Practice Test Exam Solution. Bootstrapping forward curve.
Here is abbreviated partial solved solution to the practice exam question posed earlier. The practice exam question was used in Derivatives Pricing course taught to MBA students earlier in August 2012. The solution is presented in two parts. The first part of the solution focuses on the bootstrapping forward curve and zero curve part of the exam. The second part of the solution walks through the actual IRS pricing exercise. This post covers the first partial solution.
The practice exam question had three parts and hence the solution to the practice test question will also be presented in three parts. Before you proceed further please take a quick look at our free interest rate swaps pricing guide. We will only present the basic outputs from the solved solution sheet. You will need to review the interest rate swaps pricing study note to actually build the sheet yourself on your laptop.
Practice Test Exam Question Solution – A review of solution approach
Practice Test Part I dealt with building and plotting Zero and Forward curve using the data given and the bootstrapping approach applied on the given interest rate yield curve given below. The original yield curve was a 5 step curve that gave 5 annualized rates for 5 years.
Figure 1 Practice Exam Question – Given Interest Rate Yield Curve
Practice Test Part II dealt with using the given zero and forward curves to price an Interest Rate Swap (IRS) and calculate the swap rate. The trick was extending the original curve to the new 10 step, semi annual yield curve implied by the 10 step notional amortization schedule given below.
Figure 2 Practice Test Question – The notional amortization schedule
Practice Test Part III dealt with marking to market (MTMing) the interest rate swap for an updated curve a few months later. Once again the trick is to extend the 5 step curve to a 10 step semi-annual curve which can then be used for pricing the interest rate swap.
Figure 3 Practice Test Question – the revised interest rate yield curve
Practice Test Exam Question Solution – Pricing Interest Rate Swap – Part I – Building the Forward Curve
The first step is to break the individual bonds down to their annual cash flows and treat each cash flow as an individual zero coupon bond as explained in the interest rate swap pricing study guide. The output from the first forward curve building step is shown below
Figure 4 Practice Test Question – Solution – Breaking the bonds down to individual cash flows
The second step is to use the individual zero coupon cash flows to boot strap the present value of the principal cash flow at maturity for each of the par bonds as explained in the interest rate swap pricing study guide. The output from the boot strapping step is shown below.
Figure 5 Practice Test Question – Bootstrapping discount factors
The third step is to apply the zero curve and forward curve formula to calculate the relevant zero and forward rates from the table above. As explained in the interest rate swap pricing study note. The output from the zero and forward rate calculation step is shown below.
The final and forth step is to plot the Par, Forward and Zero rates in the required graphical format using MS Excel built in graphic tools.
Figure 6 Practice Test Question – Plot of Par, Zero and Forward curves
Most students make it to this stage without any incident. Some however apply the wrong formula and end up with inconsistent results. If you are not happy with the results just apply a simple calculation cross check. If you calculate the relevant accumulation factors for each given year using your calculated Par, Zero and Forward curves they should all result in the same values as shown below.
The Par Curve Accumulation factor was already calculated above. For the Zero curve the accumulation factor is (1 + zero rate) raised to (compounding period). The accumulation factor for the forward curve is a similarly recursive calculation. Calculate the accumulation factor using the forward curve for the first year. Then use that in your calculation for calculating the accumulation factor using the forward curve for the second year.
If you see a value in the error term row, you have made a mistake in the calculations above. Review and check them.
Figure 7 Practice Test Question – Calculation Cross Check
We have now successfully completed the first part of the test question. We now move on to part II.
Practice Test Exam Question Solution – Pricing Interest Rate Swap – Part II – Extending the Forward Curve
The second part of the question requires us to use the interest rate yield curve above to price a 10 step, semiannual interest rate swap. And this is where confusion reigned supreme.
Step One – Interpolated semiannual yields
We first need to break the original annual yields to maturity down into interpolated semiannual yields as shown below
Figure 8 Practice Test Question – Interpolated semiannual yields.
Then using the same sequence of steps above we break the new implied semiannual bonds down to their zero coupon components, boot strap the curve, calculate the rates and plot the graphs.
Step Two – Zero Coupon Bonds
Figure 9 Practice Test Question – Zero Coupon Bonds by tenor
Step Three – Bootstrapped Discounted Cash Flows for Zeros
Figure 10 Practice Test Question – Bootstrapped Discounted Cash Flows
Step Four – Periodic and Annualized Par, Zero and Forward Rates
Figure 11 Practice Test Question – Model Output – Par, Zero and Forward Rates
Common Mistakes in Part II of the Test Exam
The most common mistakes that killed points for students in the second part of the practice test and exam questions were:
a) Not calculating the interpolated semiannual yields and using the original 5 annualized yield to price the 10 step interest rate swap
b) As a result of (a) above building a 5 x 5 or a 5 x 10 grid versus a 10 x 10 grid as shows in step two and step three above
c) Using annual rates rather than period rates in the 5 x 10 and 10 x 10 grid. Periodic rates are annualized rates divided by 2 since the coupon on the underlying loan is paid semi annually
d) Using periodic rates but then not annualizing them before using them in the IRS pricing later on in Part III of the practice test exam.
The correct solution and required graphical presentation if you didn’t suffer from mistake (a), (b), (c) and (d) above would look like this:
We continue our coverage of this exam question in our next post.
Pricing Interest Rate Swaps – The actual Final Exam question in full
Here is the original post that presented the question earlier today.
I teach the Derivative Pricing and Risk Management courses to EMBA and MBA students in Dubai and Singapore. In a recent exam one question caused a lot of heart ache and pain in exam takers. While most student understood the gist of the question, they still made a number of small mistakes that cost them valuable points in the final score.
If you have an interest rate swap pricing exam or test coming up, here is the solved solution that you can use as a practice exam or practice test question. For best results, first try the exam and the practice and test question and then work through the solution. Best of luck.
Pricing Interest Rate Swaps – Practice Exam – Test prep question
You are given the following term structure of interest rates for US$ using Yield to Maturity (YTM) of Par bonds that pay interest on a semi-annual basis
Figure 12 Test Prep – Practice Exam – The Interest Rates Yield Curve
Using these Par Bond Yields please answer the following questions:
7. Plot the following graph shown in Figure 1 using the above term structure and assuming that that coupon/interest is paid on a Semi Annual basis. This implies that you would need to build a 10 x 10 grid at semi-annual intervals for 5 years. Your graphical plot will show projected rates at 10 points at 10 half years. (15 Points)
Figure 13 Practice Exam – Sample Par, Zero & Forward curve plot
Please see the next page for question number 8
8. A client has recently entered into a 4.5 year floating rate loan for US$ 400,000,000? The loan will be effective six months from now and will use the following repayment schedule. (35 points)
Figure 14 Practice Exam Question – Notional Principal for pricing Interest Rate Swaps
The client has asked for a quote for the an effective interest rate risk hedge that would offset the risk of rising interest rates.
What would be the swap rate at cost or breakeven basis for this structure?
b) Would the client be paying fixed or receiving fixed
c) What would be the swap rate if the loan starts at time 2.5 with 10,000,000 and ends at time 3 with an outstanding principal of 10 million.
d) Six month later the interest rates term structure has changed as shown below. Taking the original Swap structure and assuming that the Swap was purchased at the original breakeven Swap rate, what is the MTM (Mark to Market value of the Swap).
Figure 15 Practice Exam question – Revised Interest Rate Yield Curve