Question

1.Briefly explain the Pure Expectations Hypothesis (PEH). Explain why a risk premium related to maturity is not consistent with the PEH. Is the historical empirical evidence (historical data) consistent with the PEH?

2.Define and explain Key Rate Durations. Why do we look at Key Rate Durations for portfolios only? How could two portfolios have the same weighted Modified Duration, but very different Key Rate Durations?

3.Use the following information to calculate the no arbitrage price for a Treasury note futures contract. P = 103.65, Coupon Rate = 4.50%, borrowing and lending rate = 3.00%, t = 0.35. The current quoted futures prices for the same contract is 106.20. State whether you would execute a Cash and Carry or Reverse Cash and Carry trade to create arbitrage profit. Calculate the profit per contract from the trade.

4.Explain the difference between Option Adjusted Spread (OAS) and Z-Spread (Static Spread). Separately discuss how a call and a put option impact the relationship between the two spread measures.

5.One of the key functions of a credit analyst is deriving pro forma projections for future financial data. Explain why an analyst runs a pro forma and then runs a downside or "worst case scenario".

6.Briefly explain the major differences between reduced form and structural models of default risk. Include the inputs, outputs, and uses.

Answer 1

1.The relationship between yields on financial assets of different maturities is a subject that has interested economists and policy makers for decades. The most commonly discussed explanation of this relationship is the expectations theory of the term structure. The "pure" expectations hypothesis (PEH) states that, in equilibrium, the expected returns from different investment strategies with the same horizon should be equal. For example, the expected return from investing in an n-period bond should equal the expected return from investing in a one-period bond over n successive periods. If this theory holds then long-term rates can be (approximately) expressed as a weighted average of current and expected short-term rates. More importantly, it suggests that if policy makers wish to alter long-term rates through their influence on short-term rates they must succeed in altering the market's expectations of future interest rates.

Campbell (1986), however, has defended the empirical applications of the expectations theory on two grounds. First, he argues that CIR consider a more restrictive form of the theory than is considered in the empirical literature. In particular, CIR's discussion is directed to the "pure'' expectations theory which states that risk premia are zero whereas most empirical applications consider the less restrictive expectations hypothesis (EH) which allows for constant risk premia. Campbell shows that the propositions derived from this less restrictive theory are not necessarily incompatible with each other or with arbitrage pricing equilibrium. Furthermore, Campbell shows that any inconsistencies are of second order and may often be ignored in empirical studies.

2.Key rate duration measures how the value of a security or portfolio changes at a specific maturity point along the entirety of the yield curve. When keeping other maturities constant, the key rate duration can be used to measure the sensitivity in a security's price to a 1% change in yield for a specific maturity.

The Formula for Key Rate Duration

Where:

• P- = a security's price after a 1% decrease in its yield
• P+ = a security's price after a 1% increase in its yield
• P0 = the security's original price

Calculating Key Rate Duration

As an example, assume that a bond is originally priced at $1,000, and with a 1% increase in yield would be priced at $970, and with a 1% decrease in yield would be priced at $1,040. based on the formula above, the key rate duration for this bond would be:

KRD=($1,040−$970)/(2×1%×$1,000)=$70/$20=3.5

where:KRD = Key rate duration​

It can be difficult to interpret an individual key rate duration because it is very unlikely that a single point on the treasury yield curve will have an upwards or downwards shift at a single point while all others remain constant. It's useful for looking at key rate durations across the curve and looking at the relative values of key rate durations between two securities. So it is used for portfolio mainly.

For example, assume bond X has a one-year key rate duration of 0.5 and a five-year key rate duration of 0.9. Bond Y has a key rate durations of 1.2 and 0.3 for these maturity points. It could be said that bond X is half as sensitive as bond Y on the short-term end of the curve, while bond Y is one-third as sensitive to interest rate changes on the intermediate part of the curve.

Modified duration is a formula that expresses the measurable change in the value of a security in response to a change in interest rates. Modified duration follows the concept that interest rates and bond prices move in opposite directions. This formula is used to determine the effect that a 100-basis-point (1 percent) change in interest rates will have on the price of a bond. Calculated as:

Modified Duration=Macauley Duration​ /1+YTM/n

Macauley Duration=weighted average term to maturity of the cash flows from a bond

YTM=yield to maturity

n=number of coupon periods per year​

Which is different from Key rate duration.

3.Cash and carry: Say that crude oil is selling now for $50 but the future price for October 2020 is $90. You expect that it costs $35 to store and deliver the oil in October if you buy it now. That’s what you do, you buy now and sell it in October for $90. Profit: $90-85 minus interest rates and other expenses.You can do the same for all types of futures if it’s cheaper to buy today and ‘carry’ it to the delivery date and deliver it for a price that’s higher than the implied price now. Reverse cash and carry: The other way around & it difficult to sell short oil in real life but you can do this bonds and stock indexes. You make money because you do NOT carry it.

4.The Option Adjusted spread is simply the Z- Spread excluding the premium to compensate for the option risk. Thus, the OAS is the spread above the treasury curve that compensates for credit and liquidity risk only. Z-spread is the all-in spread, meaning spread from the risk profile AND from the call risk.

Whether you are planning to purchase a put or call option, it pays to know more than just the impact of a move of the underlying on your option's price. Often option prices seem to have a life of their own even when markets move as anticipated. A closer look, however, reveals that a change in implied volatility is usually the culprit.

While knowing the effect volatility has on option price behavior can help cushion against losses, it can also add a nice bonus to trades that are winning. The trick is to understand the price-volatility dynamic—the historical relationship between directional changes of the underlying and directional changes in volatility. Fortunately, this relationship in equity markets is easy to understand and quite reliable.

5.Worst case scenario – considers the most serious or severe outcome that may happen in a given situation. An example – when calculating the net present value, one would take the highest possible discount rate and subtract the possible cash flow growth rate or the highest expected tax rate.

6.Basically the key difference is in the assumptions/needs of the two models. Structural is akin to an option analogy (and assumes that a company's assets trade like a stock). Structural is more theory based/missing key components (but it gives you a basic understanding of what you're looking for). Reduced form models builds upon this and uses historical trading data based on debt that trades in the firm; it also incorporates macro conditions + company specific conditions.