8006 Exam I: Finance Theory Financial Instruments Financial Markets - 2015 Edition Questions and Answers

A utility function provides a description of an individual or a firm's risk attitude. It expresses how risk seeking or risk averse a firm or an individual is. The utility function would explain differences between risk seeking and risk averse behavior, for example, as an individual becomes richer, he may seek (or shun) risk more than before. A utility function incorporates all of this, and therefore Choice 'd' is the correct answer.

The futures price of the contract will be the future value of the spot price, calculated at a net rate equal to the cost of funding the futures position, less any dividends or other distributions. Also note that when rates are continuously compounded, Future Value = Present Value x (exp(rate x time)).

Therefore in this case the futures price for the S&P = 1000 * exp((5%-2%)*3/12) = 1007.53

The value of a business derives itself from the value of its cash flows, and not its capital structure. However, the availability of tax deductions on debt allow increasing the value of the business to equity holders by using the tax shield.

In the absence of taxes, there is no such advantage. Even if debt is nominally priced lower than the cost of equity, substituting debt for equity makes the remaining equity more risky and increasing the cost of equity to offset any advantage gained from the lower cost of debt. Changing the capital structure does not change the value of the firm, and this in essence is the conclusion of the Modigliani-Miller theorem.

The most loss the fund manager can bear without facing a margin call is the loss that will make his margin balance account no lower than $4,000. This means he can have a loss of upto $1,000 before a margin call is triggered, implying that prices can fall by $10 per ounce (=$1,000/100 ounces per contract) without triggering a margin call. The margin call will be to top the margin up to the $5,000 initial margin.

The basis point value is equal to $102 x 5 /10000 = $0.051. Therefore Choice 'd' is the correct answer.

The basis point value, also called the PV01, is the dollar value change in the price of a bond as a result of a 1 basis point (=1/10000) change in interest rates. Since the modified duration is 5, it means a 1% change in interest rates produces a 5% change in the price of the bond, and a 1 bps change will therefore produce a 0.05% change in the value of the bond. Since the bond is currently valued at $102, the change works out to $102 x 0.05% = $0.051.

The efficient markets hypothesis states that all known information is captured in the prices of a security. It does not explain the shape of the yield curve.

The expectations hypothesis, the LPT and the market segmentation theory are all attempts to explain the shape of the term structure of interest rates.

Therefore Choice 'c' is the correct answer as it does not explain the shape of the yield curve.

The Sharpe ratio is the ratio of the excess returns of a portfolio to its volatility. It provides an intuitive measure of a portfolio's excess return over the risk free rate. The Sharpe ratio is calculated as [(Portfolio return - Risk free return)/Portfolio standard deviation].

The Treynor ratio is similar to the Sharpe ratio, but instead of using volatility in the denominator, it uses the portfolio's beta. Therefore the Treynor Ratio is calculated as [(Portfolio return - Risk free return)/Portfolio's beta]. Therefore Choice 'a' is the correct answer.

Jensen's alpha is another risk adjusted performance measure. It considers only the 'alpha', or the return attributable to a portfolio manager's skill. It is the difference between the return of the portfolio, and what the portfolio should theoretically have earned. Any portfolio can be expected to earn the risk free rate (rf), plus the market risk premium (which is given by [Beta x (Market portfolio's return - Risk free rate)]. Jensen's alpha is therefore the actual return earned less the risk free rate and the beta return.

Refer to the tutorial on risk adjusted performance measures for more details

This question touches upon a couple of issues that relate to hedging foreign exchange exposures using futures contracts. Firstly, many futures contracts on exchanges are available only at specific maturity dates, for example, the IMM dates. They may or may not coincide with actual liabilities for a running business. Also, futures contracts are standardized, ie their notional amounts are fixed rounded sums, and they can only be traded in whole numbers. This often means business using futures for hedging end up having a close enough, but not perfect hedge.

For our importer in the question, clearly he has to buy USDs so he can make his payment. Since each contract gives him USD 100k, he should buy 11 contracts that will get him very close to the amount he finally needs. Also, the contract expires a month later than his liability is due. This means he should offset the contract by closing it out in August soon as he has made his payment. This will allow him to stay hedged till August. If he does not sell out of the contracts, he will become exposed to a long position for one month till the contract settles, a risk which is unnecessary for him.

Therefore Choice 'b' is the best answer.

The highest possible transformation line, ie the transformation line with the maximum slope, is the transformation line joining the risk free rate on the y-axis and the portfolio with the maximum Sharpe ratio on the efficient frontier. This line is called the 'capital markets line'. Investors can pick any point on this line according to their risk appetite, and doing so would maximize the return they can obtain for their desired level of risk. The capital markets line is tangential to the efficient frontier. The Sharpe ratio stays constant throughout the CML. Therefore statement IV is not true, while the rest are.

