FINC 3340 Project

Seungho Baek

Due: Thursday, Dec. 05 2019, 12:30 P.M.

This is a group project (But students can do this project individually). The number of people in each team

must be 3 – 5 people. Please make a cover page and put the list of all the names and IDs in a team. To

submit take home exam electronically, use Blackboard. DO NOT send me it via email. You need submit

only one file: a single Excel . You have 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 work.

Instruction The SHB-Brooklyn Capital Management is seeking to expand to the field of high frequency

tradings from low frequency trading strategies. Because this new trading area requires the company to

purchase new trading platforms, super fast computing equipment, and expensive real time millisecond data,

and to hire genius financial engineers from Ivy league universities and brilliant CUNY-Brooklyn alumni, it

needs substantial amount of funds. To collect funds, the president and CEO agree to issue 30 year corporate

bonds to invest in this new business. Since there are no 30 year corporate bonds trading in the fixed income

market, they are not able to figure out what is the fair value of this bond.Thus, they ask the department of

Sale and Trading Securities to report what they would like to know. Now you are retained by the department

of Sale and Trading Securities in the SHB-Brooklyn Capital Management and you are asked to implement

a term structure model and price this bond.

Problem 1 Yield Curve and Cubic Spline Approach

1. Find daily treasure yield curve rates on 11/15/2019 using Daily Treasury Yield Curve Rates provided

by the US Department of Treasury1 and report those daily rates (i.e., 1 Month, 2 Month, 3 Month, 6

Month, 1 Year, 2 Year, 3 Year, 5 Year, 7 Year, 10 Year, 20 Year, 30 Year) and plot the yield curve.

2. Using the cubic interpolation method, ^ r(t) = r0 + at + bt2 + ct3,

(a) Find r0, a, b, and c. solving the below equation:

min

γ0;a1;b1;c1

P30 t=1(r1(t) – r^1(t))2 = P30 t=1 e2 i

(b) Given the estimated model, predict 0.5 year, 1.5 year, 2.5 year, 3.5 year, …, 29.5 year, and 30

year yield rates.

(c) Plot two series of graphs in a figure displaying the original yield rates and the estimated yield

rates over the entire time horizon.

Problem 2 Establish Vasicek term structure model

1. Based on the yield rates on Nov.15, 2019, let us develop another yield curve model applying the Vasicek

model as below.

r^(t) = -1

t

[ln(A(t)) – B(t) × r0]

where:

A(t) = e( (B(t)-t)( κκ22µ- σ22 ) – σ2B4κ2(t) )

B(t) = 1-eκ-κt

1http : ==www:treasury:gov=resource -center=data-chart-center=interest-rates=P ages=T extV iew:aspx?data = yield

1

(a) Find out three parameters including κ, µ, and σ solving the below equation:

min β1;β2;β3;λ | 30 X (r(t) – r^(t))2 = 30 X e |

t =1 | t =1 |

2

i

(b) Estimate 0.5 year, 1.5 year, 2.5 year, 3.5 year, …, 29.5 year, and 30 year yield rates.

(c) Plot two series of graphs in a figure displaying the original yield rates and the estimated yield

rates over the entire time horizon.

Problem 3 Establish CIR term structure model

1. Based on the yield rates on Nov.15, 2019, let us develop another yield curve model applying the CIR

model as below.

r^(t) = -1

t

[ln(A(t)) – B(t) × r0]

where:

A(t) = (2κ+( 2κγe +(γκ++γλ+)(λe)γt 2t -1))2σκµ 2

B(t) = (2κ+(κ2( +γeγt +λ-)( 1)eγt-1))

γ = p(κ + λ)2 + 2σ2

(a) Find out three parameters including κ, µ, and σ solving the below equation:

min β1;β2;β3;λ | 30 X (r(t) – r^(t))2 = 30 X e |

t =1 | t =1 |

2

i

(b) Estimate 0.5 year, 1.5 year, 2.5 year, 3.5 year, …, 29.5 year, and 30 year yield rates.

(c) Plot two series of graphs in a figure displaying the original yield rates and the estimated yield

rates over the entire time horizon.

Problem 4 Establish Nelson-Siegel term structure model

1. Based on the yield rates on Nov.15, 2019, let us develop another yield curve model applying the

Nelson-Siegel parsimonious approximation specified as below.

r^(t) = β1 + (β2 + β3)(1 – e-λt

λt ) – β3e-λt

(a) Find out four parameters including β1; β2; β3; and λ solving the below equation:

min β1;β2;β3;λ | 30 X (r(t) – r^(t))2 = 30 X e |

t =1 | t =1 |

2

i

(b) Estimate 0.5 year, 1.5 year, 2.5 year, 3.5 year, …, 29.5 year, and 30 year yield rates.

(c) Plot two series of graphs in a figure displaying the original yield rates and the estimated yield

rates over the entire time horizon.

(d) Report what you have observed from those four estimated models (Cublic Spline, Vasicek, CIR,

and Nelson Siegel). Which model would be more accurate? Provide your evidence supporting

your answers.

Problem 5 Bond Pricing

1. Using the fitted spot rates from the Nelson Siegel model and the zero rates as of Nov.15, 2019 from

the US Department of Treasury, price a 6% 30 year semi-annual coupon bond. Assume that the par

value of this bond is $ 100.

2

2. Compute a bond yield to maturity based on the bond price.

3. Calculate a duration (D) and a convexity (C) of this bond.

4. Using the obtained duration and convexity, compute a changed bond price when the yield increased

by 15 basis point.2

5. Compare the approximation of the changed bond price using duration and convexity from problem 4

with the exact changed bond price for the changed yield. 3

2Use these equations: ∆P = -P0 · D∆y + 0:5 · P0 · C(∆y)2 and P1 = P0 ± ∆P where P0 is the bond price prior to the yield

increased and P1 is the new bond price after

3Simply, with the changed yield compute the price of this bond. For clarification, the changed yield is denoted by y1 and

the original yields from problem 2 is denoted by y0. and ∆y = 0:1%. Then the changed yield is written as y1 = y0 ± ∆y

3