Chapter 16 - Properties of Interest Rates (FRM Part 1 - Book 3)
Chapter 16 - Properties of Interest Rates
Chapter 16 - Properties of Interest Rates
1. Which of the following is considered a risk-free interest rate?
A. Repo Rate
B. LIBOR
C. Treasury Rate
D. OIS Rate
Treasury rates are government borrowing rates in its own currency and are considered risk-free.
2. What is one major drawback of LIBOR that led to its phase-out?
A. It is based on estimates and prone to manipulation
B. It is too complex to understand
C. It is based only on actual transactions
D. It is a daily average of Treasury bond yields
LIBOR is based on estimates provided by banks, not actual transactions, making it susceptible to manipulation.
3. Which of the following rates is derived from actual overnight repo transactions?
A. LIBOR
B. Treasury Rate
C. OIS Rate
D. SOFR
SOFR is based on actual overnight repurchase agreement transactions.
4. Which interest rate increases as the credit risk of the underlying instrument increases?
A. All interest rates with embedded credit risk
B. Only Treasury rates
C. Only SOFR
D. Only OIS rates
Interest rates tend to rise with increased credit risk in the underlying financial instrument.
5. What does a repurchase agreement (repo) involve?
A. Buying a bond and holding it indefinitely
B. Selling a security with a promise to repurchase it later at a higher price
C. Issuing government debt to the public
D. A fixed-term deposit with a central bank
A repo agreement involves selling a security with the agreement to repurchase it later at a higher price, implying interest.
6. What happens to the coupon payment of an inverse floater when the reference rate (e.g., LIBOR) decreases?
A. It remains unchanged
B. It increases
C. It decreases
D. It becomes zero
Inverse floaters move opposite to the reference rate; if LIBOR decreases, the coupon increases.
7. What is the rate at which large financial institutions borrow from each other overnight in the United States?
A. Repo Rate
B. SOFR
C. Federal Funds Rate
D. LIBOR
The federal funds rate is the overnight borrowing rate between large U.S. financial institutions.
8. In an Overnight Indexed Swap (OIS), what is the floating leg based on?
A. Geometric average of overnight rates
B. LIBOR estimates
C. Treasury Bill average yield
D. Repo rate for the previous day
The floating leg of an OIS is calculated as the geometric average of the federal funds overnight rate over the period.
9. Which rate is increasingly preferred by derivative traders for short-term risk-free reference?
A. Treasury Rate
B. Federal Funds Rate
C. LIBOR
D. OIS Rate
OIS rates are preferred by traders as a better proxy for short-term risk-free rates, reflecting opportunity cost more accurately.
10. Why are Treasury rates considered too low to serve as practical risk-free rates by traders?
A. Because they have high default risk
B. Because regulatory demand inflates prices and suppresses yields
C. Because they are not actively traded
D. Because they are based on outdated benchmarks
Treasury prices are driven up due to regulatory demand, lowering yields and making them appear lower than true opportunity cost.
11. What type of compounding is most commonly used in derivative pricing frameworks?
A. Continuous compounding
B. Monthly compounding
C. Quarterly compounding
D. Annual compounding
Derivative pricing models typically assume continuous compounding, which simplifies mathematical modeling in continuous time.
12. What is the formula to convert a discretely compounded rate (R) to a continuously compounded rate (Rc)?
A. Rc = ln(1 + R)
B. Rc = (e^R − 1)/m
C. Rc = ln(e^R)
D. Rc = m × ln(1 + R/m)
The formula Rc = m × ln(1 + R/m) converts a discrete rate compounded m times per year to an equivalent continuous rate.
13. What does the expression AeRc × n represent?
A. Present value under simple interest
B. Future value with discrete compounding
C. Future value with continuous compounding
D. Discount factor using LIBOR
AeRc × n is the formula for calculating future value when interest is compounded continuously.
14. As the number of compounding periods per year (m) increases, what happens to the difference between discrete and continuous compounding?
A. The difference increases
B. The difference decreases
C. The difference becomes negative
D. There is no change
As compounding frequency increases, the future value of discrete compounding approaches that of continuous compounding, reducing the difference.
15. What is the formula to convert a continuously compounded rate (Rc) to a discretely compounded rate (R) compounded m times a year?
A. R = Rc / m
B. R = ln(1 + Rc)
C. R = m × (eRc/m − 1)
D. R = eRc
To convert from continuous to discrete compounding, use the formula R = m × (eRc/m − 1).
16. What are spot rates also commonly referred to as?
A. Zero rates
B. LIBOR rates
C. Forward rates
D. Swap rates
Spot rates are also called zero rates, as they correspond to the yield of zero-coupon bonds.
