Percent Change

Time Series

Easy

Problem

Theory

Percent change measures the relative change between consecutive observations in a time series, expressed as a percentage. It shows how much a value has increased or decreased relative to its previous value.

\text{PC}_t = \frac{y_t - y_{t-1}}{y_{t-1}} \times 100\%

where y_t is the current value and y_{t-1} is the previous value.

The Formula

For a time series y_1, y_2, ..., y_T:

\text{PC}_t = \frac{y_t - y_{t-1}}{y_{t-1}} \times 100\%

Without percentage: Drop the 100 factor to get decimal form.

r_t = \frac{y_t - y_{t-1}}{y_{t-1}}

Alternative form:

\text{PC}_t = \left(\frac{y_t}{y_{t-1}} - 1\right) \times 100\%

Worked Example

Time series: [100, 105, 102, 110, 108]

Period 2:

\text{PC}_2 = \frac{105 - 100}{100} \times 100\% = \frac{5}{100} \times 100\% = 5\%

Period 3:

\text{PC}_3 = \frac{102 - 105}{105} \times 100\% = \frac{-3}{105} \times 100\% = -2.857\%

Period 4:

\text{PC}_4 = \frac{110 - 102}{102} \times 100\% = \frac{8}{102} \times 100\% = 7.843\%

Period 5:

\text{PC}_5 = \frac{108 - 110}{110} \times 100\% = \frac{-2}{110} \times 100\% = -1.818\%

Result: [NaN, 5%, -2.857%, 7.843%, -1.818%]

Interpretation

Positive percent change: Value increased.

Example: +5% means value grew by 5% from previous period.

Negative percent change: Value decreased.

Example: -3% means value declined by 3% from previous period.

Zero percent change: No change.

Magnitude: Larger absolute value indicates larger relative change.

Comparison to Absolute Change

Absolute change:

\Delta y_t = y_t - y_{t-1}

Percent change:

\text{PC}_t = \frac{\Delta y_t}{y_{t-1}} \times 100\%

Example: Stock price increases from $10 to $ vs $100 to $

Absolute change: Both +$1

Percent change: +10% vs +1%

Percent change provides scale-independent comparison.

Lag-k Percent Change

One-period lag (standard):

\text{PC}_t = \frac{y_t - y_{t-1}}{y_{t-1}} \times 100\%

k-period lag:

\text{PC}_t(k) = \frac{y_t - y_{t-k}}{y_{t-k}} \times 100\%

Example applications:

• k=1: Daily change
• k=5: Weekly change (for daily data)
• k=12: Year-over-year change (for monthly data)

Year-Over-Year Example

Monthly sales data: [100, 105, 110, 95, 102, 108, 115, 120, 112, 118, 125, 130]

Year-over-year percent change (k=12):

Not defined until month 13.

Month 13 (year 2, month 1): Sales = 110

\text{YoY} = \frac{110 - 100}{100} \times 100\% = 10\%

Interpretation: Sales in month 13 are 10% higher than same month previous year.

Compounding Effects

Percent changes do not add:

If value increases 10% then decreases 10%, final value is NOT original value.

Example:

Start: $100

After +10%: $100 \times 1.10 = $

After -10%: $110 \times 0.90 = $

Lost 1% overall, not 0%.

Compounding formula:

y_t = y_0 \prod_{i=1}^{t} (1 + r_i)

Log Returns Alternative

Problem: Percent changes are asymmetric.

+50% followed by -50% does not return to original value.

Log returns:

\ell_t = \ln\left(\frac{y_t}{y_{t-1}}\right) = \ln(1 + r_t)

Advantage: Log returns are additive.

\sum_{i=1}^{t} \ell_i = \ln\left(\frac{y_t}{y_0}\right)

Approximation: For small r_t, \ell_t \approx r_t

Finance: Log returns preferred for modeling and aggregation.

Stationarity

Non-stationary series: Mean and variance change over time (e.g., stock prices).

Percent change series: Often stationary.

Transformation: Converting levels to percent changes can induce stationarity.

Application: Many time series models (ARIMA) require stationary data. Percent changes often satisfy this.

Example: Stock prices have unit root (non-stationary). Returns are stationary.

Percent Change and Volatility

Volatility: Measured as standard deviation of percent changes.

\sigma = \sqrt{\frac{1}{T-1} \sum_{t=2}^{T} (r_t - \bar{r})^2}

High volatility: Large swings in percent changes.

Low volatility: Small, stable percent changes.

Application: Risk assessment in finance. Higher volatility means higher risk.

