Quiz: Yield Curves

Last reviewed: October 4, 2026

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5 questions to check your understanding of the chapter “Yield Curves”. 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.

Question 1
What kind of calculation is bootstrapping a yield curve?
1. A.Backing out the discount factor for each maturity from swap fixed rates and other rates
2. B.Reading the yields on zero-coupon bonds directly from the market, one maturity at a time
3. C.Finding the market fixed rates of swaps from the discount factor for each maturity
Answer and explanation
Answer: A
Bootstrapping finds the discount factor for each maturity from rates such as the fixed rates of traded interest rate swaps, in order from the shortest maturity: 1 year, 2 years, 3 years, and so on. The discount factors also give the spot rates, and the resulting curve (the yield curve) is used to discount future payments and to calculate forward rates.

The yields on zero-coupon bonds, which pay no interest before maturity, cannot be observed directly in the market for many maturities. So the discount factors are backed out from the fixed rates. Revaluing swaps at their fixed rates using the discount factors is how you check that the curve you built matches the market rates. That calculation runs in the opposite direction to bootstrapping.
Learn more: Bootstrapping a Yield Curve
Question 2
As a hypothetical example, the 1-year swap fixed rate is 2.00% and the 2-year rate is 3.00%, both paid once a year. If the 1-year discount factor is 1 ÷ 1.02, or about 0.980392, which value is closest to the 2-year discount factor found by bootstrapping?
1. A.0.970874
2. B.0.942596
3. C.0.942318
Answer and explanation
Answer: C
The 2-year swap can be treated as a bond that, on a principal of 100, pays 3 after 1 year and 103 after 2 years. At the fixed rate that can be traded in the market, the present value of this bond equals the principal, 100. Using the 1-year discount factor, the present value of the 3 paid after 1 year is 3 × 0.980392, or about 2.941176, so the present value of the 103 paid after 2 years is the remainder, about 97.058824. The 2-year discount factor is 97.058824 ÷ 103, or about 0.942318.

0.942596 is the value you get if you discount the 3 paid after 1 year by 1 ÷ 1.03 instead of the 1-year discount factor. The present value of the 3 then falls to about 2.912621, which leaves a larger present value for the 103 paid after 2 years, so the discount factor becomes 1 ÷ (1.03 × 1.03), or about 0.942596. In bootstrapping, the payment after 1 year is discounted by 0.980392, the factor found from the 1-year fixed rate. 0.970874 is 1 ÷ 1.03, which discounts the payment after 2 years for only 1 year.
Learn more: Bootstrapping a Yield Curve
Question 3
As a hypothetical example, suppose that, with continuous compounding, the 2-year discount factor is 0.960789 (zero rate 2%) and the 5-year discount factor is 0.740818 (zero rate 6%). If the gap between 2 and 5 years is filled by interpolation that joins the logarithms of the discount factors with a straight line, which value is closest to the 3-year discount factor?
1. A.0.881027
2. B.0.887465
3. C.0.904837
Answer and explanation
Answer: A
When you join the logarithms of the discount factors with a straight line, you draw a line between the logarithms of the 2-year and 5-year discount factors and read the discount factor from the point on that line. The logarithm is about −0.040000 at 2 years and about −0.300000 at 5 years. 3 years is 1 year past the 2-year point. The logarithm there is −0.040000 + (−0.300000 − (−0.040000)) ÷ 3, or about −0.126667. The 3-year discount factor is $e^{-0.126667}$ , or about 0.881027.

0.887465 is the value from joining the discount factors themselves with a straight line: 0.960789 + (0.740818 − 0.960789) ÷ 3. 0.904837 is the value from joining the zero rates with a straight line. The 3-year zero rate is 2% + (6% − 2%) ÷ 3, or about 3.3333%, so the discount factor is $e^{-0.033333 \times 3} = e^{-0.1}$ , or about 0.904837. Joining a different quantity with a straight line gives a different 3-year discount factor, even with the same 2-year and 5-year inputs.
Learn more: Curve Interpolation
Question 4
As a hypothetical example, the continuously compounded zero rate is 5% at 2 years and 3% at 5 years. Which way of joining the two points makes the forward rate negative just before 5 years?
1. A.Joining the discount factors with a straight line
2. B.Joining the logarithms of the discount factors with a straight line
3. C.Joining the zero rates with a straight line
Answer and explanation
Answer: C
When you join the zero rates with a straight line, the annual rate $r(t)$ (in %) at maturity $t$ is $r(t) = 19/3 - (2/3)\,t$ . The forward rate $f(t)$ is the slope of $r(t)\,t$ , so $f(t) = 19/3 - (4/3)\,t$ , which reaches 0 at about 4.75 years. $r(t)\,t$ is 10% at 2 years, about 15.04% at 4.75 years, and 15.00% at 5 years. It peaks partway through the interval and then falls, so the forward rate is negative just before 5 years. At 5 years it is about −0.33%.

The discount factor falls from about 0.904837 at 2 years ( $e^{-0.05 \times 2}$ ) to about 0.860708 at 5 years ( $e^{-0.03 \times 5}$ ). If you join the discount factors with a straight line, the forward rate stays positive, because the discount factor falls between these two points. If you join the logarithms of the discount factors with a straight line, the logarithm is −0.10 at 2 years and −0.15 at 5 years, so the forward rate from 2 to 5 years is constant at (0.15 − 0.10) ÷ 3, or about 1.67%. With the same values at the points, the sign of the forward rate changes with how you join them.
Learn more: Curve Interpolation
Question 5
An interest rate swap whose floating rate is TIBOR, with yen cash exchanged as collateral, is valued with the multi-curve framework. What is the curve used to discount future payments built from?
1. A.A curve built from the fixed rates of swaps whose floating rate is TIBOR
2. B.A curve built from the fixed rates of yen OIS
3. C.A curve built from the difference between the fixed rates of TIBOR swaps and yen OIS
Answer and explanation
Answer: B
In the multi-curve framework, the discounting curve is set by the collateral terms. Cash collateral in yen earns the yen overnight rate, so future payments must be discounted at that interest rate on collateral, and the curve built from the fixed rates of yen OIS becomes the discounting curve.

Building the discount factors from the fixed rates of TIBOR swaps is the single-curve approach, which uses the same curve for discounting and projection. In the multi-curve framework, TIBOR is used for the projection curve that forecasts floating payments. The difference between the fixed rates of TIBOR swaps and OIS corresponds to the basis spread, and it is not an input to the discounting curve. The TIBOR forward rates are found from the condition that a TIBOR swap has zero value when discounted with the OIS discount factors.
Learn more: The Multi-Curve Framework

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Question 5