Please make a cover sheet. In a cover sheet, list all the names and IDs in your group (Provide a list of names who discussed about your attempt in the last page of your report.). To submit this electronically, use Blackboard. DO NOT send me it via email. Please submit one R-script file that shows your analysis procedure and a summary report via Blackboard. NO LATE SUBMISSION ALLOWED. If you cannot submit it in time, you will get zero automatically.

Part 1:

## Estimation and Forecast

You have been retained by the fixed income desk of Baek-Brooklyn Investment Group to provide a forecast about future short term interest rates, namely, the 3 month t-bill rate. You decide to use two sources of data: historical interest rate data and current forward rates. The data necessary for this forecasting exercise are contained in the Excel file (Project1 2020.XLSX), which you can find on the course web site.

This dataset contains daily observations of the 3 month t-bill rate until April 12, 2017, as well as the Treasury Strip Price data on April 13, 2017. You must write a report including all relevant information and computations, and provide a forecast for an horizon ranging between 6 months and 5 years. Follow the steps below.

- Let us denote by rt the Bond Equivalent Yield (BEY) on day t. The data on DTB3 in Project1 2020.XLSX are quoted on a discount basis. Use the below formulas to obtain a time series of Bond Equivalent Yield (BEY) (n=90). Refer to the below equations and you will easily compute a time series of bond equivalent yield.

Pt = 100[1 − n

360 × dt]

dt = 100 − Pt

100 × 360

n

rt = BEY = 100 − Pt

Pt × 365

n

where t is the time to maturity, dt is a discount yield at time t, n is the number of days to maturity, and P is the price of the T-bill at time t. Note that the amount of denomination is $100.

- Estimate the AR(1) process1 for interest rates.

rt+1 = α+ βrt + εt+1 (1)

where εt+1 ∼ N(0, σ2). 1AR represents an auto-regressive linear model

1

- Let α̂, and β̂ be the estimated parameters from (1). Use (1) together with the most current interest rate available on Project1 2020.XLSX, call it rTODAY , to make a forecast of future interest rates rTODAY+T . Provide forecasts for horizons T = 6 month, and 1, 2,. . ., 5 years (a plot would suffice). Explain how you make the forecasts. (Tip: When you make the forecasts, assume there are 252 (business)days in one year).
- Using the Treasury Strip Prices which also contained on STRIPS 04132017 in Project1 2020.XLSX, compute the current yield curve and forward rates. To compute a zero rate, zt, and a forward rate, ft at time t based on the average price of bid and ask price at time t, use the following equations.

Pt = PBidt + P

Ask t

2

zt = ( 100

Pt ) 1 t − 1)

ft = [ (1 + zt)

t

(1 + zt−1)t−1 ]1/τ − 1

where t is the time to maturity, Pt is the average price of the T-bill at time t, and τ is a time interval between t and t − 1.2 Note that the amount of denomination is $100. Note that zt and ft are not annual interest rates but periodic interest rates per quarter.

- Compare the forecasts of future interest rates that are implicit in the forward rates to those obtained in step 3 above. Plot the forecasts and the corresponding forward rates. Discuss your findings.

Part 2:

### Bootstrapping

Elon Musk, the CFO of Tesla, has been planning to issue 5 year and 15 year option-free corporate bonds (semi-annual) to raise its capital for further R&D. He has just contacted Baek-Brooklyn Investment Group, LLC. to get information on what the fair values of two bonds should be if the bonds are issued on 01/31/2020. Thus, Betty, the head of the fixed income desk, is now managing this task. She now asks you to find spot rates prior to price two bonds.

- Find daily treasure yield curve rates (par interest rates) on 01/31/2020 3 and report those par 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, and 30 Year Par Rate).4 Note that all the rates for the respective tenors are expressed in their bond equivalent yields (BEY).

2Note that τ is 0.25 for this problem. 3Note that Daily Treasury Yield Curve Rates have been provided by the US Department of Treasuryhttp://www.

treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield 4These rates are commonly referred to as “Constant Maturity Treasury” rates, or CMTs. Yields are interpolated by the Treasury from the daily yield curve. This curve, which relates the yield on a security to its time to maturity is based on the closing market bid yields on actively traded Treasury securities in the over-the-counter market.