The stock has a beta of 1.2, therefore intuitively it can be expected to earn more than the broad market index. It will earn the risk free rate, ie 3%, and 1.2 times the equity risk premium of 5% (8% - 3%). The expected returns from the stock therefore are 3% + (8% - 3%)*1.2 = 9%

Macaulay duration is the weighted average of the present value of payments received on a bond, weighted by the year in which the payments are received. If the yield curve changes shape, it will change the present values of the coupons and therefore also the Macaulay duration. Statement I is therefore not correct. However, if the bond were to be a zero-coupon bond, changes in the curve will not affect its duration so long as the interest rate at maturity stays constant.

Modified duration = Macaulay Duration/(1 + rate/compounding frequency). Therefore modified duration will change if the compounding frequency changes, assuming the nominal rate of interest for the coupons are identical. Therefore statement II is correct.When rates are expressed as continuously compounded rates, modified duration and Macaulay duration will be identical. This intuitively follows from the formula for modified duration: Modified duration = Macaulay Duration/(1 + rate/compounding frequency). For continuously compounded rates, the compounding frequency approaches infinity, so the denominator approaches 1, leaving modified duration equal to the Macaulay duration. Therefore statement III is correct.

Convexity is higher for a bond that has its payments spread out compared to a bond that has its payments concentrated at a single point in time. For a bond with a higher coupon, the higher coupons have the effect of spreading the present value of the bond over a longer period, when compared to a bond with lower coupons where the payment is comparatively more concentrated at the maturity. Therefore Convexity is higher for a bond with a lower coupon when compared to a similar bond with a higher coupon. Statement IV is not correct.

Note that intrinsic value of an option is the difference between the value of the underlying and the strike price of the option.

European options can only be exercised at expiry, and Bermudan options only at certain dates during the life of the option. Therefore the option may be valued at less than intrinsic value if the earliest possible exercise date is not very close. An American option however can be exercised at any time prior to expiry, which means that its value can never fall below its intrinsic value. Because if it did, arbitrageurs would buy the option and immediately exercise it to get a risk free profit. It does not matter whether the option is a call or a put - therefore the correct answer is Choice 'c'.

The bond equivalent yield for a treasury bill can be calculated as [(Future value - Present value)/Present value x 365/days to maturity]. In this case this works out to ($100-$98)/$98 * 365/90 = 8.28%

[Why do we use 365 days and not 360? And is there a different way to calculate treasury bill yields?

The answer to this question is that there are alternate ways to calculate these yields. For the exam, I would suggest that you calculate according to one method, see if you get an answer that matches, and pick that. I found the below on the NY Fed's website - a very clear example: (http://newyorkfed.org/aboutthefed/fedpoint/fed28.html)

The Discount Yield Method

The following formula is used to determine the discount yield for T-bills that have three- or six-month maturities:

Discount yield = [(FV - PP)/FV] * [360/M]

FV = face value

PP = purchase price

M = maturity of bill. For a three-month T-bill (13 weeks) use 91, and for a six-month T-bill (26 weeks) use 182

360 = the number of days used by banks to determine short-term interest rates (the investment yield method is based on a calendar year: 365 days, or 366 in leap years).

Example

What is the discount yield for a 182-day T-bill, auctioned at an average price of $9,659.30 per $10,000 face value?

Discount yield = [(FV - PP)/FV] * [360/M]

FV = $10,000 PP = $9,659.30 M = 182

Discount yield = [(10,000) - (9,659.30)] / (10,000) * [360/182]

Discount yield = [340.7 / 10,000] * [1.978022]

Discount yield = .0673912 = 6.74%

For the 13-week bill, the same formula would be used, dividing 360 by a maturity of 91 days rather than 182 days.

The Investment Yield Method

When comparing the return on investment in T-bills to other short-term investment options, the investment yield method can be used. This yield is alternatively called the bond equivalent yield, the coupon equivalent rate, the effective yield and the interest yield.

The following formula is used to calculate the investment yield for T-bills that have three- or six-month maturities:

Investment yield = [(FV - PP)/PP] * [365 or 366/M]

Example

What is the investment yield of a 182-day T-bill, auctioned at an average price of $9,659.30 per $10,000 face value?

