Risk-factor returns

Quantitative Risk Management in R

Alexander McNeil

Professor, University of York

Risk-factor returns

Changes in risk factors are risk-factor returns or returns
Let ($Z_t$) denote a time series of risk factor values
Common definitions of returns ($X_t$)
- $X_t = Z_t - Z_{t-1}$ (simple returns)
- $X_t = \dfrac{Z_t - Z_{t-1}}{Z_{t-1}}$ (relative returns)
  - 0.02 = 2% gain, -0.03 = 3% loss
- $X_t = \ln(Z_t) - \ln(Z_{t-1})$ (log-returns)

Properties of log-returns

Resulting risk factors cannot become negative
Very close to relative returns for small changes:
- $\ln(Z_t) - \ln(Z_{t-1}) \approx \dfrac{Z_t-Z_{t-1}}{Z_{t-1}}$
Easy to aggregate by summation to obtain longer-interval log-returns
Independent normal if risk factors follow geometric Brownian motion (GBM)

Log-returns in R

sp500x <- diff(log(SP500))
head(sp500x, n = 3)  # note the NA in first position

                 ^GSPC
1950-01-03          NA
1950-01-04 0.011340020
1950-01-05 0.004736539

sp500x <- diff(log(SP500))[-1]
head(sp500x)

                  ^GSPC
1950-01-04  0.011340020
1950-01-05  0.004736539
1950-01-06  0.002948985
1950-01-09  0.005872007
1950-01-10 -0.002931635
1950-01-11  0.003516944

Log-returns in R (2)

plot(sp500x)

Let's practice!

Quantitative Risk Management in R