## 3 -Ways to Bootstrap Spot Rates for the Treasury Yield Curve Note: I have added a fourth method of calculating spot rates for the Treasury yield curve. The workbook is the same. For a brief explanation of the forth method:

http://pistulka.com/Other/?p=2812

I have used what I am calling the “spreadsheet method” of bootstrapping spot rate from the U.S. Treasury yield curve in two pervious posts. The posts are U.S. Treasury Forward Rate Curves & Forward Rates Part 3: Spot Rates. I used the “spreadsheet method” because it seemed easier to understand the process of creating a Treasury spot curve. In this post I will offer three ways to get the same results:

2. User defined function
3. Excel formulas
All three methods will use the same Treasury Yields, from the six month bill out to the 30-year bond. In order to create a spot rate for each six month period (60 spot rates), I used linear interpolation between each given yield. I used the following Treasury yields: It is assumed, for the purpose of these calculations, that all bonds are par bonds. For example, the 3-year note at a yield of 1.19% is assumed to have a coupon of 1.19%, and pays interest of \$5.95 every six months on \$1,000. Also, although Treasury notes pay on an actual/actual basis, I am not using dates, so I am assuming each 6-month period has an equal number of days. Each calculation assumes \$100 in principal. Basically, starting at six months, each coupon payment is present valued using the corresponding spot rates that were previously calculated. In other words, we move to the right calculating  new spot rates, and use those spot rates to discount the next bond’s coupon payments.
The present valued coupons are summed, and subtracted from the original \$100 in principal. Take for example the 3-year note in the above image. There were five coupon payments that were present valued. That means that in present value terms, all five payments are as if they were received at the initial investment. They are therefore subtracted from the \$100 initial principal investment. The future payment is the last coupon, including the \$100 in principal:
.
Present value = \$100 – \$2.938273 = \$97.06173
Future Value = \$100 + \$.595 = \$100.595
Solve for Annual Return (FV/PV) = (\$100.595/97.06173)^(1/3)-1 = 1.19898%
Convert to Semiannual Basis =2*((1+1.19898%)^(1/2))-2 = 1.19541%
For more detailed information you can read Forward Rates Part 3: Spot Rates.
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User Defined Function:
The UDF is VBA and does the same calculations as the “Spreadsheet” method. The functions name is “spotrate”
=spotrate ((E\$5:F6)
The range the function is asking for is two columns, the years and yield columns down to the 30-year bond (red box). Note that the row 5 remains an absolute. The function can be entered at the 1-years sport rate and copied down. There is a big advantage to using the UDF, in that it does not need the shorter term spot rates to be calculated first, in order to calculate a specific spot rate. The UDF calculates those sport rates each time it is run. Function spotrate(yearsandrates As Range)
Dim myrange(100)
last = yearsandrates.Count / 2
myrange(1) = yearsandrates(1, 2)
If last = 1 Then Spot = myrange(1): spotrate = Spot: Exit Function
For i = 2 To last
c = 0
For j = 1 To i – 1
c = c + (yearsandrates(i, 2) / 2 * (1 / (1 + (myrange(j) / 200)) ^ (yearsandrates(j, 1) * 2)))
Next j
myrange(i) = (((100 + yearsandrates(i, 2) / 2) / (100 – c)) ^ (1 / (yearsandrates(j, 1) * 2)) – 1) * 200
Spot = myrange(i)
Next i
spotrate = Spot
End Function
.
Excel Formula
The formula does exactly what the spreadsheet method does, but without the additional columns used in the spreadsheet method. The three year spot rate formula looks like this:
_
=IF(\$E10=0.5,F10,(((100+\$F10/2)/(100-SUMPRODUCT((1/(1+\$G\$5:\$G9/200))^(\$E\$5:\$E9*2)*(\$F10/2))))^(1/(\$E10*2))-1)*200)
_
The formula should be pasted in the first cell in the “Spot Rates” column and copied down. Note that the purple part of the formula needs a 6-month bill to start. The 6-month (.5 year) will not need to be calculated, because it is already a spot rate, in that it does not require any reinvestment of interest payments. The green portion of the formula solves for the present value of all the preceding coupon payments. The brown portion of the formula solves for the rate that equates the difference between the original investment less the present value of each coupon, and the last principle and coupon payment.
_
Summary
All three methods give the same results. The formula method takes 60 columns to calculate what the formula does in one column. The UDF requires VBA, but can calculate the spot rate for one term, without the need for the previous spot rates to be calculated first.
. Don Pistulka

Retired Credit Union CFO - Finance
Background: over 40 years in investments, asset/Liability management, banking, securities trader.
Worked for: California Credit Union, WesCorp, CalFed S&L, Crocker Bank, Carroll McEntee, Federal Home Loan Bank Board (D.C.), Western Asset Management, Security Pacific National Bank.

1. Parva

Hi Don,

Thanks very much for the sharing. Can you please provide the derivation of the formula used.

Regards,
Parva

1. Don Pistulka

Hi Parva,
I’m not sure what you are asking for. Each of the four methods I show in the workbook use the same methodology, but come at the problem four different ways. In the Excel workbook, I show the math for calculating the spot rate for the three-year notes on both the “Spreadsheet” method and the “Fourth Method”. Maybe you can tell me what you mean by the derivation.

Don

2. Giovanni Politi

Hello Don, thank you for sharing.

How can I calculate the 2month spot rate having 1 month and 3 month spot rates? I am a bit confused about the right procedure in excel.

1. Don Pistulka

Giovanni,
A spot rate is a zero coupon rate. In other words, if the security pays all interest at maturity (no coupon payments) it is already a spot rate. Spot rates are typically calculated using the U.S. Treasury market. The 1 and 3 month Treasury securities are Treasury bills, that have only one payment, at maturity. Therefore, there is no need to calculate a spot rate.

Don

3. Flora

Hello, thank you very much! Is there any way to do it in C#? Thanks.

1. Flora,
I don’t know C#.
Don

4. Li

Hi Don,

Thanks very much for the sharing. I am wondering what is the difference between linearly interpolated discount factors after bootstrapping and discount factors from linearly interpolated and bootstrapped interest rate? Thanks for sharing your opinion.

Best,
Li

1. Don Pistulka

Li,
The reason I need to interpolate the Treasury curve when bootstrapping spot rates is that I don’t have access to live Treasury yields for each 6-month period out to the 30-year bond. I only have access to the current coupons (less than a dozen yields for the 60 spot rates I am calculating).The more live Treasury yields you have out the curve that mature either on or close to the six month increment that I used, the better your accuracy will be. My examples for bootstrapping are more about explaining the process, than being the most accurate. If I had access to a provider like Bloomberg for live Treasury security yields, I could build a more accurate spot curve.

I hope that helps.
Don

5. Donghyun Lee

1. Don Pistulka
1. Don Pistulka