Session 7 - Derivatives - FRAs & Eurodollar Futures

During this class we learned what a zero-coupon rate is and how to calculate discount factors (have a look at the Excel file).

We also reviewed how to calculate forward discount factors and forward rates avoiding potential arbitrage opportunities. A forward rate is a rate to be applied to a forward starting loan/deposit. The idea behind this concept is that if I do a 1) 12m deposit or 2) a 6m deposit and I reinvest the proceeds in a new forward starting 6m deposit, the outcome should be the same.

Additionally, we had a look at how FRAs (Forward Rate Agreements) and Eurodollar Futures work.

The presentation can be found here.

Session 6 - Derivatives - 2014

During this session we reviewed how to price forwards on currencies. Please, have a look at the Excel file. The idea is exactly the same as with other asset classes: if the forward is too expensive (if the forward price should be 1.3062 and it is trading at 1.28 - I can buy less USD for the same amount of EUR), I will sell USD forward; if the forward is too cheap (if the forward price should be 1.3062 and it is trading at 1.32 - I can buy more USD for the same amount of EUR), I will buy USD forward. There is only one no-arbitrage possibility: F = S * e ^ (rd - rf).

Additionally, we had a look at how we should price forwards on consumption assets. In this case, we must take into account any potential convenience yield (for instance, to avoid shortages of the product that could affect our production line) and, also, any storage cost.

Remember the logic behind of the formula:

F = S * e ^ (+ any potential cost - any potential income)

The first day we buy/sell a Forward, the MTM of the position (value) is equal to zero. Remember we do not have to pay anything when we buy/sell a Forward. However, as time goes by and as the underlying asset and interest rates move, the MTM of the Forward will change. Basically, we will compare the price at which we can buy/sell at Maturity with the current Forward price and we will bring the difference to present value.

Finally, in spite of the fact that we will assume both Futures prices and Forward prices to be the same during the course, we reviewed why they are not (correlation between interest rates and price of the underlying asset, different interest rates, credit risk). You can find the presentation here.

Session 5 - Derivatives - 2014

We started this session reviewing how simple interest rates and compounded interest rates work. Remember that for this course we will use continous compounding (discount factor = e ^ (-r*T); future value = e ^ (r*T)). It is interesting to remember also how to work with Natural Logs. Remember that Ln e ^(r*T) = r*T.

When we value Futures, we consider an Eonia curve (Eonia is a riskless interest rate and Futures are riskless due to the margining process). When we value forwards, we consider a Euribor curve. Additionally, we would have to charge an additional spread that depends on the creditworthiness of the counterparty (CVA).

We valued Forward for an asset producing no income assuming a no-arbitrage hypothesis. If the forward is too expensive, we can sell (short) the forward and ask for a loan to buy the underlying asset. At maturity, we give the asset to the person who bought the forward from us and we make a riskless profit. So, if the forward is expensive, many arbitrageurs will enter into this strategy, taking the price of the forward down.

If the forward is too cheap, we buy the forwards and short the underlying asset. We will make a deposit with the amount that we obtain by shorting the asset. At maturity, we buy the asset from the person who sold us the forward and we give it back through the short contract. We would make a riskless profit. So, if the forward is cheap, many arbitrageurs will enter into this strategy, taking the price of the forward up.

The only possible value to avoid any potential arbitrage opportunity would be F = S*e ^ (r*T). Please review the Excel file. Khan Academy explains this potential arbitrage opportunity here and here.

We applied the same logic to assets producing a discrete income and to assets producing a yield. You can find the presentation here.

Session 4 - Derivatives - 2014

During this session we reviewed what type of dealers we can find in the Derivatives Markets (hedger, speculators, arbitrageurs) and we also had a look at how we could hedge with Futures. We gave an example of how a hedger/arbitrageur can become a speculator.

We focused on how to hedge a position. Hedging means buying or selling a financial instrument to offset potential losses/gains that may be incurred by a companion investment. 