17. What type of interest rate is appropriate for discounting a single cash flow occurring at a specific future time?
A. Coupon bond yield
B. Par yield
C. Spot rate
D. Forward rate
Spot rates are used to discount individual future cash flows to present value since they match the timing of each cash flow.
18. What does the bootstrap method primarily aim to calculate from market bond prices?
A. Coupon rates
B. Spot (zero-coupon) rates
C. Par yields
D. Nominal rates
The bootstrap method allows us to extract zero-coupon (spot) rates from coupon bond prices by discounting known cash flows sequentially.
19. In the bond pricing formula under continuous compounding, what does the negative sign on the exponent indicate?
A. Inflation is being considered
B. Interest is paid in arrears
C. It is a floating rate bond
D. The cash flow is being discounted to present value
The negative exponent reflects discounting of future cash flows back to present using continuous compounding.
20. Why is the value of a coupon bond considered the sum of zero-coupon bonds in the bootstrap method?
A. Coupon bonds mature at multiple dates
B. Each individual cash flow can be discounted separately using a spot rate
C. Spot rates are easier to observe than bond prices
D. Bonds are always priced using par value
The bootstrap method treats a coupon bond as a sum of zero-coupon bonds by discounting each cash flow individually at the appropriate spot rate.
21. What is the yield of a bond?
A. The coupon rate of the bond
B. The spot rate of the bond
C. The single discount rate that equates the bond’s price to its present value
D. The difference between coupon and market price
The bond yield is the internal rate of return (IRR) that equates the present value of all future cash flows to the bond's current market price.
22. What is the par yield of a bond?
A. The spot rate that discounts all cash flows to zero
B. The rate that makes the bond’s price equal to its par value
C. The yield when bond is priced above par
D. The average of all zero-coupon rates
The par yield is the coupon rate that results in the bond being priced exactly at its face (par) value.
23. What happens when a bond’s coupon rate equals its yield?
A. The bond trades at par
B. The bond trades at a discount
C. The bond trades at a premium
D. The bond price becomes zero
When the coupon rate equals the bond’s yield to maturity, the bond is priced at par.
24. Which of the following is true when a bond trades above par value?
A. Its coupon rate is less than its yield
B. Its coupon rate equals its par yield
C. Its yield equals its spot rate
D. Its coupon rate is greater than its yield
If the bond trades above par, it means the fixed coupon payments are more attractive than current market yields, so the coupon rate exceeds the yield.
25. According to the professor's note, what can be said about the results of discounting methods in exam scenarios?
A. Only one method will give the correct answer
B. The result depends on the duration of the bond
C. Both discrete and continuous discounting give similar results
D. Use of continuous compounding is always required
The professor notes that both discounting methods yield similar results and, in exam settings, selecting the closest answer is acceptable.
26. What does bootstrapping help derive in bond pricing?
A. Yield to maturity
B. Par yield curve
C. Bond duration
D. Spot rate curve
Bootstrapping uses coupon bond prices to compute theoretical spot rates or zero-coupon yields for various maturities.
27. Which of the following best describes a forward rate?
A. A rate derived directly from coupon bonds
B. An implied future interest rate between two future periods
C. The actual interest rate agreed today for the next year
D. A one-year zero-coupon rate
A forward rate is an interest rate implied by today's spot rate curve for a loan or investment starting at a future time.
28. What is the FRA payoff formula if receiving the fixed rate RK?
A. L × (RK − R) × (T2 − T1)
B. L × (RK + R) × (T2 − T1)
C. L × RK × (T2 + T1)
D. L × R × (T2 − T1)
If you are receiving the fixed rate RK, your profit is based on the difference between RK and the actual market rate R over the FRA period.
29. Why is the FRA value discounted back to T1 even though interest is due at T2?
A. Because settlement occurs at T2 in all markets
B. Because FRAs are always valued using spot rates
C. Because FRAs are settled at the beginning of the period
D. Because continuous compounding is not allowed
Although the theoretical interest applies over the period T1 to T2, FRA settlements occur at T1, requiring discounting to present value at that time.
30. If the forward rate RForward is lower than the agreed fixed rate RK in an FRA where you are receiving RK, what will happen?
A. You will receive a positive cash flow
B. You will pay the counterparty
C. The FRA has zero value
D. There will be no settlement
Since you locked in a higher fixed rate RK and the market forward rate is lower, you benefit from the difference and receive a cash flow.
31. Which theory suggests that the forward rate is an unbiased predictor of future spot rates?
A. Expectations Theory
B. Liquidity Preference Theory
C. Market Segmentation Theory
D. Preferred Habitat Theory
The Expectations Theory assumes that forward rates reflect the market's expectations of future spot rates and do not include risk or liquidity premiums.
