Term Structure Modelings

FINC 3340 Term Structure Modelings

Seungho Baek

Due: Wednesday, Nov. 29, 2017, 06:30 P.M.

Please make a cover page (name and ID) in Sheet1 in your EXCEL file. To submit this homework electron- ically, use Blackboard. DO NOT send me it via email. Please submit your homework in a single EXCEL file for this time. ONLY an EXCEL FILE will be accepted. You have to write up with your own words. Do not copy from others’ work. In the last page, you have to show all your references for your homework in the last sheet in your Excel.

Part 1 Revise NS model

Instruction: In the last class, we implemented the cubic spline model and the Nelson-Siegel model and estimated yield rates as in NS MODEL.xlsx. Using the estimated model, we priced a 5 year semi-annual bond with a coupon rate of 10%. However, we found that there is an estimation error in the Nelson-Siegel model.

Problem 1. Find out where the error comes. Next correct the error. And then estimate the yield rates.

Problem 2. Using the modified yield rate, compute the price of the bond.

Problem 3. Do you see an arbitrage opportunity? (In other words, is there any price discrepancy between the cubic and the NS?)

Part 2 Term Structure Modeling

Instruction: You have been retained by the treasury department of SHB-Brooklyn corporation to price derivative securities. The CEO and CFO of the company plan to issue 5 year and 15 year corporate bonds to raise capital and ask you to suggest fair values of the newly issued bonds. Thus, you are now plan to implement two term structure models, Cubic-Spline and Nelson-Siegel (NS) yield curve model, and price bond instruments using these models.

Problem 1. Go and visit U.S department of treasury website 1 and obtain the yield rate on 11/17/2017 from U.S. department of treasury.

1) Plot the yield rates over time to maturity and explain the shape of the curve.

2) Using the cubic spline model, which is given by

y(t) = r0 + at + bt 2 + ct3 (1)

estimate 0.5 year, 1 year, 1.5 year, 2 year, . . ., 29.5 year, 30 year yield rates.

3) Plot a combo chart that shows how well the estimated yields from the cubic spline model fit the original yields.

4) Compute two semi annual coupon bonds, 5 year semi-annual coupon bond with a coupon rate of 3% and 15 year semi-annual coupon bond with a coupon rate of 5.5% assuming a face value of $ 1,000 and continuously compounding rates.

1https : //www.treasury.gov/resource−center/data−chart−center/interest−rates/Pages/TextV iew.aspx?data = yield

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Problem 2. In problem 1, you have used the cubic spline method to find the yield rates. For this time, let us use the other model, i.e. Nelson-Siegel model which specified as in equation (2).

y(t) = β1 + β2( e−λt

λt ) + β3(

1 − e−λt

λt − eλt) (2)

1) Estimate 0.5 year, 1 year, 1.5 year, 2 year, . . ., 29.5 year, 30 year yield rates.

2) Plot a combo chart that visually describes how well the estimated yields from the cubic spline model fit the original yields.

3) Compute two semi annual coupon bonds, 5 year semi-annual coupon bond with a coupon rate of 3% and 15 year semi-annual coupon bond with a coupon rate of 5.5% assuming a face value of $ 1,000 and continuously compounding rates.

4) Compare the bond prices from problem1 and bond prices from the NS model. Is there any difference in those prices?

Problem 3. Look at sum of squared errors, SSE,

SSE =

T∑ i=1

e2 = ∑ t=1

T(r(t) − ˆr(t))2 (3)

for the cubic model and the NS model and then find out which model would be more accurate.

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