Quiz: Interest Rate Exotics

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7 questions to check your understanding of the chapter “Interest Rate Exotics”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.

1. Question 1

   Why does valuing a CMS require what is called a convexity adjustment?

   1. A.It is based on the difference between a long-term and a short-term swap rate and is affected by their correlation
   2. B.It has several exercise dates, and valuing the right to exercise before maturity needs computational techniques
   3. C.Its market is illiquid, and the actual sale price may be well below the theoretical price
   4. D.The expected swap rate at the payment date differs from the forward swap rate implied by today's market
   Answer and explanation

   Answer: D

   For a CMS, the expected swap rate at the payment date generally does not equal the forward swap rate implied by today's market. This is why valuation requires what is called a convexity adjustment.

   The other three are reasons why interest rate exotics other than the CMS, or structured products, are difficult to value or trade. The effect of the correlation between two rates is a risk specific to products with more than one underlying asset, and products with a single underlying asset do not have it. A right that can be exercised before maturity is found in Bermudan-style products. Valuing this right with the LIBOR market model, one of the term structure models, requires computational techniques such as least-squares Monte Carlo. Low liquidity is a problem when selling structured products, not a reason for a valuation adjustment.

   Learn more: Overview of Exotic Rate Products

2. Question 2

   A CMS references the 5-year swap rate, pays interest once a year, and sets each payment by the swap rate on its observation date. For the payment observed in 5 years and the one observed a year later, which swaps does it reference?

   1. A.Both payments reference the 5-year swap starting in 5 years
   2. B.The 5-year swap starting in 5 years, and the 5-year swap starting in 7 years
   3. C.The 5-year swap starting in 5 years, and the 5-year swap starting in 6 years
   Answer and explanation

   Answer: C

   On each observation date, a CMS uses the rate of a swap of the same tenor starting on that date, so each payment refers to a different swap. The payment observed in 5 years is set by the 5-year swap rate for a swap starting in 5 years, and the payment observed one year later by the 5-year swap rate for a swap starting in 6 years.

   It is a mistake to think that every payment references the same swap. Starting in 7 years would assume observation dates 2 years apart, but with interest paid once a year the gap is 1 year.

   Learn more: Constant Maturity Swaps (CMS)

3. Question 3

   As a hypothetical example, consider a call on the difference between the 10-year swap rate and the 2-year swap rate. The term is 1 year, the strike equals the expected value of the difference, and discounting is ignored. Both swap rates follow normal distributions, and each has a constant standard deviation of 100bp a year (1bp is 0.01%). If the correlation coefficient between the two swap rates falls from 0.9 to 0.1, what happens to the premium of this option?

   1. A.It rises
   2. B.It falls
   3. C.It does not change
   Answer and explanation

   Answer: A

   Let the standard deviation of the difference be $\sigma_{\text{diff}}$, the standard deviations of the two swap rates $\sigma_1$ and $\sigma_2$, and the correlation coefficient $\rho$. Then $\sigma_{\text{diff}}^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\,\sigma_1\sigma_2$.

   With a correlation coefficient of 0.9, the standard deviation of the difference is about 44.7bp, and with 0.1 it is about 134.2bp. The lower the correlation, the more independently the two rates move, the larger the standard deviation of the difference, and the higher the premium.

   This is the opposite of the view that a lower correlation lowers the premium. Even when each rate has the same standard deviation, a change in the correlation changes the standard deviation of the difference, and so the premium. If the difference exceeds the strike, the buyer receives the excess. If it does not, the buyer loses the premium.

   Learn more: CMS Spread Options

4. Question 4

   As a hypothetical example, a company has bought a payer swaption (a right to start a swap in which the holder pays the fixed rate and receives the floating rate) with a notional amount of 100 million yen, a strike rate of 2%, and a maturity of 5 years. It can be exercised on one date a year, 1 to 4 years from now. The market fixed rate for the 2-year swap starting in 3 years is 3.5%, and the company exercises then. By how much are its fixed payments, as a total before discounting, lower than at the market level?

   1. A.1.5 million yen
   2. B.3 million yen
   3. C.4.5 million yen
   4. D.6 million yen
   Answer and explanation

   Answer: B

   A Bermudan swaption can be exercised on several dates, and the later the exercise date, the shorter the swap that starts. If the company exercises in 3 years, the swap that starts runs 2 years, to year 5.

   The company pays the strike rate of 2% against a market rate of 3.5%, a gap of 1.5% a year. Per year, that is 100 million yen × 1.5% = 1.5 million yen. Over 2 years, it is 1.5 million yen × 2 = 3 million yen.

