This article investigates the statistical properties of the recently introduced quantile periodogram for time series with time-dependent variance. The asymptotic distribution of the quantile periodogram is derived in the case where the time series consists of i.i.d. random variables multiplied by a time-dependent scale parameter. It is shown that the time-dependent variance is represented approximately additively in the mean of the asymptotic distribution of the quantile periodogram. It is also shown that the strength of the representation is proportional to the squared quantile of the i.i.d. random variables, so that a stronger characterization is expected at upper and lower quantile levels if the time series is centred at zero. These properties are further demonstrated by simulation results. The series of daily returns from the Dow Jones Industrial Average, which is known to exhibit heteroscedastic volatility, serves to motivate the investigation.