32. Which term structure theory best explains the existence of a consistently upward-sloping yield curve?
A. Expectations Theory
B. Liquidity Preference Theory
C. Market Segmentation Theory
D. Rational Expectations Theory
Liquidity Preference Theory adds a liquidity premium to longer-term bonds, causing the yield curve to be upward-sloping even if future short rates are expected to stay constant.
33. What is a key assumption of Market Segmentation Theory?
A. Investors are indifferent to maturity preferences
B. Future spot rates influence bond pricing directly
C. Bond markets operate independently across different maturities
D. Long-term interest rates always exceed short-term rates
Market Segmentation Theory assumes that investors and borrowers confine their activity to specific maturity segments, and supply-demand dynamics within each segment determine interest rates.
34. Why is the Expectations Theory often criticized in real-world observations?
A. It fails to explain why yield curves are mostly upward-sloping
B. It overestimates the role of central banks
C. It ignores the impact of inflation
D. It assumes perfect capital markets
The Expectations Theory is challenged because it suggests that upward and downward sloping yield curves should occur equally, but in practice, yield curves are upward-sloping more frequently.
35. According to Liquidity Preference Theory, what causes the upward slope of the yield curve?
A. Rising inflation expectations
B. Increasing default risk
C. Government policy shifts
D. Investors demanding a premium for longer-term investments
Liquidity Preference Theory argues that because investors prefer shorter maturities, issuers must offer higher yields (liquidity premium) for longer maturities, leading to an upward-sloping yield curve.
36. What is the duration of a bond?
A. The time until the first coupon payment
B. The time to maturity of the bond
C. The time until the bond's market value reaches zero
D. The weighted average time until the bond's cash flows are received
The duration of a bond is the weighted average time until the bond's cash flows are received.
37. What happens to the bond's price if its duration is 5 years and the yield increases by 1%?
A. The bond price will decrease by approximately 5%
B. The bond price will increase by approximately 5%
C. The bond price will remain unchanged
D. The bond price will decrease by approximately 10%
According to the duration formula, the change in bond price is approximately the product of the negative duration and the change in yield. A 1% increase in yield will result in a price decrease of approximately 5% (duration × change in yield).
38. What is the formula for Macaulay duration?
A. Σ (t_i × [c_i × e^yt_i])
B. Σ (t_i × [c_i × e^yt_i]) / B
C. Σ (t_i × [c_i × e^(-yt_i)]) / B
D. Σ (t_i × [c_i × e^(-yt_i)])
The Macaulay duration formula involves discounting the cash flows by the continuously compounded yield and taking the weighted average of the times to those cash flows.
39. If the yield of a bond increases, what happens to the bond's price?
A. The price of the bond increases
B. The price of the bond decreases
C. The price of the bond remains unchanged
D. The price of the bond fluctuates unpredictably
When the yield increases, the price of the bond decreases. This is due to the inverse relationship between bond prices and interest rates.
40. What is modified duration?
A. A measure of bond price sensitivity to changes in interest rates
B. The weighted average time until the bond's maturity
C. The sum of all bond cash flows
D. The percentage change in bond price for a 1% change in yield
Modified duration measures the percentage change in a bond's price for a 1% change in interest rates. It is derived from Macaulay duration.
41. What is dollar duration?
A. The price of the bond in dollars
B. The duration in terms of currency units
C. The change in the dollar price of a bond for a 1% change in yield
D. The product of the bond's modified duration and its price in dollars
Dollar duration is the product of the modified duration and the bond price in dollars. It measures the dollar change in the price of the bond for a given change in yield.
42. If a bond has a higher duration, what does it indicate about its price sensitivity?
A. The bond's price is less sensitive to yield changes
B. The bond's price is more sensitive to yield changes
C. The bond has a lower coupon rate
D. The bond's price is unaffected by yield changes
A higher duration means that the bond’s price is more sensitive to interest rate changes. Bonds with longer durations are more sensitive to yield fluctuations.
43. What is the formula for modified duration when the yield is expressed as a semiannually compounded rate?
A. Modified duration = Duration × (1 + y / m)
B. Modified duration = Duration / (1 + y / m)
C. Modified duration = Duration / (1 + y / 2)
D. Modified duration = Duration × (1 + y)
Modified duration is calculated as the bond's duration divided by (1 + y/m), where y is the yield and m is the number of compounding periods per year.
44. What happens to the difference between Macaulay duration and modified duration as the number of compounding periods (m) increases?
A. The difference becomes larger
B. The difference stays the same
C. The difference decreases
D. The two measures become equal
As the number of compounding periods (m) approaches infinity (i.e., continuous compounding), the difference between Macaulay and modified duration vanishes, making them equal.
