Finance Fixed Income Assignment | Online Assignment

Its a Finance Fixed Income FinalFINC-3380 Take Home Final
Seungho Baek
Due: Tuesday,
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 le or
one excel 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.
Part1: Estimation and Forecast
You have been retained by the xed 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 le (Project1 2020.XLSX), which you can nd
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 nd this data exploration procedure?
2. 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
1
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.
3. Estimate the AR(1) process1 for interest rates.
rt+1 = + rt + “t+1 (1)
where “t+1 N(0; 2).
4. From now, let us the new data set. First, let ^, and ^ , 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 suce). (Tip: When you make the
forecasts, assume there are 252 (business)days in one year.)
5. 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 =
PBid
t + PAsk
t
2
zt = (
100
Pt
)
1
t 1)
ft = [
(1 + zt)t
(1 + zt1)t1 ]1= 1
where t is the time to maturity, Pt is the average price of the T-bill at time t, and 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.
6. 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 ndings.
7. 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?
1AR represents an auto-regressive linear model
2Note that is 0.25 for this problem.
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Part2: Simulations
The Baek-Brooklyn Investment Group is also interested in obtaining interest rates scenarios at
various maturities in the future. In this part of the exercise you must use Monte Carlo simulations
for your scenario analysis. The following is the rst step into the use of simulations, and so we
proceed slowly. Follow the steps below.
1. Use a random number generator to obtain plots analogous to Figures 1 and 2 as below.
2. Use the parameter estimates obtained in problem 3 of part 1, and simulate forward(starting
from rTODAY ) the interest rate, according to process (1), up to 5 years into the future.
Start with plotting only 10 paths of interest rates. That is, for each simulation run j, for
j = 1; : : : ; 10, you start from rrunj
0 = rTODAY . You then simulate a set of random normal
draws “run j
1 ; “run j
2 ; : : : ; “run j
T (from a normal N(0; 2) and compute
rrunj
1 = ^ + ^ rrun j
0 + “run j
1 (2)
rrunj
2 = ^ + ^ rrun j
1 + “run j
2
rrunj
3 = ^ + ^ rrun j
2 + “run j
3
…
rrunj
T = ^ + ^ rrun j
T1 + “run j
T
You do this 10 times, and plot the various paths.
3. Perform the exercise above again, but this time simulate the interest rate 100, or better yet,
500 times. Obtain a histogram of the interest rate at T = 6 month and T = 5 years.
Tips for Excel Users
1. There are two ways to obtain draws from a standard Normal distribution (i.e.mean = 0, std.
dev. =1):
a) You may use Yi = rand() to obtain draws from a uniform distribution (between 0 and 1)
and approximate a draw X from a standard normal distribution using
X =
X12
i=1
Yi 6
3
b) You may use Y =rand() to obtain a draw from a uniform distribution (between 0 and 1)
and then use the command NORMINV, that is
X = NORMINV (Y; 0; 1)
2. To obtain draws from a N(; ) you recall that if X N(0; 1), then
” = + X N(; )
3. To obtain draws
Xi =

1 with probability p
(n + 1)=2 with probability 1 p
you can still use Xi = rand() to generate a random realization between 0 and 1 and then use
the IF() function to obtain Xi: That is, set Xi =IF(Yi > 1 p, 1,1).
Part3: Bootstrapping and Smoothing Interest Curve
Elon Musk, the CFO of Tesla, has been considering to issue four dierent xed 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
oating rate note
FRN2: 7 year
oating rate note (capped
oater)
FRN3: 7 year
oating rate note(
oored
oater)
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 xed income desk,
is now managing this task. She now asks you to nd 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).
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 xed 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.
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2. Using the cubic spline model, which is given by
ct =
0 + 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
t 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 matu-
rities.
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 ndings. (Hint: plot three interest rates against
the following time interval: 0.5 year, 1 year, …, 29.5 year, 30 year.)
Part4: 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 xed income
securities. Let us use these rates to price them.However, you need to nd 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 re
ect 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
ows 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
oater.
A rate of
oor is 1.9% per annum for the 7 year
oored
oater.
Simply, all the obtained zero and forward rates are expressed in BEY(though those rates are actually
continuously compounded interest rates).
1. To re
ect this credit quality of the company to its bonds, you need to adjust interest rates by
adding credit spread. Go to FRED website and nd 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.
5https://fred.stlouisfed.org/series/BAMLC0A4CBBB
5
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 dierent callable bonds with the forward rates using the
backward induction method and compare them.
8. Calculate the prices for the three dierent
oating rate notes with the forward rates using
the backward induction method and compare them.
Part5: Arbitrage Bond Pricings with binomial interest rate tree
Although you computed the callable options bond and
oating rate bonds based on a simple induc-
tion method, the previous pricing method are not able to considered various callable/capped/
oored
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
oating rate notes. Because the developed Ho-Lee model can t 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
oating rate notes.
1. Select any ve interest paths in the binomial tree and plot them. Compare your gure and
coin-tossing gure in part 2. Does your graph look like the coin-tossing gure? If so, what
does that imply?
2. Adjust all the rates in the tree nodes prior to price these securities by adding the OAS spread
to each forward rate.
3. Compute the three call option bonds using the modied interest rate tree.
4. Calculate the respective eective durations and eective convexities for the three callable
bonds. Interpret all the values.6
5. Compute the values of the three
oating rate notes using the modied interest rate tree.
6. Compare your results from part 5 and the results in part 4 and provide your ndings.
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E. Duration =
PV PV+
2PV0(curve shift)
E. Convexity =
PV+ + PV 2PV0
PV0(curve shift2)
where PV is the price of the bond when the bench mark yield curve is shifted down by Curve, PV+ is the price
of the bond when the bench mark yield curve is shifted up by Curve, PV0 is the current price of the bond, and
Curve is the magnitude of the parallel shift in the bench mark yield curve (in decimal).
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