Deck 25: Estimating the Yield Curve

Question 1

In selecting the placement of knot points in implementing cubic splines, it is common to

A) Space the knot points equally.
B) Use more knot points where data is sparse and fewer where data is plentiful.
C) Use fewer knot points where data is sparse and more where data is plentiful.
D) Allocate the knot points randomly across the spectrum to minimize estimation bias.

Answer 1

Use fewer knot points where data is sparse and more where data is plentiful.

Question 2

Which of the following is not a feature of the exponential splines fitting approach?

A) All its parameters may be estimated simultaneousy using OLS regression.
B) The most popular form is the cubic exponential one.
C) There is an additional parameter $m$ over that of the cubic spline method.
D) The parameter $m$ is the long-maturity limit of the forward rate.

Answer 2

All its parameters may be estimated simultaneousy using OLS regression.

Question 3

The Nelson-Siegel-Svensson model is
(a) A special kind of splining technique.
(b) Adds an exponential spline to the Nelson-Siegel model.
(c) An alternative approach to splines in estimating yield curves.
(d) A superior approach to splines in estimating the yield curve.

Answer 3

C.

Question 4

Which of the following is NOT a benefit of the spline method of estimating discount functions across a spectrum of maturities?
(a) Splines allow fitting curves flexibly to a high degree of fit.
(b) Splined curves are continuous.
(c) Spot and forward rates obtained from splined curves are always smooth.
(d) Splined curves allow fitting linear functions of maturity between knot points.

Question 5

Which of the following is NOT a property of a cubic spline?

A) It is flexible concerning the number of knot points to be used.
B) It uses third-order polynomials in time $t$ .
C) There are three parameters in each function between pairs of knot points.
D) It is capable of accommodating a wide variety of shapes for the yield curve.

Question 6

Under logarithmic interpolation, if he $t _ { i }$ -tear yield is $y \left( t _ { i } \right)$ , $i = 1,2$ , then the interpolated yield for $t$ lying between $t _ { 1 }$ and $t _ { 2 }$ is given by $y ( t ) = y \left( t _ { 1 } \right) \left[ 1 + \ln \left( 1 + x \cdot \left( t - t _ { 1 } \right) \right) \right]$ where $x$ is a parameter. Suppose the yield at one year is 4% and the yield at two years is 5%. Then, the closest yield at one and a half years, using logarithmic interpolation in time $t$ , is

A) 4.47%
B) 4.50%
C) 4.53%
D) 4.55%

Question 7

One of the deficiencies of the bootstrap method is that it only returns discount functions that are on discrete dates. One approach to address this problem for cashflows that do not fall on these dates is to split them into allocations to near dates for which discount functions are available. Suppose we have discount functions $d ( t )$ , where $d ( 1 ) = 0.95$ and $d ( 2 ) = 0.90$ . Assuming continuous compounding, how will a cashflow at $t = 1.5$ years of $100 be allocated to $t = 1$ and $t = 2$ years such that the present value and duration of cashflow remains the same? Assume that the forward rate between 1 and 2 years is constant. The allocation of cashflows is:

A) $48.63 at one year and $51.33 at two years.
B) $47.33 at one year and $52.63 at two years.
C) $46.19 at one year and $53.87 at two years.
D) $50 at one year and $50 at two years.

Question 8

Let two time points, $t _ { 1 }$ and $t _ { 2 }$ , on a yield curve be given, and let $y \left( t _ { 1 } \right)$ , $y \left( t _ { 2 } \right)$ be the yields at these maturities. You want to draw an interpolating curve between these maturities and are considering three alternatives:
-Linear (L): $y ( t ) = y \left( t _ { 1 } \right) + x \cdot \left( t - t _ { 1 } \right)$ .
-Exponential (E): $y ( t ) = y \left( t _ { 1 } \right) e ^ { x \left( t - t _ { 1 } \right) }$ .
-Logarithmic (G): $y ( t ) = y \left( t _ { 1 } \right) \left[ 1 + \ln \left( 1 + x \cdot \left( t - t _ { 1 } \right) \right) \right]$ .
Since the interpolated curves will not coincide perfectly except at the two end-points, interpolated yields will be higher under some methods versus the others. What is the rank-ordering of size of interpolated yields?

A) $L < E < G$
B) $E < G < L$
C) $G < L < E$
D) $E < L < G$

Question 9

The Nelson-Siegel algorithm is primarily used for fitting a smooth curve to the following:
(a) Yields.
(b) Zero-coupon rates.
(c) Forward rates.
(d) Discount functions.

