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# key rate exposure

#### ajsa

##### New Member
Hi David,

Is key rate exposure the dollar key rate duration? How is it measured? linear regression?

Also could you explain the advantages/disadvantages of key rate exposure technique in mult-factor hedging? why is the face value than the current market value of a hedging bond used?

Thanks.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi asja,

where I simply followed Tuckman's motivating example of a 30-year mortgage:

(per the video tutorial), each key rate is "shocked" with a +1 bps shift (and the neighboring rates), so if you consider 5-year shift, we are taking a flat 5% yield curve and shocking up all the rates from 2 years to 10 years (linear interpolate from +0 to +1 bps from 2 to 5 years, and interpolate from +1 bps to +0 from 5 to 10 years). So, this is as if the yield curve were a string and we pulled up the 5 year rate slightly....

Then the $10.65 key rate 01 is just like a DV01, it is the dollar change for the 2 year key rate shift. But while DV01 assumes a parallel shift of the entire yield curve, the KR01 gives the dollar change for just the shock to the segment (in this case, from 2 to 10 years). And the key rate duration = KR01 / Initial bond price * 10,000 Note this is equivalent to: mod duration = DV01/Price * 10,000 so in key rate duration, we have an analog to modified duration; i.e., what is the % change in the bond price given a one unit shift in the key rate Also could you explain the advantages/disadvantages of key rate exposure technique in mult-factor hedging? why is the face value than the current market value of a hedging bond used? The key advantage is: it gives the chance to be more realistic than single-factor duration, and implicity then, allows some consideration for non-parallel shifts. From Sanjay Nawalkha (emph mine): "key rate durations can manage interest rate risk exposure arising from arbitrary nonparallel shifts in the term structure of interest rates. ... The key rate duration model describes the shifts in the term structure as a discrete vector representing the changes in the key zero-coupon rates of various maturities. Interest rate changes at other maturities are derived from these values via linear interpolation. Key rate durations are then defined as the sensitivity of the portfolio value to the given key rates at different points along the term structure. These duration measures can be used in decomposing portfolio returns, identifying interest rate risk exposure, designing active trading strategies, or implementing passive portfolio strategies such as portfolio immunization and index replication. Similar to the duration vector models, an appealing feature of the key rate model is that it does not require a stationary covariance structure of interest rate changes (unless performing a value at risk or VaR analysis). Hence, it doesn’t matter whether the correlations between changes in the interest rates of different maturities increase or decrease or even whether these changes are positively or negatively correlated. Also, the model allows for any number of key rates, and therefore, interest rate risk can be modeled and hedged to a high degree of accuracy" why is the face value than the current market value of a hedging bond used? because Tuckman quotes DV01 per 100 face value David Last edited: #### ajsa ##### New Member Hi David, Thanks for the explanations! 1. so is KR01 the "key rate exposure"? 2. what are the disadvantages of key rate exposure technique in mult-factor hedging? 3. "And the key rate duration = KR01 / Initial bond price * 10,000 Note this is equivalent to: mod duration = DV01/Price * 10,000" The KR01 uses face value and DV01 uses market value? Thanks again! #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi asja (sorry for delay) 1. KR01 is the key rate 01 (1 basis point) the key rate equivalent of DV01; I think "key rate exposure" is just generic for "the bond's exposure to a shift/shock in the key rate" something we measure with the KR01 2. About the disadvantages of a multi-factor disadvantages, I do not see where Tuckmans lists them (??). But Sanjay Nawalkha (again, my favorite) has some good review of this: 2a. Hedge implies more instruments and more transaction costs: Though using more risk measures will lead to better immunization performance, doing so will require higher transaction costs and may even require explicit or implicit short positions. Hence, the number of risk measures should be carefully selected after running many yield curve scenarios with transactions cost analysis. 2b. Selections may become arbitrary: "However, unlike the duration vector models, which require at most three to five duration measures, the number of key rate durations to be used and the corresponding choice of key rates remain quite arbitrary under the key rate model. For example, Ho (1992) proposes as many as 11 key rate durations to effectively hedge against interest rate risk. 2c. Sill uses linear approximations: "Further, unlike the duration vector model, where the higher order duration measures serve as linear as well as nonlinear risk measures, the key rate durations give only the linear exposures to the key rates. To measure nonlinear exposures to the key rates, key rate convexity measures are required. Hedging against a large number of key rate durations and convexities, implies large long and short positions in the portfolio, which can make this approach somewhat expensive in terms of the transaction costs associated with portfolio construction and rebalancing." 