Risk and Derivatives

Since we do not know what will happen in the future, the cashflows envisaged in the analyses described above are estimates or expectations. Actual outcomes are always variable, and they are more likely than not to be different from expected results. For example, a perfectly balanced coin would have a 50-50 chance of landing with "HEADS" up when it is flipped. Accordingly, when such a coin is flipped ten times, one would expect to get five "HEADS," 8 but that will not necessarily be the actual result. Usually, the divergence from the expected result is small. When a coin is flipped ten times, in more than 65% of cases, we should see four, five or six "HEADS." How can we quantitatively depict such a situation: the possibility of getting results around the expected result, or from the flip side, the variability of outcomes? What is the likelihood of getting seriously adverse results? In other words, how can we measure risk?

The preferred statistical measure in finance for risk is variance or standard deviation, which is the square root of variance. Variance (or standard deviation) for outcomes that are closely bunched together around the expected result will be small. If extreme results are more likely, variance will be large. When financial practitioners talk to each other, the word "volatility" is used interchangeably with standard deviation. Meanwhile, when one looks at historical data on market price movements, the distribution of the size of movements looks very much like the so-called bell curve, the normal distribution: there are more days when price changes are small than when they are big. Consequently, the evaluation and communication of variability or risk in finance often assumes that outcomes follow the normal distribution. Under that assumption, the calculation of probabilities for a certain outcome becomes a straightforward exercise. 9 One only needs to ascertain how many standard deviations away is the outcome from the mean. Once that number is obtained, probability can easily be calculated using the cumulative distribution function of standard normal distribution. 10 A related idea of risk measurement is the "value at risk" (VaR). Instead of calculating the probability of an outcome, VaR expresses the magnitude of an extreme result happening over a short time horizon 11, which is often defined as events with the probability of 5% or 1% over a day or a week. For example, if an extreme market movement which would occur just once in 100 days would erode the value of a group of financial assets (a "position"), and the magnitude of such valuation loss was greater than $1 million, the VaR of that position was $1 million. As such, VaR is tied to a certain probability (once in 100 days in this example), and such probability is expressed as the "confidence level," which means the probability of an extreme result not happening. Given that people are concerned with adverse results, the probability is usually "one-sided" (99% in this example). Once we have a yardstick for risk, we could begin managing risk in line with our preferences, or our appetite for risk. If the probability of losing a certain amount, or the VaR at a certain confidence level was too high, we could reduce our exposure to (holdings of) risky assets. In order to do that, we can obviously sell risky assets and switch to less risky, or even risk-free, assets. Alternatively, we can try to take positions that would move in the opposite direction. The problem is that these operations could be very costly because of various transaction costs. The solution devised by financial practitioners was the use of derivatives. 12

Derivatives are financial instruments (contracts) whose value is based on (or derive from) other underlying financial assets. Parties to a derivative contract agree to buy/sell some financial asset at a future date (forwards) 13, to swap specific cashflows (swaps), or to obtain a right to or assume an obligation to buy/sell some financial asset at (by) a future date (options). Since there is no need for parties to a derivative contract to take positions in the underlying asset, it is less costly and thus enables more efficient risk management.

Q10: Regarding UL8, what is the expected value of a game, if, with each coin flip, you would get $10 for "HEADS," but would lose $5 for "TAILS?" Q11: Regarding UL9, describe briefly the problem raised by assuming normal distribution. Q12: Regarding UL10, assuming current USDJPY exchange rate at 105 and annual volatility (standard deviation) at 12%, what is the probability of finding the USDJPY exchange rate below 95 one year from now? What is the probability if the period was three months instead of one year (260 trading days, both answers rounded to the nearest unit)? Q13: Regarding UL11, if an investor was exposed to SP500 stock index and her exposure was valued at $100 million, what is the daily VaR at the confidence level of 99% (one-sided) of that position when the daily volatility is 1% (round to the nearest thousand dollars)?

Q14: Regarding UL12, what is another way of controlling risk not mentioned in the paragraph? Q15: Regarding UL13, forward contracts for an asset should be priced so that the same result would be reached from entering into a forward contract or from buying that asset now and holding it until the date of sale/purchase agreed in the contract. If crude oil is trading currently at $40.00 per barrel what should be the forward price one year from now? Assume that one-year interest rate for US dollars is 1% and storage cost (to be paid at the end of storage period) for a barrel of oil for one year is $1.00 (round to two decimal places).