Investment yield = [(FV - PP)/PP] * [365/M]

FV = $10,000 PP = $9,659.30 M = 182

Investment yield = [(10,000 - 9,659.30) / (9,659.30)] * [365/182]

Investment yield = [340.70] / 9,659.30] * [2.0054945]

Investment yield = .0707372 = 7.07%

For the 13-week bill, the same formula can be used, dividing 365 (or 366) by a maturity of 91 days.

Statement I is correct - since forward prices are determined as (Spot - PV of dividends)*e^(rt), an increase in dividends will reduce forward prices.

Statement II is incorrect as forward prices will be determined by near term interest rates, specifically by the borrowing rate for the period of the contract, and will stay unchanged if near term interest rates do not change.

Statement III is incorrect. Futures exchanges require both buyers and sellers to deposit initial margins as prices can move adversely for either of them.

Statement IV is incorrect, as the margin deposited when a contract is entered into is called initial margin. Margin calls thereafter resulting from movements in prices are called variation margin.

Statement V is incorrect. Most futures exchanges distinguish between hedgers and speculators and require different margins from each.

Statement VI is incorrect. Interest rate futures behave almost identically to their bond counterparts, and carry both duration and convexity.

There are two ways to think about this question: First, because the asset manager thinks that interest rates are going to decline, his profit will be maximized if he buys the bond with the greatest duration. The zero coupon bond has the greatest duration among the alternatives listed. Therefore Choice 'c' is the correct answer.

The second way to look at this question is to consider what is called 'reinvestment risk'. Yields to maturity calculations have an implicit assumption that any cash flows received prior to maturity are investible at the ytm rate. Thus, an yield to maturity calculation for a coupon bearing bond assumes that coupon payments get invested at the same rate as the calculated ytm from the time they are received till the time the bond matures. In an environment where interest rates fall, this assumption will not hold true. Reinvestment risk is the risk that coupons will not be able to earn the ytm that the bond was brought at - and the investor will be at a loss to the extent the money does not earn the ytm rate till the end of the investment. One way to reduce reinvestment risk is to minimize coupon receipts so the money stays invested as part of the bond, and the zero coupon bond helps to do exactly that.

The principle of maximum expected utility requires maximizing the expected utilities of the different possible outcomes of a gamble weighted according to the probabilities of their occurrence. This is very difficult to apply in practice in the financial markets where utility functions and various other inputs for maximizing expected utility are not known. Markowitz suggested the mean-variance criterion as a simplification of the principle of maximum expected utility, and it can be shown that the mean-variance gives a good approximation when the range of outcomes under consideration does not exceed plus or minus one coefficient of risk tolerance. (Recall that the coefficient of risk tolerance is the value of x where the gambler is indifferent between equal probabilities of winning x or losing x/2.)

A cash or nothing call is one that pays a fixed amount if the underlying exceeds a certain threshold, or nothing. An asset or nothing call is the same as the cash or nothing call except that instead of paying a fixed amount, it 'pays' the asset. Therefore the payoff profile of a long asset or nothing call has the same payoff as a long cash or nothing call and a long vanilla call.

This is best considered by building payoff diagrams which will show how the two positions are economically equivalent. Choice 'd' is the correct answer, and the rest are not.

A higher coupon brings forward the cash flows of a bond, thereby lowering duration. Therefore statement I is true. The lower the duration the lower the convexity. Therefore a higher coupon leads to lower duration and therefore lower convexity. Thus statement II is true.

Bonds do not have negative modified duration, therefore statement III is false.

If the bond is callable, it leads to negative convexity as a fall in interest rates is likely to induce the issuer to call the bond and re-issue at the new lower interest rates (much like refinancing mortgages when interest rates fall), therefore the upside from a fall in rates is limited. This creates 'negative convexity'. Thus statement IV is correct.

The key point in this question is that the bond is zero coupon, and can only be called at face value. Since the bond is zero coupon, its value will always be less than its par value at any time during its existence (as any interest rate will be a positive number). Therefore the issuer will never exercise the call. Thus the bond will have a duration equal to what an equivalent non-callable bond would have.

Since zero coupon bonds have a duration equal to their maturity, the bond's duration is 10 years.

Since the stock is selling for $100 spot, its 3 month forward price for you would be $100*(1 + 6%*3/12) = $101.50. Since the forward is priced at $104, it is 'rich' compared to the spot and should be sold. A profitable trade would be to buy the spot and sell the forward.

Imagine you borrow $100 and buy the stock, and simultaneously sell the forward. At the end of 3 months, you will receive $104 in return from the forward, and be obliged to return the initial $100 borrowed together with interest of $1.50 for the three months. You would make a riskless profit of $2.50 per share in three months. Thus Choice 'c' is the correct answer.

