# Estimating the VaR by scaling volatility with the square root of time

Assume that the investment horizon of an exposure in voyage route C4 of the BCI is 30 days and the risk manager is interested in quantifying the 30-day ahead VaR made at time period t (i.e. forecast НгК,+30). Assume the one-day ahead forecast of the volatility obtained from a GARCH model fitted to the freight return data to be 5% and the one-day average return to be —0.2%. Then the 30-day 95% VaR can be estimated through volatility scaling by the square-root-of-time in order to obtain the 30-day VaR as follows:

per metric tone.

Multiplying with the dead weight tonnage of 150,000mt gives a VaR of \$76),575. Thus, the maximum reduction of freight income that the risk-taking agent may incur due to adverse movements in the freight rate markets over the next 30 days with a confidence level of 95% is estimated to be \$76,575.

Regarding medium-term risk, Kavussanos and Dimitrakopoulos (2011) proposed an empirical scaling law, which determines the appropriate scaling factor to be used in order to estimate market risk more accurately. This approach is based on the theory of self-similarity and assumes that irrespective of scaling, the characteristics of the process of freight rate returns remain the same. According to this approach, VaR is estimated by:

where, T is the investment horizon of the underlying exposure and h a scaling exponent obtained by:

with eVaRl+T and eVaR t being the historical simulation VaR of the T-day and one-day returns, respectively.

# Estimating medium-term VaR

A shipowner is considering fixing a Capesize vessel in route C4 Richards Bay to Rotterdam on 8 August 2018 and she is interested in quantifying the market risk for this exposure. Assuming that the investment horizon for this exposure is 30 days, she may utilise GARCH modeling to forecast a 99% VaR with volatility scaling and quantile scaling by applying the empirical scaling law. This exercise is performed in the following case.

## Case 1: VaR estimation with volatility scaling

In order to forecast volatility, a GARCH(1,1) model is fitted to 1,000 daily returns preceding 28 August. The estimation results of the GARCH model produced are presented in Table 8.4.

Table 8.4 Estimation of a GARCH (1,1) model on 1,000 daily returns of C4 route

 Variable Coefficient Std. Error s-Statistic Prob. Constant 0.0000 0.0000 2.3328 0.0197 GARCHjl} 0.1325 0.0510 2.5982 0.0094 ARCH{1} 0.8353 0.0600 13.9295 0.0000

The forecast of one-day ahead volatility according to this model is 0.614% and the mean daily return over the sample period is —0.0001529. In order to estimate the 30-day 99%-VaR with volatility scaling, the following equation is utilised:

The corresponding 99%-VaR of the underlying exposure is forecasted at SI 1,047 which indicates that the maximum reduction in freight income to be sustained on the vessel employed on route C4, corresponding to an investment horizon of 30 days and a confidence level of 99%, is \$11,047.

## Case 2: KiR estimation by applying the scaling law

In order to calculate the 30-day VaR we first need to estimate the one-day ahead VaR (i.e. FaR ,) using the GARCH method and then scale this with It,

where

The one-day 99% VaR according to the volatility estimate of the GARCH model is given by:

The next step involves calculating the scaling exponent /; for determining the relationship between the historical simulation 99%-VaR (c VaR, ,+30) and the historical simulation 99%-one-day VaR of the sample of daily returns (eVaR(ll+]). According to calculations, the eVaRlll+}0, i.e the 1% quantile of the historical sample of 30-day freight returns, is \$0.89538 per ton and the eVaRn+l, i.e the 1% quantile of the historical sample of one-day freight returns, is \$0.06325 per ton. Thus, the scaling exponent It is calculated as:

Then, the 30-day 99% VaR is obtained by:

## Case 3: Estimating the portfolio’s risk for freight rate exposures

When all assets of a portfolio have the same investment horizon, a case which is unlikely, the estimation of market risk is straightforward and multivariate extensions of the methods discussed in previous sections may be utilised for estimating the VaR. However, the investment horizon of a portfolio of vessels is typically different for each vessel. Estimating VaR for each vessel and aggregating VaRs to estimate the overall VaR of the portfolio may be misleading as risk forecasts refer to different investment horizons. On the other hand, using returns of different frequency and combining these to estimate a variance-covariance matrix for the portfolio in order to calculate VaR is problematic as they refer to different time periods. An alternative is to set the investment of the portfolio to the maximum of the investment horizons of all vessels and within this horizon calculate the total returns of the portfolio. Assuming normally distributed returns, a series of total portfolio returns can be used for the estimation of VaR by applying the univariate models discussed. Another option is to use simulation techniques to simulate freight rate price paths for each vessel and then aggregate returns over a longer-term investment horizon.

## Summary

This chapter provided an overview of market risk, which constitutes an important segment of the overall risk being borne by participants in seaborne shipping. As shown in this chapter, the risk manager has a plethora of approaches available in order to quantify appropriately market risk, such as the standard deviation, the semi-variance and the mean absolute deviation, to more advanced probabilistic risk metrics such as Value-at-Risk (VaR) and expected shortfall (ES). Furthermore, this chapter showed that modeling volatility is a central issue when calculating the VaR and ES measures. Thus, a range of volatility models lying into the non-parametric, parametric and semi- parametric approaches have been discussed. In addition, the chapter focused on the very important issue of evaluating the performance of the developed VaR model through the procedure of backtesting. Finally, a series of practical examples were presented applying the previously discussed models and approaches in the shipping markets. Overall, this chapter focused on the latest developments on market risk measurement and discussed market risk measurement from the perspective of seaborne shipping. The following chapter focuses on the very important issue of bunker price risk management, which refers to the cost side of the computation of the cash-flows, instead of the revenue side (freight rates).

## Notes

• 1 We may define VaR in terms of the realised P&L by omitting the hat symbol in Equation 8.2 and use realise ex-post P&L instead.
• 2 For a risk metric p, translation invariance requires for all positions X G G and a real number a, we have /o(X + ap) = p(X) — Q, positive homogeneity requires that for allA>Oand all X G G, /a(AX) < Ap(X) and monotonicity that for all X and Y G G with X < У, we have p(X) > p(V).
• 3 The risk metric of expected shortfall is also known as “tail conditional expectation”, “conditional VaRand “worst conditional expectation".
• 4 Violation of the VaR is another term used for denoting that the realised loss is lower (exceeds) the estimated VaR.
• 5 The
• 6 Symmetric elliptical distributions such as the normal and the student-? distribution comprise a dominant subset of such distributions.
• 7 The continuously compounded returns at time t is calculated as: r, = (77—) where/’ is the price of

the asset at time t.

• 8 “Earnings at risk" is a more appropriate term for expressing the risk due to adverse freight rate movements as a freight price increase will result in a decrease in the income of the producer rather than a realised loss.
• 9 Bootstrap consists of making random draws with replacements from a sample.