Question 10

The Nelson-Siegel model is extended to what is known as the Nelson-Siegel-Svensson model. The primary additional property that is added by this extension is:
(a) It allows the curve to have a double hump.
(b) It increases the smoothness of the fitted curve.
(c) It mathematically simplifies the earlier model.
(d) It adds an additional name to the moniker of the model.

Question 11

Under exponential interpolation, if he $t _ { i }$ -tear yield is $y \left( t _ { i } \right)$ , $i = 1,2$ , then the interpolated yield for $t$ lying between $t _ { 1 }$ and $t _ { 2 }$ is given by $y ( t ) = y \left( t _ { 1 } \right) e ^ { x \cdot \left( t - t _ { 1 } \right) }$ where $x$ is a parameter. Suppose the yield at one year is 4% and the yield at two years is 5%. Then, the closest yield at one and a half years, using exponential interpolation in time $t$ , is

A) 4.47%
B) 4.50%
C) 4.53%
D) 4.55%

Question 12

You are given two discount bonds of one-year and two-year maturities, with prices of 95 and 90, respectively. A third bond of three-year maturity and an annual coupon of 8% is trading at par. What is the three-year continuously-compounded zero-coupon rate?
(a) 7.7%
(b) 7.8%
(c) 7.9%
(d) 8.0%

Question 13

Exponential splines
(a) is another name for the polynomial spline approach used in yield-curve estimation. (b) Are more parsimonious than polynomial splines in using fewer parameters in yield-curve estimation.
(c) Posit a different functional form for the yield curve than do polynomial splines.
(d) Satisfy both properties (b) and (c) listed above.

Question 14

In the cubic splines technique, which of the following is a benefit of adding more knot points to the algorithm?
(a) It enhances the smoothness of the forward curve obtained from the fitted function.
(b) It makes implementation easier as the number of parameters increases.
(c) It increases flexibility in being able to fit a greater variety of shapes for the discount function.
(d) It enables fitting the curve to more bonds and allows incorporating illiquid bonds as well.

Question 15

A major advantage of the cubic spline approach is that it can be estimated using ordinary least squares (OLS) regression. Which of the following is NOT related to this estimation feature?
(a) OLS may be used because the price of bonds is the just the product of cashflows and discount functions summed up over all cash flows.
(b) OLS enables the use of all the bonds in the data set, even though there are more bonds than parameters to be estimated.
(c) OLS works for polynomials of any order in the splining function.
(d) OLS may be used because it satisfies the smoothness condition required for the forward curve in the estimation.

Question 16

The Nelson-Siegel-Svensson model is
(a) An alternative approach to splines in estimating yield curves.
(b) An approach to estimating yield curves that does not require choosing knot points.
(c) Generally a more parsimonious approach to yield curve estimation than splines.
(d) An approach to yield curve estimation that shares all of the properties listed above.

Question 17

In the Nelson-Siegel framework, which of the following statements is valid? Let the notation used to describe the fitting function be the following: $A ( t ) = a + b e ^ { - t / d } + c ( t / d ) e ^ { - t / d }$

A) $\alpha$ determines the slope.
B) $b$ determines the location of the hump.
C) $C$ determines the hump shape.
D) $d$ determines the long-term rate.

Question 18

Which of the following is a required property of the cubic spline approach to estimating the yield curve and discount functions $d ( t )$ ? Recall that the approach fits functions $d _ { k } ( t )$ to $k$ regions between $( k + 1 )$ knot points.

A) $d ( 0 ) > 1$
B) $d _ { k } \left( T _ { k + 1 } \right) = d _ { k + 1 } \left( T _ { k + 1 } \right)$
C) $d _ { k } ^ { \prime } \left( T _ { k + 1 } \right) > d _ { k + 1 } ^ { \prime } \left( T _ { k + 1 } \right)$
D) $d _ { k } ^ { \prime \prime } \left( T _ { k + 1 } \right) < d _ { k + 1 } ^ { \prime \prime } \left( T _ { k + 1 } \right)$ where $d ^ { \prime } ( t )$ and $d ^ { \prime \prime } ( t )$ denote first and second derivatives of the function $d ( t )$ with respect to $t$ .

Question 19

Which of the following is NOT a valid restriction or deficiency of the bootstrap method of estimating the discount function?

A) The method requires that all bonds have cashflows on the same dates.
B) If bond cash flows are written as arrays (i.e., as $\left( c _ { 1 } , c _ { 2 } , \ldots \right)$ on the chosen dates $t _ { 1 } , t _ { 2 } , \ldots$ ), the method requires that the cash flow arrays of bonds used in estimating the discount function be linearly dependent.
C) It requires choosing specific bonds on each date and is sensitive to the choice of bonds.
D) After bootstrapping, It is difficult to value bonds that have cashflows on dates that are in between discount function maturities.