# Fixed income mapping

##### Member
Hi David,

Assume a flat yield curve with spot rate of 4.0% at all maturities and normally distributed yield volatility of 1.0%. We are mapping a two-bond portfolio. Both bonds have a $100 million face value and pay an ANNUAL 4% coupon. One bond has a one year maturity; the other has a five year maturity. If we use PRINCIPAL MAPPING what is the portfolio's 95% value at risk (VaR)? a.$2.7 million
b. $4.9 million c.$7.5 million
d. $9.5 million Your answer was: D- The 3-year zero-coupon bond (the primitive), under annual compounding, has modified duration = 3/(1+4%) = 2.885. The Returns (%) VaR = 1% yield volatility * 1.645 deviate * 2.885 mod duration = 4.75% Risk (i.e., Returns VaR) The 95% VaR = 4.75%*$200 MM = $9.49 million. I did not understand the formula of the modified duration. I can see that we need to get the average maturity, however, no clue as to the equation 3/(1+4%). Can you please explain. Thanks Imad #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Imad, The formula, which is really important, is given by: modified duration = Macaulay duration/(1+yield/k), where k is the number of (compound frequency) periods per year. If the yield is 4% and k = 1 ("pay an annual coupon"), then because the Mac duration of a zero coupon bond equals its maturity (very common assumption on exams b/c otherwise the Mac duration is tedious to calculate), the mod duration = 3.0 years Macaulay / (1+ 4% yield / k = 1 period per year under annual discrete compounding). ... Mac duration is the weighted average maturity of the bond, but we generally need modified duration for sensitivities (as Modified duration is a function of the first partial derivative, dP/dY) and hedging, and mod duration is slightly less than Mac duration under discrete compound frequencies. Thanks, #### Imad ##### Member Thanks David, I am aware of the formula but I can understand that when we "principal map", we assume a zero coupon bond with the average of maturity, right? Also, can you please elaborate about the following formula: The Returns (%) VaR = 1% yield volatility * 1.645 deviate * 2.885 mod duration. Imad #### choonho ##### Member Subscriber David, For Principal Mapping, Jorion used a simple average of maturity, not modified duration. Why does your answer calculate duration for Principal Mapping? #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @choonho Please see the source at https://forum.bionicturtle.com/threads/l2-t5-63-fixed-income-mapping.3617 Principal mapping treats the portfolio's average maturity as the risk factor. In the mapping, the two-bond portfolio is "replaced" by a zero-coupon bond with maturity equal to the average maturity of the portfolio. So, in both my question above and in Jorion's (Chapter 11) example, the principal mapping is to a zero-coupon bond with maturity of three years (i.e., the average of one and five). Although it's very difficult to trace back to his Chapter 8, his 1.484% returns VaR = 0.5234% Yield VaR * 2.835 modified duration, where 2.835 = 3 year Mac duration/(1+5.89% yield), see Table 8-4. In other words, they are the same approach: • Jorion: 0.5234% yield VaR * 3.0 [Mac duration /(1 + 5.81% yield)] *$200 MM = $2.97 mm • My Question: [1% yield volatility * 1.645 deviate] * 2.885 mod duration *$200 MM = \$9.49 million, where yield vol * deviate = yield VaR.
So it is a mapping approach of mapping the portfolio to a zero-coupon bond with a 3.0 year maturity, but we still estimate the risk by asking "what is the modified duration of such a [primitive] bond?" I hope that helps. BTW, please note that my 63.2 is still incorrect in the source because it calculates an incorrect modified duration for the underlying bonds, and therefore for the portfolio. Whereas principal mapping maps a zero-coupon bond to the portfolio's average maturity (three years, in this case), duration mapping maps the zero-coupon bond to the portfolio's duration, which given these bonds pay coupons, must be somewhat less than 3.0 years. Thanks!

#### emilioalzamora1

##### Well-Known Member
Has this something to do with adjoining vertices? The portfolio duration is 2.885, so we we first need to compute the 2-year and 3-year maturtiy VAR because the modified duration of 2.885 lies between 2 and 3? This could be total nonsense, but it's my first intuition from Jorion's Appendix 'Assigning Weights to Vertices'.

