This is a query posted in Risk Management Forum in my FB Account

I request all the FRM- 1 aspirants to understand the concepts in core readings. Conceptual understanding makes our life and exam also easy. I will discuss the discrete and continuous rates once I complete the quiz on forward and futures prices.

Please look into the question:

1 – Year zero coupon bond is trading at a price of $95.2381 and the YTM is 5%.

2 – Year Treasury note with 6% coupon is trading at a YTM of 5.5% and its price is $100.9232

3-Year Treasury note with 7% coupon is trading at a YTM of 6% (Price is $102.6730)

Question: Assume annual coupon payment and discrete compounding. Use a bootstrapping method to determine 2Year and 3 Year spot rates

Explanation:

According to John C Hull (Chapter 4), we use already available zero rates of zero coupon bonds to obtain the zero rates of coupon bonds.

Why the question is saying to use discrete? It is quite simple. First to obtain zero rates, generally we first obtain discrete rates and those discrete rates are converted to continuous rates. In this case, no need to convert to continuous rates, as they have asked us to calculate the zero rates using discrete method.

Calculating the zero rate for two years: Use the zero rate available for 1 year to calculate the zero rate of 2-year. The two year T-note is paying a coupon of 6% and its price is $100.9232 ð  (6/1.05) + 106/(1+r)2 = 100.9232 ð  5.7143 + 106/(1+r)2 = 100.9232 ð  [106/(1+r)2] = 95.2089 ð  95.2089 x (1+r)2 = 106 ð  (1+r)2 = 106/95.2089 ð  (1+r)2 = 1.11334 ð  (1+r) = 1.05515 ð  r = 5.515% ð  This is the zero rate of two years

Calculating the zero rate for three years period Use the zero rates available for one year and two years to obtain the zero rate for three year period. The coupon rate of T-note of 3years is 7% and its price is $102.6730; ð  ($7/1.05) + ($7/1.055152) + $107/(1+r)3 = $102.6730 ð  $6.67+$6.2874+$107/(1+r)3 = $102.6730 ð  107/(1+r)3 = 89.7156 ð  89.7156 x (1+r)3 = 107 ð  (1+r)3 = 1.192658 ð  (1+r) = (1.192658)1/3 ð  (1+r) = (1.192658)0.3333 ð  (1+r) = 1.0605 ð  r = 6.05%

Zero rates: 1 year-5%, 2 year- 5.515% and for three years-6.05%