3. The KR01 uses face value and DV01 uses market value? No, i don't think i said "face value" anywhere ... Tuckman's example uses market value (price), just like we use estimated market price for regular duration, and my XL example uses market price (or, model price as estimate of market price, if you like) thanks, David #### Matt John ##### New Member Fixed income attribution refers to the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time. For example, the risks affecting the return of a bond portfolio include the overall level of the yield curve, the slope of the yield curve, and the credit spreads of the bonds in the portfolio. A portfolio manager may hold firm views on the ways in which these factors will change in the near future, so in three separate risk decisions he positions the assets in the portfolio to take advantage of the expected forthcoming market movements. If all views subsequently prove to be correct, then each decision will generate a profit. If one view is wrong, it will generate a loss, but the effect of the other bets may compensate. The overall performance will then be the sum of the performance contributions from each source of risk. Attribution is therefore an extremely useful tool in verifying a fund manager’s claims to possessing particular investment skills. If a fund is marketed as being interest-rate neutral while providing consistent returns from superior credit research, then an attribution report will confirm this claim. Conversely, if the attribution report shows that this same manager is making non-zero returns from interest rate movements, then his exposure to interest rate risk is clearly not zero and his investment process clearly differs from his stated position. Fixed income attribution therefore provides a much deeper level of information than is available from a simple portfolio performance report. Typically, such a report only shows returns at an aggregated level, and provides no feedback as to where the investor’s true skills lie. For these reasons, fixed income attribution is rapidly growing in importance in the investment industry. In itil practice exams, key rate exposure is also discussed; Thanks #### ajsa ##### New Member Hi David, 3. In the 2008 Note2 p139, "The second step is to solve a simultaneous equation. In this case, there are four unknowns: F2, F5, F10, and F30. These are the face values of the corresponding bonds that solve the equations". Then I wonder why face value than market value should be used to "Compute the appropriate hedging positions for given a specific key rate exposure profile" Thanks. #### jwanderson54 ##### New Member In the equations (ex: .035/100 X F5 +.015/100 X F10 =14.74) What are F5 and F10 and how do I find those? I need mathematics broken down like you would to a 5th grader to comprehend them. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber @jwanderson54 I don't know where you are specifically referencing with those numbers sorry (cc @Nicole Seaman ) , but this is an instance of seeking to identify the hedging portfolio (of bonds) to an underlying exposure (bond or portfolio of bonds) that has a key rate exposure (at some maturity, let's call it X) of$14.74; for example, if X = 5 years, then your formula is solving for the face values of hedging bonds (a 5-year and a 10-year bond) which hedge an underlying bond (exposure) that drops in value by $14.74 if there a one basis point shock to the 5-year key rate. The hedging portfolio seeks to neutralize key rate exposures, so this formula has at least two hedging securities: a 5-year bond with an X-year (e.g., 5-year) key rate of 0.035 (per$100 face value) and a 10-year bond with X-year (e.g., 5-year) key rate of 0.015 (per $100 face value). The formula wants the face amounts of each of these hedging securities, that together give a total key rate exposure of$14.70, which will hedge the underlying bond. Symbolically, this is just an single equation in two variables: 0.00035*F(5) + 0.00015*F(10) = 14.74. It can't be solved without an additional equation. Probably the 10-year key rate exposure lets us solve for F(10). F(10) is the face amount of the bond in the hedging portfolio which gives us the sought-after KR01s in the hedging portfolio.

For example, say the solution gets you F(10) = $30,000. This means that the hedging portfolio includes$30,000 of face value in a 10-year bond, and with respect to neutralizing the X-year key rate above, it contributes 0.015/100*F(10) = 0.015/100*30000 = \$4.50 toward neutralizing a shock to the X-year key rate., while the 5-year hedging bond contributes the other 14.74 - 4.50 = 10.24.

It's very difficult without an example, and ideally, to be honest, this is best understood in a spreadsheet (IMO). Thanks,

#### jwanderson54

##### New Member
Thanks I was over thinking it