: FIRE, savings plan, withdrawal and leveraged ETF

Market value divided by the book value of equity. Fama and French sort stocks into value and growth by its inverse (book-to-market).

Source: Fama and French (1992)

More: Factor investing: understanding the Fama-French factors

Share price divided by earnings per share, usually over the last twelve months. A P/E of 20 means: the price equals 20 years of earnings. Fluctuates more with the business cycle than the CAPE.

More: Shiller P/E (CAPE): how expensive is the stock market?

Difference between the forward rate for a future period and today's key interest rate, often expressed in rate steps of 0.25 percentage points. Positive means the market is pricing in hikes, negative means cuts, in each case including term and liquidity premiums.

Example: Forward rate in one year 2.20%, deposit rate 2.50%: −0.30 percentage points ÷ 0.25 = −1.2 rate steps, i.e. a little more than one cut.

More: Market expectations: what the yield curve, inflation-linked bonds and options reveal about the future · Methodology: Macro dashboard and regime analysis: methodology

Breaks the common variation of many assets down into independent components. If the first one explains almost everything, the portfolio essentially holds just one risk.

More: Correlation and diversification: why mixing assets reduces risk

Loans that funds and other non-banks extend directly to companies, mostly mid-sized ones, without an exchange and without syndication through banks. They usually carry floating rates and are senior secured, are held to maturity and are valued with models rather than market prices.

Source: Cai and Haque (2024); IMF (2024)

More: Private credit and public credit compared: returns, risks and what the research shows

The probability that the true Sharpe ratio lies above a threshold, given the estimated Sharpe ratio, the length of the series, and the skewness and kurtosis of the returns. With the expected best Sharpe ratio of many trials as the threshold, it becomes the deflated Sharpe ratio.

Source: Bailey and López de Prado (2012)

More: Deflated Sharpe ratio: how much skill is in a Sharpe ratio?

The probability that the configuration that is best in the backtest performs only below average out of sample. Estimated over many splits of the data into training and test parts.

Source: Bailey et al. (2016)

More: Probability of backtest overfitting: does the best variant deliver what the backtest promises?

The probability that, with the planned withdrawals, wealth runs out before the end of the planning horizon. The counterpart of the success rate in a Monte Carlo simulation.

Source: Milevsky and Robinson (2000)

More: Monte Carlo simulation: how safe is your plan?

Statistical model for yes/no events: the probability is the cumulative distribution function Φ of the standard normal distribution evaluated at a linear function of the explanatory variable, P = Φ(α + β · x). The parameters are estimated by maximum likelihood; a related model is the logit model with the logistic function.

Source: Estrella and Trubin (2006)

More: Market expectations: what the yield curve, inflation-linked bonds and options reveal about the future · Methodology: Macro dashboard and regime analysis: methodology

Excess return of profitable companies, measured for example by gross profit relative to total assets. It complements value well, because profitable firms tend to be on the expensive side.

Source: Novy-Marx (2013)

Financing a single venture through a dedicated special purpose vehicle: lenders are repaid only from the project's cash flows, without recourse to the owners' other assets. The debt is tiered by seniority (senior before mezzanine); covenants such as lock-ups, cash sweeps and reserve accounts protect the lenders.

Example: For bank loans to projects, the cumulative default rate after ten years was 5.0% according to Moody's (Basel definition), and the average recovery rate after a default was 81.5%.

Source: Moody's (2020); Blanc-Brude and Hasan (2016)

More: Infrastructure investing: equity and debt, revenue models, valuation and research

Description of how people actually decide under uncertainty: they evaluate gains and losses relative to a reference point, are loss-averse and overweight small probabilities.

Source: Kahneman and Tversky (1979)

Equity portfolio plus bought puts as insurance. Limits the loss, but costs noticeable return over the years, because protection is expensive on average.

More: Option pricing: where fair option prices come from

Substitute series for the period before a fund's launch, such as an index or an older fund with a similar strategy. Extends backtests, but is only an approximation.

Compares a fund's cash flows with an investment in an index at the same points in time. In the KS-PME of Kaplan and Schoar, distributions (plus NAV) and calls are each divided by the index level and set in relation to each other; above 1, the fund did better. Direct alpha turns this into an excess return per year.

Example: In the sample fund: 102.42 / 85.51 = 1.20, direct alpha 3.55% a year.

Source: Kaplan and Schoar (2005); Sorensen and Jagannathan (2015); Gredil et al. (2023)

More: Measuring private markets: J-curve, cash flows, smoothing and public proxies

Listed investment with a risk similar to that of a private asset class, used as a yardstick or substitute series; for buyouts, for example, small value stocks with leverage. Unlike a series built from estimated fund values, a proxy is not smoothed and shows volatility and correlation more realistically.

Example: Small value index with a D/E of 0.5 as a proxy for buyouts with a D/E of 1.5: leverage L = 2.5 / 1.5 = 1.67.

Source: Phalippou (2014); L’Her et al. (2016); Stafford (2022)

More: Measuring private markets: J-curve, cash flows, smoothing and public proxies

Market manipulation in which perpetrators buy shares cheaply, drive up the price with promotion, rumors or sham trades, and then sell to the buyers they have lured in. Afterward the price usually collapses.

Source: Leuz et al. (2017); Aggarwal and Wu (2006)

More: Pump and dump and penny stocks: how the scam works

Put option: makes money when the underlying falls below the strike price. Bought puts hedge an equity portfolio against crashes, but cost premium on an ongoing basis.

More: Option pricing: where fair option prices come from

Model-free relationship between a European call and put with the same strike and maturity: call minus put equals the discounted forward minus the discounted strike. If the market deviated from it, there would be an arbitrage; conversely, parity yields the forward.

Example: Price and strike 100, one year, 5% interest rate, 20% volatility: call 10.45 − put 5.57 = 100 − 100 × e^(−0.05) = 4.88.

Source: Hull (2021)

More: Option pricing: where fair option prices come from · Methodology: Option pricer: how Portfolio Lab values options

Selling puts on an index and holding the notional amount in safe assets. Similar profile to a covered call: steady premiums, but losses in crashes.

Traded (or open) puts divided by calls. High values indicate a lot of hedging and therefore fear; contrarian investors often see this as a buying opportunity.

Source: Pan and Poteshman (2006)

Methodology: Sentiment: methodology for the fear and greed index, put/call ratio and COT