Handling Zero and Negative Values

Division by zero:

If y_{t-1} = 0, percent change undefined.

Negative values:

Percent change still defined but interpretation complex.

Example: Change from -10 to -5.

\text{PC} = \frac{-5 - (-10)}{-10} \times 100\% = \frac{5}{-10} \times 100\% = -50\%

Confusing: Value improved (less negative) but percent change is negative.

Solution: Use absolute change for series with zero or negative values.

Seasonality Detection

Seasonal patterns visible in percent changes:

Compare percent changes at same seasonal period.

Example: Monthly retail sales.

December often shows large positive percent change (holiday shopping).

January often shows large negative percent change (post-holiday).

Analysis: Consistent patterns in percent changes indicate seasonality.

Moving Average of Percent Changes

Smooth percent change series:

\text{MA}(\text{PC}_t) = \frac{1}{w} \sum_{i=0}^{w-1} \text{PC}_{t-i}

Interpretation: Average rate of change over recent periods.

Application: Identify sustained trends vs temporary spikes.

Example: Average percent change over last 12 months indicates overall growth rate.

Cumulative Percent Change

Cumulative effect of percent changes:

y_t = y_0 \prod_{i=1}^{t} (1 + r_i)

Cumulative percent change:

R_{\text{cum}} = \frac{y_t - y_0}{y_0} \times 100\% = \left(\prod_{i=1}^{t} (1 + r_i) - 1\right) \times 100\%

Example: Daily returns of +2%, -1%, +3%

R_{\text{cum}} = (1.02 \times 0.99 \times 1.03 - 1) \times 100\% = (1.04 - 1) \times 100\% = 4\%

Growth Rate Interpretation

Percent change as growth rate:

g_t = \frac{y_t - y_{t-1}}{y_{t-1}}

Continuous compounding approximation:

y_t \approx y_{t-1} e^{g_t}

Long-term growth:

If growth rate is constant g:

y_t = y_0 (1 + g)^t

Doubling time:

t_{\text{double}} \approx \frac{\ln(2)}{\ln(1+g)} \approx \frac{0.693}{g}

Example: 7% annual growth doubles in approximately 10 years.

Outlier Detection

Percent change highlights anomalies:

Sudden large percent changes indicate unusual events.

Threshold method:

Flag |\text{PC}_t| > k \sigma where \sigma is standard deviation of percent changes and k=3 typically.

Example: Stock price jumps 20% in one day (earnings surprise, merger announcement).

Application: Anomaly detection, event study analysis.

Forecasting with Percent Changes

Naive forecast:

\hat{\text{PC}}_{t+1} = \bar{\text{PC}}

Use historical average percent change.

Level forecast:

\hat{y}_{t+1} = y_t (1 + \hat{\text{PC}}_{t+1})

Multi-step:

\hat{y}_{t+h} = y_t (1 + \hat{\text{PC}})^h

Assumes constant growth rate.

Better models: ARIMA on percent changes, exponential smoothing on levels.

Symmetric Percent Change

Standard percent change asymmetry:

Change from 100 to 150: +50%

Change from 150 to 100: -33.3%

Different magnitudes for same absolute change.

Symmetric percent change:

\text{SPC}_t = \frac{y_t - y_{t-1}}{(y_t + y_{t-1})/2} \times 100\%

Denominator: Average of current and previous values.

Example:

100 to 150: \frac{50}{125} \times 100\% = 40\%

150 to 100: \frac{-50}{125} \times 100\% = -40\%

Now symmetric in magnitude.

Annualized Percent Change

Convert to annual rate:

r_{\text{annual}} = \left(1 + r_{\text{period}}\right)^{n} - 1

where n is number of periods per year.

Example: Monthly return of 1%

r_{\text{annual}} = (1.01)^{12} - 1 = 1.1268 - 1 = 0.1268 = 12.68\%

Interpretation: If 1% monthly growth continues, annual growth is 12.68%.

Percent Change in Indexes

Index construction:

Set base period to 100.

I_t = \frac{y_t}{y_0} \times 100

Percent change in index:

\text{PC}_t = \frac{I_t - I_{t-1}}{I_{t-1}} \times 100\%

Same as percent change in original series.

Advantage of indexing: Easy comparison across multiple series with different units.

Percent Change and Correlation

Correlation of levels vs returns:

Levels may be spuriously correlated due to trends.

Returns often have different correlation structure.

Example: Two stock prices both trending upward (high correlation in levels).

Daily returns may be uncorrelated (price movements independent).

