Total RAFE (T-RAFE) — Sharpe-Gap Upper Bound

Combines RAFE and C-RAFE into the upper bound on the Sharpe-ratio gap \(\Delta = SR^{*} - SR(\hat w)\) of a plug-in mean-variance portfolio: \(\Delta \le c \cdot \mathrm{RAFE} + SR^{*} \cdot \mathrm{C\text{-}RAFE}\).

Usage

compute_trafe(
  mu_hat,
  mu,
  Sigma,
  Sigma_hat,
  c = 1,
  SR_star = NULL,
  variant = c("none", "cc05", "cc0", "cv", "i")
)

Arguments

mu_hat

Numeric vector of forecast means.

mu

Numeric vector of realised means (same length as mu_hat).

Sigma

Realised covariance matrix. The argument order matches the paper's replication code, so compute_rafe(mu_hat, mu, Sigma) does the same thing here as it does there.

Sigma_hat

Forecast covariance matrix.

c

Scalar weighting for the mean channel. Defaults to 1, the published definition of T-RAFE in Equation (22). Set c = NULL to apply the data-dependent rule of Equation (4) (1 if RAFE <= SR_star, else 2), which yields a guaranteed upper bound on the Sharpe-ratio gap.

SR_star

Optional oracle Sharpe ratio. If NULL, computed as \(\sqrt{\mu^\top \Sigma^{-1} \mu}\) from mu and Sigma_inv (or Sigma).

variant

Covariance restriction applied to both channels; see restrict_cov(). SR_star is always computed from the unrestricted Sigma, matching the published tables.

Value

A length-1 numeric, carrying the components as attributes rafe, crafe, c and SR_star.

Details

Equation (22) of Salcher, Stöckl & Hanke (2026) defines T-RAFE with no constant on the mean channel, i.e. \(c = 1\), and that is the default here so that compute_trafe() returns the published metric.

Equation (4) of the same paper gives the bound, which carries a data-dependent constant: \(c = 1\) when \(\mathrm{RAFE} \le SR^{*}\) and \(c = 2\) otherwise. Pass c = NULL to apply that rule and obtain a quantity guaranteed to dominate the Sharpe-ratio gap. The two coincide whenever \(\mathrm{RAFE} \le SR^{*}\).

References

Salcher, L., Stöckl, S., & Hanke, M. (2026). Lost in Translation? Risk-Adjusting RMSE for Economic Forecast Performance. Journal of Forecasting. doi:10.1002/for.70134

Examples

set.seed(1)
mu        <- rnorm(5) / 10
Sigma     <- diag(5)
mu_hat    <- mu + rnorm(5, sd = 0.02)
Sigma_hat <- Sigma + diag(5) * 0.05

tr <- compute_trafe(mu_hat, mu, Sigma, Sigma_hat)
tr
#> [1] 0.03668315
#> attr(,"rafe")
#> [1] 0.02742747
#> attr(,"crafe")
#> [1] 0.04761905
#> attr(,"c")
#> [1] 1
#> attr(,"SR_star")
#> [1] 0.1943693
attributes(tr)[c("rafe", "crafe", "c", "SR_star")]
#> $rafe
#> [1] 0.02742747
#> 
#> $crafe
#> [1] 0.04761905
#> 
#> $c
#> [1] 1
#> 
#> $SR_star
#> [1] 0.1943693
#>