These market yields are calculated from composites of indicative, bid-side market quotations (not actual transactions) obtained by the Federal Reserve Bank of New York at or near 3:30 PM each trading day. The CMT yield values are read from the yield curve at fixed maturities, currently 1, 2, 3 and 6 months and 1, 2, 3, 5, 7, 10, 20, and 30 years. This method provides a yield for a 10 year maturity, for example, even if no outstanding security has exactly 10 years remaining to maturity. Yield curve rates are usually available at Treasury’s interest rate web sites by 6:00 PM Eastern Time each trading day.

2

- Convert into the respective spot rates using the treasury rates, applying the bootstrapping method. Report those spot 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, and 30 Year Spot Rate).
- Draw the par interest rate curve and the spot interest rate curve in a plot. Compare two lines and provide your findings.

Part 3:

## Term Structure Modeling of Spot Rates with the Cubic Spline Method

From the previous part, you have the respective spot rates for 1 month, 2 month, 3 month, 6 month, 1 year, 2 year, 3 year, 5 year, 7 year, 10 year, 20 year, and 30 year zero coupon bond security. However, in order to price 5 year and 15 year semi-annual coupon bonds, you must obtain spot rates for 0.5 year, 1 year, 1.5 years, 2 years, 2.5 years,. . ., 29.5 years, 30 years. You are now asked by Betty to implement a term structure model using a simple cubic-spline method.

- Using the cubic spline model, which is given by

rt = γ0 + at+ bt 2 + ct3

estimate 0.5 year, 1 year, 1.5 year, 2 year, . . ., 30 year spot rates and report these rates. Note that rt represents the spot rate for maturity t.

- Plot a combo chart and explain how well the estimated spot rates from the cubic spline model fit the original yields. Note that you must provide supporting evidence as to explain the goodness-of-fit of the model. 5

Part 4:

### Bond Pricing

You have all the spot rates to compute 5 year and 15 year bonds. However, you need to find appropriate coupon rates for two securities. From the daily treasury yield rates, you can easily compute a coupon interest rate for the 5 year security, whereas you cannot directly get a coupon interest rate for the 15 year security.

- Find and report the 5 year and 15 year par rates as of 01/31/2020. To find the 15 year par rate, you need to estimate the rate using the cubic spline method.
- Plot a combo chart and explain how well the estimated par yields from the cubic spline model fit the original yields. Again, you must provide supporting evidence as to explain the goodness-of-fit of the model.
- Compute two bond prices using the obtained nominal spot rates and par rates.
- Find two YTMs (yield to maturities) for the respective bonds.
- Tesla has recently had a B- rating (i.e. highly speculative) from S&P Global Rating. To reflect this credit quality of the company to its bonds, you need to adjust bond prices by adding credit spread. Go to FRED website and find the ICE Bank of America Merrill Lynch US corporate BBB Option-Adjusted Spread on 01/31/2020.6

5 More specifically, provide the sum of squared errors: SSE = min γ,a,b,c

∑12 j=1(rj − r̂j)

2 = ∑12 j=1 e

2 j

6 https://fred.stlouisfed.org/series/BAMLC0A4CBBB

3

- Compute adjusted bond prices using the respective yield to maturities and the credit spread. The formulas for the adjusted bond prices is given by

P5Y R = 10∑ t=1

C/2

(1 + Y TM+spread2 ) t

+ M

(1 + Y TM+spread2 ) 10

P15Y R = 30∑ t=1

C/2

(1 + Y TM+spread2 ) t

+ M

(1 + Y TM+spread2 ) 30

where C represents the annual coupon payment, and M represents the maturity value (i.e. $100).

- Go to FRED website and find the 5-Year High Quality Market (HQM) Corporate Bond Spot Rate7 and the 15-Year High Quality Market (HQM) Corporate Bond Spot Rate on Jan. 2020.8
- Compute the 5 year and 15 year bond prices with those HQM corporate bond rates. Note that use the same coupon rates from the previous.
- Compare the adjusted 5 Year and 15 year bond prices and the 5 year and 15 year HQM bond prices. Discuss your findings.

7https://fred.stlouisfed.org/series/HQMCB5YR 8https://fred.stlouisfed.org/series/HQMCB15YR