#### bpdulog

##### Active Member
Has this something to do with adjoining vertices? The portfolio duration is 2.885, so we we first need to compute the 2-year and 3-year maturtiy VAR because the modified duration of 2.885 lies between 2 and 3? This could be total nonsense, but it's my first intuition from Jorion's Appendix 'Assigning Weights to Vertices'.

They are using an interpolated value

#### flex

##### Member
need to detail: is it avg of maturity weighted by security(instrument)'s weight (share) in the portfolio?

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#### Nikita Gusev

##### New Member
Hi, David

Could you please explain what kind of zero coupon bond we take for mapping? I mean Is it OK to replace corporate portfolio risk with government zero? There is no information of quality type of zero bond for mapping in any source.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Nikita Gusev I think that's a great question, by which I mean to suggest that I don't necessarily perceive there is a single correct answer. But I think the theory (in the FRM at least) is that mapping is essentially the exercise of simplifying an intractably complex reality by defining a portfolio (or position etc) in terms of its sensitivity to a limited number of primitive risk factors, or put another way, by expressing its value as a simple(r) function of some small set of primatives.

I was just earlier answering a question about option delta/bond duration, as they are both first partial derivatives. I would like to remind that when we estimate an option's price change by multiplying a worst expected stock price change by delta, we are "mapping" the option value (as the exposure) to the stock price change (as the underlying factor): ~Δc = ∂c/∂S * ΔS. But, if somebody else wants to be more accurate by adding a sensitivity to volatility, with ~Δc = ∂c/∂S * ΔS + ∂c/∂σ * Δσ, it's not like one of is right or wrong!

So, philosophically, knowing that we cannot (and do not want to) map to all possible risk factors, we would like to map to a limited number of primitive risk factors that, in the ideal, are somewhat "reusable" (ie., common to our various exposure and positions). Further, and in that vein, we probably want to parse out visibly different risks, in this case: we probably want to parse out interest rate risk (market risk) from credit risk (as measured by credit spread). So for me personally, the best answer to your question is risk factors that capture only market risk, and therefore my vote would go to what Fabozzi calls "default-free theoretical spot rates;" aka, risk-free zero rates, typically captured in practice with US Treasury zeros. In general, you do want to map to zeros. And then credit spreads would be separate risk factors. (And i do recall there is some earlier GARP question where the answer wants you to parse the credit spreads from the risk-free interest rate shifts).

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#### Nikita Gusev

##### New Member
Thank you much, David!

#### Collen

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

What I understand is that, for duration mapping, the portfolio is replaced by a zero-coupon bond with maturity = duration of the portfolio.

My question is; how did you obtain a modified duration of 2.885 years for the portfolio consisting of 2 COUPON bonds in question 63.2?

Thanks
Collen

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @collen I apologize but it's a mistake (that I didn't fix) and you (your implication) is correct. Per the source (https://forum.bionicturtle.com/threads/l2-t5-63-fixed-income-mapping-jorion.3617/), if they are both 4% par bonds (c = y = 4%) with maturities of 1.0 and 5.0 years, then I get Macaulay durations of {4.6299, 1.0000} and modified durations of {4.4518, 0.9615} such that 50%/50% portfolio Mac duration is 2.815 years and portfolio modified duration is 2.707 years. Sorry for the confusion! thanks, David

#### Collen

##### New Member
HI @collen I apologize but it's a mistake (that I didn't fix) and you (your implication) is correct. Per the source (https://forum.bionicturtle.com/threads/l2-t5-63-fixed-income-mapping-jorion.3617/), if they are both 4% par bonds (c = y = 4%) with maturities of 1.0 and 5.0 years, then I get Macaulay durations of {4.6299, 1.0000} and modified durations of {4.4518, 0.9615} such that 50%/50% portfolio Mac duration is 2.815 years and portfolio modified duration is 2.707 years. Sorry for the confusion! thanks, David

Thanks for the clarification @David Harper CFA FRM

Cheers
Collen