Note that the treasury yield is irrelevant, what matters is the rate at which you can finance the trade which is 6% per annum.

Only the butterfly spread has a payoff profile that benefits when prices do not move much. The collar benefits during declining markets, the straddle and the strangle benefit from sharp movements in the markets. Therefore Choice 'c' is the correct answer.

Backwardation is said to occur when futures prices are lower than the current spot prices. This would happen only when carrying costs are negative. Carrying costs are equal to interest, plus storage costs and less any 'convenience yield'. The existence of large convenience yields may explain backwardation in commodity futures prices. Therefore Choice 'd' is the correct answer.

Contango is the 'normal' market situation where forward prices are higher than spot prices. Storage costs explain contango, not backwardation. Risk free rates, or the cost of funding for the futures position, are always positive and do not explain backwardation.

T-bill futures 'discount' can be converted to a price for the bill using the formula Price = [1 - discount * number of days/360]. In this case, this works out to (1- 10% *90/360) * 100 = $97.50. Choice 'd' is the correct answer.

The delta of an at-the-money call option is close to 0.5. It is close to 1 when it is deep in the money. It is close to 0 when it is deep out of the money. It is never negative. Therefore Choice 'b' is the correct answer.

Gamma risk is unique to options. Therefore gamma risk can only be hedged using options, and Choice 'a', ie an option position that offsets any existing gamma in the portfolio, is the only way to hedge gamma.

Positions in stock or bank deposits do not carry gamma and therefore cannot be used to hedge gamma risk.

This question is based upon a weather derivative contract traded on the CME in the US. For each day, 'Heating-Degree-Days' (HDD) is calculated as equal to 65 degrees Fahrenheit minus the daily average temperature. The daily average temperature is based upon the temperature reported by the Earth Satellite Corporation using a specified automated weather station. Based upon daily values of HDD, an aggregated number called the 'CME degree days index' is calculated for each contract month. In other words, the index for a particular month is just the aggregation of the 'HDD' value for each of the days of that month. Each contract settles at the end of the month at a value equal to $20 x Degree Days Index. (In a similar way, 'Cooling Degree Days' are also calculated and a futures contract offered, except that CDD is equal to the average daily temperature minus 65 degrees). (Source: CME's website at CMEGroup.com)

In the given question, we are interested in hedging against the possibility of the temperature being too low. This means we should buy the HDD futures contract (the lower the temperature, the higher the difference of the average temperature from 65 degrees, and the higher the settlement). Therefore Choice 'b' is the correct answer. The lower the actual temperature turns out to be, the higher the payout to the grower. It would not be wise to wait till January to buy the contract as by then the prices of the contract would have already risen if the grower's fears of a colder January appear to be coming true. He can hedge his exposure by immediately locking in the January prices.

The duration matched hedge is the best hedge for the bond manager. Matched durations in the bond portfolio and the hedge positions will ensure that any decline in the bond portfolio due to interest rate changes is matched by a rise in the futures position. Hedging dollar-to-dollar will not work unless the bonds and the futures contracts are identical in duration terms. For example, if the duration of the portfolio is 7 years, and each futures contract has a duration of 3.5 years, then selling $1m in value will only cover half the risk. Choice 'b' is the right answer.

Systematic risk refers to market risk that cannot be diversified away. Specific risk relates to the unique risk from the securities selected in the portfolio, and these can be diversified away by adding other securities to the portfolio. Diversifiable risk is risk that can be diversified away, ie the same as specific risk. Portfolio risk is the total risk of the portfolio that includes both specific and systematic risk.

Futures initial margins are determined by the exchange. SPAN is the name of a framework the CME uses to determine margins. Only Choice 'd' is correct.

Since the 10 year rate has stayed the same, but the curve has gone from being flat to upward sloping, it means the shorter term interest rates have gone down. Recall that Macaulay duration is the PV of the payments received on a bond over its life weighted by the year each payment is received in. Also remember that a bond's current price is nothing but the summation of the different PVs of its various payments. It effectively gives the real average life of the bond - for example if a hypothetical (and impossible) bond derives half of its value from a payment today and the other half from a payment in 10 years, the investor can look at the average as being 5 years (0*50% + 10*50%). That is what is what we are calculating when deriving Macaulay duration.

In this case, a reduction in the near term interest rates increases the PV of the payments received prior to the payment at 10 years. This increases the value of the bond, and also increases the weightage of the earlier payments, decreasing the Macaulay duration (by shifting it closer to now and further behind from the maturity date).

Statement I is true. Consider the Black Scholes PDE 202.23.e1

and substitute delta = 0 (as this is a delta neutral portfolio), and we get the result 202.23.e.