Usually, if the underlying asset of the Future and the asset that we want to hedge are the same, to determine the amount of contracts that we should buy/sell we should divide the amount (units) of the underlying asset that we want to hedge (in the example, 2,000,000 gallons of jet fuel) by the amount (units) of each contract (in the example, 42,000).

However, if the underlying asset of the Future and the asset that we want to hedge are not the same, we will have to do cross-hedging. When we cross-hedge, two questions arise: 

1) What is the hedge ratio that we should apply to determine the number of contracts that we have to buy/sell?

- We use a linear regression where the variable to be explained (y) is the change in price of the asset that we want to hedge and the exogenous variable (x) is the change in price of the Future. With this analysis, we will determine how "y" moves when "x" moves.

- The hedge ratio will be equal to the beta parameter of the regression line (beta = correl. coef * sigma "y" / sigma "x"). Beta means how many units "y" moves when "x" moves by 1 unit.

- In our example, beta = 0.77 (meaning that when "x" moves by 1, "y" moves by 0.77). Then, the number of contracts that we will have to long (I am short the underlying asset, if prices go up, my P&L would be lower) would be 0.77 * 2MM / 42k.

2) How good will the hedge be?

- To determine how good the hedge will be, we must calculate R^2 (R^2 = (Covariance / (Sigma "y" * Sigma "x"))^2.

- Values above 0.75 should indicate that the hedge is quite good.

In the second part of the class, we reviewed how to extrapolate cross-hedging to hedge equity portfolios or single stocks with Futures on an Index.

Why would I want to hedge an equity portfolio?

Typically, if I am long an equity portfolio I would be convinced of the potential positive performance of the portfolio. Then, if I am thinking about hedging, maybe I should sell my portfolio and buy later... There are three reasons to hedge:

- Transaction costs (produced by selling and buying again) may be high.

- Hedging with a Future on an Index would eliminate systematic risk (market risk). I would only be exposed to the relative performance of the portfolio vs. the Index.

- The investment is designed for the long run, while I want to hedge the short run (maybe we are waiting for bad news).

The process of cross-hedging is very similar to the prevous case. First, we must determine the beta of the portfolio. To calculate the beta of the portfolio, we calculate a value-weighted average of the betas of the securities in the portfolio. The betas of the securities that compose the portfolio will be available in Reuters or Bloomberg as beta is a fundamental component of CAPM. 

To hedge completely the portfolio, we should long/short a number of contracts equal to beta * Value of Portfolio / Euro Value of Future. Remember that to obtain the Euro Value of the Future we must use the Future multiplier. If I hedge completely my portfolio the beta of my new portfolio (old portfolio + Future) will be equal to zero (it does not matter how the market moves, my new portfolio will not move).

I can also change the beta of the portfolio (reduce/increase it). For instance, if the beta of my portfolio is 1.003 and I want to take it to 2, I will have to long a number of contracts equal to (Objective beta - Current beta) * Value of Portfolio / Euro Value of Future.

You can find the presentation used in class here. You can find the Excel file used in class here. You can find how beta minimizes the variance of the new portfolio here.

Session 3 - Derivatives - 2014

In today's session, we reviewed how Futures Markets work, how to define a Futures contract and what are its main characteristics, how the price of the Future moves in relation to the price of the underlying asset as expiry approaches and what arbitrage opportunities exist, how the margin process works and what is the intuition behind the margin calculator in MEFF and, finally, what is the main regulation affecting these markets.

Remember that margins are always calculated at a portfolio level.

You can find the presentation here.

You have access to the margin calculator and to a description of EMIR regulation in the links section of the blog.

Clase 7 curso 2013-2014 - FRAs y Futuros de Eurodollar

Durante esta clase vimos como determinar tipos de interés forward por medio de una demostración que asume que no puede haber arbitraje. Fijar un tipo de interés forward nos puede servir para cubrir el coste financiero de una póliza de crédito que aún no está dispuesta, pero que sé que dispondré de ella en el futuro. El tipo de interés forward se puede cubrir por medio de un FRA (instrumento OTC) o por medio de futuros.

Podéis encontrar la presentación aquí.