# Hedge Ratio

# Introduction

This is a series of short articles to discuss the tools to manage risk in the commodity markets using **R**. My goal is to show how can we optimize the hedging strategy using commodities contracts - futures and options.

Load some packages.

```
library(tidyverse)
library(quantmod)
library(zoo)
library(ggplot2)
library(Quandl)
library(lubridate)
library(ggthemes)
library(reshape)
library(riskR)
library(stargazer)
library(tinytex)
```

## Forward-Spot relationship

-The relationship between the forward and the spot price, assuming no-arbitrage, can be describe as follows Geman (2005): \begin{align} f ^T (t) = S(t)e^{(r-y)(T-t)} \end{align} where $r$ is the continuously compound interest rate at instant t and maturity $T$, and $y$ is the convenience yield.

We call backwardation when $(r-y) < 0$, the forward curve is a decreasing function of maturity.

The contango sitaution is when $(r-y) > 0$, the forward curve is an increasing function of maturity.

Extract the Soybean prices from Quandl.

```
soy = Quandl("TFGRAIN/SOYBEANS")
soy_f = soy %>%
mutate(soy, Futures = soy$`Cash Price`-soy$Basis)
dd= soy_f[,c(1:2, 6)]
```

Spread Futures vs Spot price.

Let’s assume the

*Basis*as only the spread between*Spot*and*Future*price.Supposing you want to hedge a Cash Price position (Spot) with a Future contract (F) in the Chicago Mercantile Exchange (CME).

Build ggplot.

```
dd1= melt(dd,id=c("Date"))
dd1 %>%
ggplot(aes(x=Date, y=value,color=variable))+geom_line()+
theme_wsj()+
xlab(" ")+ylab(" ")+
scale_colour_manual("",
breaks = c("Cash Price", "Futures"),
values = c("Cash Price"="darkblue", "Futures"="red"))+
theme(legend.text = element_text(size = 8, colour = "grey10"))
```

Hedge Ratio.

- The hedge ratio is a measure that compares a financial asset to a hedging instrument. The measurement indicates the risk of a shift in the hedging instrument (Lien,2016). \begin{align} H^* = \rho \frac{\sigma_S}{\sigma_F} \end{align}

\begin{align} H_{mv}= \frac{Cov(\Delta S_t, \Delta f_t)}{Var(\Delta f_t)} \end{align}

- In the commodity markets is common to use Futures contracts to hedge the Spot price. If a producers/exporters want to hedge their production, for example, then they would sell Futures contracts; if a buyers/importers want to hedge their position in the futures markets, then they would buy futures contracts. In this sense, the hedge ratio indicates the level of risk a producer/exporter are exposed.

Time series - zoo.

```
data.z = zoo(dd[,-1], as.Date(dd[,1], format="%Y/%m/%d"))
S = data.z[,"Cash Price",drop=FALSE]
F = data.z[,"Futures",drop=FALSE]
# Estimating the returns
lS= diff(log(S))
lF=diff(log(F))
# Hedge Ratio
H = cov(lS,lF)/cov(lF)
stargazer(H, type = "text", title="Hedge Ratio", rownames = FALSE,
colnames = FALSE)
```

Now suppose a company/importer knows that it will buy 1,000,000 of soybeans in one month. The soybean futures contract unit is 5,000/bushels. So, the number a futures contract (long position) the company will buy is…

`N = H*(1000000/5000) #buy 190 futures contracts stargazer(N, type = "text", title="N of Contracts", rownames = FALSE, colnames = FALSE)`

Risk Metrics. Estimating optimal hedging ratios based on risk measures (see, Chan (2019)).

- The risk manager’s role is to mitigate the volatility by hedging the underlying asset or avoiding the deviation from the expected value.
- There are a few risk metrics to measure the uncertainty in the futures contracts. Chan (2019) created a package that computes 26 financial risk measures. Thus, we applied the function to our example (soybean hedging).

We use the Soybean Future contract (F) for hedging the Spot price (S).

```
rh = riskR::risk.hedge(lS,lF,alpha=c(0.05, 0.01), beta = 1, p=2)
stargazer(rh,type="text",font.size = 'tiny',
no.space = TRUE, column.sep.width = '4pt', title="Risk Metrics")
```

Descriptions:

- Risk measures (Standard Deviation (StD)
- Value at Risk (VaR)
- Expected Loss (EL)
- Expected Loss Deviation (ELD)
- Expected Shortfall (ES)
- Shortfall Deviation Risk (SDR) Expectile Value at Risk (EVaR)
- Deviation Expectile Value at Risk (DEVaR)
- Entropic (ENT)
- Deviation Entropic (DENT)
- Maximum Loss (ML)

# References

Chan, S., & Nadarajah, S. (2019). Risk: An R Package for Financial Risk Measures. Computational Economics, 53(4), 1337–1351. https://doi.org/10.1007/s10614-018-9806-9

Geman, H. (2005). Commodities and commodity derivatives. In J. W. & Sons (Ed.), Modeling and Pricing for Agriculturals, Metals and … (p. 419). John Wiley \& Sons.

Lien, D., Shrestha, K., & Wu, J. (2016). Quantile Estimation of Optimal Hedge Ratio. Journal of Futures Markets, 36(2), 194–214. https://doi.org/https://doi.org/10.1002/fut.21712