Since r is generally small, and σ and S are positive, theta will be negative when gamma is positive and vice versa.

Statement II is incorrect. The gamma of a call and a put are equal and do not add to 1. The relationship described applies to delta, and not gamma.

Statement III is correct because positive gamma means that the portfolio gains both from an increase and a decrease in the value of the underlying, given delta neutrality. (Generally, a positive gamma is a good thing, but like everything else, it does not come free. A trader can choose to keep their portfolio gamma positive, but in doing so he or she would be giving up premiums from options they would have sold to neutralize the gamma.)

Statement IV is correct because the only way to hedge gamma and vega is through other options positions as only options have gamma and vega. If only one tradeable option is available, it would be possible to hedge either the gamma or the vega, but not both, as achieving neutrality in one will upset the neutrality of the other. The only way to simultaneously hedge the two would be to use two different options on the underlying, and determine the number of options to be traded (using a system of simultaneous equations).

T-bill futures and fed futures are very short term futures contracts, and unlikely to provide much of a hedge against a rise in interest rates. FRAs however are customized and would be the right instrument for the borrower to hedge his interest rate risk. A long FRA position gives him the right to borrow at an agreed rate in the future. This rate will not change regardless of the changes in interest rates, and therefore Choice 'b' is the correct answer.

Bootstrapping is not one of the various approaches to try to value a convertible security. The rest of them are, and therefore Choice 'b' is the correct answer.

Treasury bond futures do not specify which bond can be used to effect delivery, but allow the seller to pick from a number of available bonds. As a result, one of these eligible bonds emerges as being the 'cheapest' to deliver, and this CTD bond is determined by the basis between the cash price of the bond and the futures spot price as adjusted by the conversion factor for this specific bond. (ie, basis = Cash Price of the Bond - (Futures Price x Conversion Factor).)

The bond with the lowest basis is generally the CTD. In the given question, the three basis calculations are:

1) 97 - (0.9 x 105) = 2.5

2) 101 - (1 x 105) = -4

3) 106 - (1.1 x 105) = -9.5

Since the third bond with a quote of 106 has the lowest basis, Choice 'b' is the correct answer.

The dividend payment has a present value of $2.20*e^( -6%*6/12) = $2.14. Therefore the forward price for delivery 9 months hence should be ($102 - $2.14)*e^(7%*9/12) = $105.25

As a bond approaches maturity, its value will be driven less by the time value of money or its coupons, and more by its redemption value, ie the par. Therefore as maturity approaches, its value converges to its par value and this phenomenon is called pull to par.

There was no event such as the CDO squeeze in 2008. (Quite on the contrary, securitized products were selling at distressed prices.).

The silver squeeze of 1979-80 (Hunt brothers), the Chicago fire of 1872 (leading to a short squeeze on wheat), and the wheat squeeze (Hutchingson) of 1866 are real historical events that led to short squeezes in commodity markets. Choice 'b' is therefore the correct answer.

For the PRM exam, you should try to remember the event broadly, and the commodity involved.

In this case, we need to determine the value today of an FRA where the bank has to pay 4% from year 2 to 3 in exchange for the then prevailing LIBOR. We do this by using the forward rate from year 2 to 3, and comparing it to the fixed rate. The forward rate is determined from the zero rates as =(1.03^3 / 1.025^2) - 1 = 4.0073%. The bank is committed to paying 4%, therefore the value of the FRA at the end of year 3 = (4.0073% - 4%) * $10m = $732.90. But this is the value at the end of year 3, and needs to be discounted to the present using the 3 year zero rate. Therefore the value of the FRA is $732.90/(1.03^3) = $670.70.

The investor's needs to sell futures contracts, as his current position is long. He needs to sell ($1,000,000 x (2.5-1)) / ($240,000 x 1.2) = 5.2 contracts, or rounded to 5 contracts.

It is important to note here that the investor wishes to retain a beta of 1, and does not want to get rid of all market exposure.

An increase in volatility increases the value of the option, and so do increases in the price of the underlying and the risk free rate. However, since a European option can only be exercised at expiry, an increase in the time to expiry may not necessarily increase the value of the option as it may increase the uncertainty around a more certain payout.

Consider the extreme case of a deep in the money European call option that has 1 day left to expiry, and a payout is certain. Now imagine the time to expiry is increased by say, 6 months. Now the payout is no longer certain as no one knows what the value of the underlying will end up at after 6 months. In such a case, the value of the option would decline. But this applies only to a European option. An American option, which can be exercised any time, will not be affected by this reasoning.