45. What is dollar duration?
A. Duration multiplied by the bond’s price in dollars
B. Modified duration multiplied by the bond’s price in dollars
C. Duration divided by the bond’s price in dollars
D. The sum of all bond cash flows in dollars
Dollar duration is the product of the modified duration and the bond’s price in dollars. It measures the dollar change in the price of the bond for a given change in yield.
46. Which of the following statements is true regarding modified duration?
A. It is used when the yield is expressed as something other than a continuously compounded rate
B. It is the same as the Macaulay duration for bonds with annual compounding
C. It is always greater than the Macaulay duration
D. It is always less than the Macaulay duration
Modified duration is used when the yield is expressed as something other than a continuously compounded rate. It accounts for compounding and adjusts the Macaulay duration accordingly.
47. What is the primary limitation of duration in estimating bond price changes?
A. It assumes bond price changes are non-linear
B. It only applies to bonds with coupon payments
C. It is only effective for large interest rate changes
D. It is only accurate for small changes in interest rates
Duration is an approximation of price changes that works well for small changes in interest rates. For larger changes, the non-linear relationship between price and yield causes errors.
48. What does convexity address in the context of bond price and yield changes?
A. It reduces the price volatility caused by interest rate changes
B. It corrects for the non-linear relationship between bond price and yield
C. It measures the curvature of the bond price/yield relationship
D. It makes the bond price/yield relationship linear
Convexity addresses the curvature in the bond price/yield relationship, correcting the errors in price estimations caused by duration when there are large changes in interest rates.
49. How does convexity affect the bond price estimate when interest rates change?
A. It always corrects the estimated price in the direction of the interest rate movement
B. It is always added to duration for option-free bonds
C. It makes the price volatility predictions more erratic
D. It always results in a lower price increase when yields fall
Convexity always adds to the duration correction for price volatility errors in option-free bonds. It reduces the drop in price for rising yields and adds to the rise in price for falling yields.
50. What is the formula for calculating the convexity effect on bond price changes?
A. Convexity effect = convexity × Î” y
B. Convexity effect = 1/2 × convexity × Î” y2
C. Convexity effect = convexity × Î” y × (1 + y/m)
D. Convexity effect = convexity × Î” y × m
The convexity effect is calculated as 1/2 × convexity × Î” y2. It measures the price change not explained by duration, correcting the errors in price estimates.
51. For option-free bonds, the convexity effect is always:
A. Positive
B. Negative
C. Zero
D. Varies with interest rate changes
For option-free bonds, convexity is always positive, regardless of whether interest rates rise or fall, which helps correct price volatility errors due to duration.
52. How can you more accurately estimate the change in a bond’s price when interest rates change significantly?
A. By combining duration and convexity
B. By adjusting the bond’s price with only duration
C. By using only convexity to estimate the price change
D. By calculating the change in the bond’s yield
When interest rates change significantly, combining both duration and convexity gives a more accurate estimate of the bond's price change. Duration captures the linear price movement, while convexity accounts for the curvature.
53. What is the formula for calculating the total change in a bond’s price, considering both duration and convexity?
A. Total change = duration × Î”y
B. Total change = (duration × Î”y) + (1/2 × convexity × Î”y²)
C. Total change = (duration × Î”y) - (1/2 × convexity × Î”y²)
D. Total change = convexity × Î”y
The formula to calculate the total change in price by combining duration and convexity is:
Total change = (duration × Î”y) + (1/2 × convexity × Î”y²).
This accounts for both the linear and non-linear effects of yield changes.
54. What does the convexity effect represent in the price change calculation?
A. The non-linear part of the price change
B. The impact of changes in the bond's coupon rate
C. The linear part of the price change
D. The change in the bond’s duration
The convexity effect represents the non-linear portion of the price change due to large movements in interest rates. It adjusts for errors made when using duration alone.
55. When does the convexity effect become more significant in estimating bond price changes?
A. When there is a small change in interest rates
B. When there are large changes in interest rates
C. When the bond has a high coupon rate
D. When the bond price is low
The convexity effect becomes more significant when there are large changes in interest rates. This is because the relationship between bond price and yield is convex, and duration alone cannot fully capture this curvature.
56. If a bond has a higher convexity, how does that affect the bond’s price movement for large changes in interest rates?
A. The bond price will experience more volatility
B. The bond price will be less sensitive to interest rate changes
C. The bond price will have a smoother change with large interest rate movements
D. The bond price will remain unchanged despite interest rate movements
Higher convexity leads to a smoother bond price change with large interest rate movements, as convexity helps to reduce the error in the estimated price change by accounting for the non-linear relationship.