Analysis: Compute correlation of percent changes to assess true relationship.

Differencing vs Percent Change

First difference:

\Delta y_t = y_t - y_{t-1}

Percent change:

r_t = \frac{\Delta y_t}{y_{t-1}}

Differencing: Additive model (absolute changes).

Percent change: Multiplicative model (relative changes).

Use differencing when: Variance is constant in levels.

Use percent change when: Variance proportional to level (heteroscedasticity).

Percentage Point vs Percent Change

Percentage point: Absolute difference in percentages.

Example: Interest rate changes from 5% to 7%.

Change is 2 percentage points.

Percent change: Relative change.

\frac{7 - 5}{5} \times 100\% = 40\% \text{ increase}

Critical distinction: Often confused in media and reports.

Reversion to Base Value

Asymmetric recovery:

Decrease of 50% requires 100% increase to return to original.

Example:

Start: 100

After -50%: 50

To return to 100: \frac{100-50}{50} \times 100\% = 100\% increase needed.

Implication: Losses are harder to recover in percentage terms.

Volatility Clustering

Financial returns: Large percent changes tend to cluster.

High volatility period: Consecutive large percent changes.

Low volatility period: Consecutive small percent changes.

ARCH/GARCH models: Capture volatility clustering.

\sigma_t^2 = \alpha_0 + \alpha_1 r_{t-1}^2 + \beta \sigma_{t-1}^2

Application: Risk management, option pricing.

Real vs Nominal Percent Change

Nominal percent change:

r_{\text{nominal}} = \frac{P_t - P_{t-1}}{P_{t-1}}

Inflation adjustment:

r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \pi} - 1

where \pi is inflation rate.

Approximation:

r_{\text{real}} \approx r_{\text{nominal}} - \pi

Example: Nominal return 10%, inflation 3%.

Real return \approx 7\%.

Interpretation: Purchasing power increase.

Geometric vs Arithmetic Mean

Arithmetic mean of percent changes:

\bar{r}_a = \frac{1}{T} \sum_{t=1}^{T} r_t

Geometric mean:

\bar{r}_g = \left(\prod_{t=1}^{T} (1 + r_t)\right)^{1/T} - 1

Relationship: \bar{r}_g < \bar{r}_a unless all r_t equal.

Use geometric for: Compounding returns over time.

Use arithmetic for: Average single-period return.

Percent Change Distribution

Empirical observation:

Financial returns (percent changes) approximately normal with fat tails.

Stylized facts:

• Mean close to zero
• Positive or negative skewness possible
• Excess kurtosis (leptokurtic)

Implications:

Standard normal assumption underestimates extreme events.

Use Student-t or other heavy-tailed distributions for better fit.

Time Aggregation

Daily to monthly percent change:

Cannot simply average daily percent changes.

Correct method:

r_{\text{monthly}} = \prod_{d=1}^{D} (1 + r_d) - 1

where D is number of days in month.

Example: Daily returns of 1%, 2%, -1%.

r = 1.01 \times 1.02 \times 0.99 - 1 = 1.0198 - 1 = 1.98\%

Not (1 + 2 - 1)/3 = 0.67\%.

Confidence Intervals

For normal returns:

\text{CI} = \bar{r} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{T}}

Example: Average daily return 0.05%, standard deviation 1.5%, 100 observations.

95% CI: 0.05\% \pm 1.96 \times \frac{1.5\%}{10} = 0.05\% \pm 0.294\%

Interpretation: Plausible range for true mean return.

Signal-to-Noise in Percent Changes

Efficient market hypothesis:

Asset returns are unpredictable (high noise, low signal).

Autocorrelation test:

If \text{Corr}(r_t, r_{t-1}) \approx 0, returns are unpredictable.

Momentum vs mean reversion:

Positive autocorrelation: Momentum (trends persist)

Negative autocorrelation: Mean reversion (reversals)

Empirical: Stock returns show weak autocorrelation, volatility shows strong autocorrelation.

Examples

Hints

1. Begin the loop at index 1 so both the current and previous values exist.
2. Check the previous value before performing the division.

Requirements

• Compute one fractional change for each consecutive pair.
• Divide by the previous value.
• Use 0.0 when the previous value is zero.
• Return a list with len(series) - 1 values.

Constraints

• series contains at least two numeric values.
• Time limit: 300 ms.

Starter Code

def percent_change(series: list) -> list:
    """
    Returns the fractional change between consecutive values.
    """
    # Write code here
    pass

Test Cases