   The 1.5 million yen figure counts only one year, and 4.5 million yen treats the swap as a 3-year swap because the exercise is in year 3 (1.5 million yen × 3). The 6 million yen figure is for the 4-year swap that starts if the company exercises in year 1 (1.5 million yen × 4).

   If the market fixed rate in 3 years is below the 2% strike, for example 1.5%, exercising would mean paying 500,000 yen a year more than the market (100 million yen × 0.5%), so the company does not exercise. If it never exercises, it loses the premium it paid when it bought the option.

   Learn more: Bermudan Swaptions

5. Question 5

   A callable swap gives the fixed-rate payer a right to terminate the swap before maturity. How does the fixed rate of this swap compare with that of a plain swap on the same terms?

   1. A.It is higher than for the plain swap, and the price of the right shows up in the fixed rate
   2. B.It is higher than for the plain swap, and the price of the right is paid up front as a lump sum
   3. C.It is lower than for the plain swap, and the price of the right shows up in the fixed rate
   Answer and explanation

   Answer: A

   In a callable swap, there is no up-front premium, and the price of the right shows up as a difference in the fixed rate. When the payer holds the right, the fixed rate is higher than for a plain swap on the same terms. In a hypothetical example, when the plain swap is at 2.0% a year, a callable swap that can be terminated from year 3 is at 2.3%.

   In a swaption, the buyer pays the premium as a lump sum, but a callable swap does not work that way, so the second choice is wrong. When the receiver holds the right, the fixed rate is lower than for a plain swap, and the third choice describes that case.

   The payer who holds the right gains if rates fall, because it can terminate and re-enter a swap at a lower fixed rate. If rates rise, the extra amount in the fixed rate remains as a cost.

   Learn more: Callable Swaps

6. Question 6

   A range accrual note accrues interest at a higher rate on days when the reference rate is inside a range set in advance, and at a lower rate on days when it is outside. In derivative terms, buying this note is the same as which of the following positions?

   1. A.Having bought a series of digital options struck at the upper and lower ends of the range
   2. B.Having sold a series of digital options struck at the upper and lower ends of the range
   3. C.Having bought barrier options with barriers at the upper and lower ends of the range
   4. D.Having sold the digital call at the upper end of the range and bought the digital put at the lower end
   Answer and explanation

   Answer: B

   The investor who receives the interest is in the same position as having sold a series of digital options. On each observation date, the investor sells a call struck at the upper end of the range and a put struck at the lower end. In return for selling them, the investor receives a higher rate for the days inside the range, which shows up in the interest.

   It is a mistake to think of the investor as having bought options because the investor receives interest at a higher rate. The investor is on the side that sells the options. There is also a form that sells barrier options at the ends of the range, but it is rare, and it is also on the selling side. Buying the put reverses the direction at the lower end of the range.

   The investor's gain or loss depends on the days the rate falls outside the range. If the rate stays inside the range every day, the interest stays at the rate for days inside. The more days outside, the more the interest is reduced by the difference in rates for those days.

   Learn more: Range Accruals

7. Question 7

   As a hypothetical example, an investor holds a snowball note that pays interest once a year. The rate for year 1 is a fixed 3.0%. From year 2 on, the rate is the previous period's rate plus an add-on of 2.0%, minus that period's floating rate, and it is 0% if the result is negative. The floating rate is 4.0% in year 2, 5.5% in year 3 and 1.0% in year 4. What is the rate for year 4?

   1. A.4.0%
   2. B.3.0%
   3. C.1.0%
   4. D.0%
   Answer and explanation

   Answer: C

   The rate for year 2 is 3.0% + 2.0% − 4.0% = 1.0%. For year 3, 1.0% + 2.0% − 5.5% = −2.5% is negative, so the rate is 0%. For year 4, the previous period's rate is 0%, so the rate is 0% + 2.0% − 1.0% = 1.0%. The 0% figure assumes that once the rate reaches 0% in year 3, it stays at 0% from year 4 on. The 3.0% figure adds the floating rate instead of subtracting it (0% + 2.0% + 1.0%), and the 4.0% figure restarts from the year 1 rate instead of the previous period's rate (3.0% + 2.0% − 1.0%).

   A snowball's rate is set by the path the rates have taken. If the floating rate keeps falling, the rate builds up. For example, if the floating rate falls to 1.0%, 0.8%, 0.5% and 0.5% in years 2 to 5, the rate for year 5 is 8.2%. If the floating rate keeps rising, the rate shrinks and may reach the 0% floor.

   Learn more: Snowballs