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Estimation and Forecast of Short-Term Interest Rates - Baek-Brooklyn Assignment

## 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.

1. Let us explore the variable to check the quality of the data. Follow the next steps:

(a) Plot the 3-month t-bill rate over the time.

(b) Plot the histogram with the 3-month t-bill. Discuss about two plots.

(c) Compute a set of simple statistics (mean, standard deviation, minimum, maximum) for

the 3-month t-bill rate.

(d) Exclude observations with missing values or zero values and create a new data set.

(e) Plot the 3-month t-bill rate over the time with a new data.

(f) Plot the histogram with the 3-month t-bill with the new data. Discuss about two plots.

(g) What do you find this data exploration procedure?

2. Estimate the AR(1) process1 for interest rates. rt+1 = α + βrt + εt+1

where εt+1 ∼ N(0, σ2

3.From now, let us the new data set. First, let ˆα, and βˆ, be the estimated parameters from (1). Use (1) together with the most current interest rate available on Project1 2020.XLSX, call it rT ODAY , to make a forecast of future interest rates rT ODAY +T . Provide forecasts for horizons T = 6 month, and 1, 2,. . ., 5 years (a plot would suffice). (Tip: When you make the forecasts, assume there are 252 (business)days in one year.)

4. Compare the forecasts of future interest rates that are implicit in the forward rates to those obtained in step 4 above. Plot the forecasts and the corresponding forward rates. Discuss your findings.

5. In the last project, you did the similar forecast with AR(1). But you did make a forecast without considering any data management. Do you think that your made a good forecast in the last project? Why or Why not?

## Part 2: Bootstrapping and Smoothing Interest Curve

Elon Musk, the CFO of Tesla, has been considering to issue four different fixed income securities:

• Bond1: 7 year option-free corporate bonds (semi-annual)

• Bond2: 7 year call option bond (callable during the period from 0.5 year to 6.5 years)

• Bond3; 7 year call option bond (only callable during the period from 3 year to 6 years)

• Bond4: 7 year call option bond (only callable during the period from 1 year to 4 years)

• FRN1: 7 year floating rate note

• FRN2: 7 year floating rate note (capped floater)

• FRN3: 7 year floating rate note(floored floater)

to raise its capital for further R&D and for boosting upr its production lines. He has just contacted Baek-Brooklyn Investment Group, LLC. to get information on what the fair values of three bonds should be if the bonds were issued on 01/31/2020. Thus, Claire, the head of the fixed income desk, is now managing this task. She now asks you to find appropriate interest rates.

1. 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).

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

ct = γ0 + at + bt2 + ct3

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

3. Plot a combo chart and explain how well the estimated spot rates from the cubic spline model fit the original yields.

4. 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, 1.5 Year, 2 Year, . . ., 29.5 Year, and 30 Year Spot Rate). (Note that 1 Month, 2 Month, 3 Month,

6 Month, and 1 Year par rates are equal to those of spot rates.)

5. Calculated implied forward rates for 0.5 year, 1 year, 1.5 year, . . ., 29.5 year, 30 year maturities.

6. Draw the par interest rate curve, the spot interest rate curve, and the forward rate curve in a plot. Compare three lines and provide your findings. (Hint: plot three interest rates against the following time interval: 0.5 year, 1 year, ..., 29.5 year, 30 year

## Part 3: Arbitrage Bond Pricings without binomial interest rate tree

You have all the spot rates and the forward rates in order to compute those 7 year fixed income securities. Let us use these rates to price them.However, you need to find appropriate coupon rates for two securities. Tesla has recently had a B- rating (i.e. highly speculative) from S&P Global Rating. Tesla has to issue its bonds at a discount for marketability. To reflect this, you suggest the followings:

• A 1.42% par plus 70 bp credit spread, equivalent to a coupon rate of 3.7% per annum,for the 7 year option free and option embedded bonds.

• The OAS spread for discounting the corresponding cash flows should be considered. (Note: Add this spread to all forward rates and zero rates to price those securities)

• A rate of cap is 2.4% per annum for the 7 year capped floater.

• A rate of floor is 1.9% per annum for the 7 year floored floater

Simply, all the obtained zero and forward rates are expressed in BEY(though those rates are actually continuously compounded interest rates).

1. To reflect this credit quality of the company to its bonds, you need to adjust interest rates by adding credit spread. Go to FRED website and find the ICE Bank of America Merrill Lynch US corporate BBB Option-Adjusted Spread on 01/31/2020.5

2. Compute the option free bond price with the zero rates.

3. Compute the yield to maturity for the option free bond using the obtained bond price and compare its coupon rate and the YTM.

4. Can Tesla issue its option free bond at discount using the OAS?

5. Next, compute the option free bond price with the forward rates applying the backward induction method.

6. Check whether two option free bond prices with the zero rates and the forward rates are same.

7. Calculate the prices for the three different callable bonds with the forward rates using the backward induction method and compare them.

8. Calculate the prices for the three different floating rate notes with the forward rates using the backward induction method and compare them.

Although you computed the callable options bond and floating rate bonds based on a simple induction method, the previous pricing method are not able to considered various callable/capped/floored scenarios over the life of the securities. Suppose that in order to take care of this shortcomings, you has implemented based on the Ho-Lee interest tree model to price option embedded bonds and floating rate notes. Because the developed Ho-Lee model can fit the current yield curve, the theoretical price of bonds from the model is equal to the market price. The generated forward rates are in ”Implied Tree.xlsx”. All the rates are semi-annual compounding rates. Using the implied tree nodes, compute three call option bonds and three floating rate notes.

1. Select any five interest paths in the binomial tree and plot them. Compare your figure and coin-tossing figure in part 2. Does your graph look like the coin-tossing figure? If so, what does that imply?

2. Adjust all the rates in the tree nodes prior to price these securities by adding the OAS spreadto each forward rate.

3. Compute the three call option bonds using the modified interest rate tree.

4. Calculate the respective effective durations and effective convexities for the three callable bonds. Interpret all the values.6

5. Compute the values of the three floating rate notes using the modified interest rate tree.

6. Compare your results from part 5 and the results in part 4 and provide your findings.