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Bootstrapping Solution

May 23, 2014

What is the Bootstrapping method?

Bootstrapping is a method for constructing a zero-coupon yield curve from the prices of a set of coupon-bearing products.As you may know Treasury bills offered by the government are not available for every time period hence the bootstrapping method is used mainly to fill in the missing figures in order to derive the yield curve. The bootstrapping method uses interpolation to determine the yields for Treasury zero-coupon securities with various maturities.

Let’s consider following example:

We have bonds/instruments with face value of $100 with maturity of 6months, 1 year, 1.5 years and 2 years. Let’s assume for simplicity that coupon for those individual bonds equal to YTM on those bonds implying that Bonds are trading at Par. i.e. $100

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Face Value100
YTM511.52
Bond Price100100100100
Coupon =YTM3.00%4.00%5.00%6.00%

As you can see for 6-month security we are going to receive only single bullet payment implying that its yield is equivalent to zero coupon bonds. Hence we can say 6 months Spot rate is 3%. Now if we consider 1 year security you can see that we are going to receive two cashflows, first one after 6 months (of $2) and another at the end of a year (of $102 = principal + coupon). Discounting this cashflow @ 4% will give us its market price of $100 however we know that first cashflow should be discounted at 3% as that is the spot rate for that tenor. So the question becomes at what rate we should discount next cashflow which we are going to get at the end of year 1 so that price comes to $100?

We can get this answer by solving simple equation:

100 = [2 / (1+3%/2)^1] + [102/(1+x/2)^2]

Solving for ‘X’ gives us value if 4.01%. This rate is effectively zero coupon rate for 1 year security and we will call it 1 year Spot rate. Now similarly we can continue this process for maturity of 1.5 year, 2 years & so on…

The formula, however to calculate next spot rate can be simplified as

spot rate calculation formula

You can also watch the youtube video for more theoretical explanation.

The attached excel file provides you with complete reconciliation and explanation on how to bootstrap.
Bootstrapping Solution

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