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# Chapt.5 Tuckman - Hedging PF with key rate exposure

#### FlorenceCC

##### Member
Hi @David Harper CFA FRM,

I have a couple of questions related to the study notes on the chapter mentioned in title.
1. Specifically, p97, I am trying to understand using the spreadsheet how we built the overall KR01s hedges to our initial portfolio positions. In particular some columns leave me puzzled:
• 10 year: why is the equation not 0.0870/100*F10 - 0.0345/100*F30-Z + 0.0010/100*F30-C= -$171? Why does your equation not take into consideration the 30yr-Z bond? • 30 year column: here again, 0.1219/100*F30-Z is not included -> what am I missing about this bond that makes it not useful or redundant? • I guess I could have the same question about the 2yr column but at the same time we already have a single original position in that 30y-Z so it would make complete sense that the hedging position would be in the only other security that shows a KR01 different from 0.(i.e. 2yr bond). 2. I understand that if a coupon bond has a maturity = key rate, and it is priced exactly at par, then effectively its yield is the par (key) rate. I am not sure why it means that it's KR01s with respect to other key rate would be zero though? Is it because key rates themselves are only affected by their 1bp change and not changes affecting other key rates? Many thanks in advance for your feedback. Florence #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @FlorenceCC You hopefully noticed that we are just following Tuckman's example in Chapter 5(I replicated in my XLS @ https://www.bionicturtle.com/topic/learning-spreadsheet-tuckman-chapter-5/, there is some slight rounding differences). We closely follow Tuckman's presentation, although I would love to someday re-format to improve for clarity because I think the tables are quite confusing. 1. This is simpler than you'd expect. The "baseline" is two set of trades where the trader neutralizes single-factor duration/DV0:$40 million short of 10y 3.5s of 5/20 hedged by $72 mm long of 5y 2.125s of 5/15; plus$100 mm short of 30y-Z 0s of 5/40 hedged by long $4.077 mm of 30y-C 4.375s of 5/40; i.e., the 30Y zeros are a trade in the underlying client-requested exposure ("the trader executed two large trades: 1. The trader shorted$100 million face amount of a 30-year STRIPS to a customer, buying about $47 million face of the 30-year bond to hedge the resulting interest rate risk. 2. The trader facilitated a customer 5s-10s curve trade by shorting$40 million face of the 10-year note and buying about \$72 million of the 5-year note."). None of this is key rate hedging; it is just DV01 hedging. Then the next step illustrates key hedging, where instead of the 30 y zero, the 30-year coupon bond is used as a hedging security. The goal is to neutralize each of the four key rates (KR01s) and only the four hedging securities are required. The 30-year zero-coupon to which you refer is one of the underlying client-initiated trades, it is not part of the hedging portfolio (that seeks to neutralize the non-zero KR01 exposures).
2. Yes, you are correct that "it because key rates themselves are only affected by their 1bp change and not changes affecting other key rates?" This is only in the case that the key rates have been defined as par yields (as opposed to zero rates or forward rates). This is notoriously difficult to grok. The premise is that we have hedging securities which are sensitive to key rate changes. If the key rates are spot/zero rates, then shocking the 2 years zero rate naturally impacts the price of a 5-year coupon-bearing bond. But if the key rates are par yields, then shocking the 2-year par yield but leaving the 5-year par yield unchanged has no effect on a 5-year bond that is priced at par (i.e., its coupon equals its yield to maturity). This is true by definition of the par yield. I have found that the most common stumbling block here is understanding the definition of the par yield. Below is a recent YouTube video on the par yield (which is think is my best explanation of it, so far). I hope that helps, thanks