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P2.T6.306. Credit spreads and spread '01 (DVCS; Malz section 7.1)

Fran

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AIMs: Define the different ways of representing spreads. Compare and differentiate between the different spread conventions and compute one spread given others when possible. Define and compute the Spread ‘01.

Questions:

306.1. The following curves are applicable to a risky 2-year bond that pays 6.0% semi-annual coupon:



The bond's market price of $103.73 can be computed by discounting its cash flows continuously at 4.0% per annum, which is represented by the flat yellow line. Specifically: $3.00*exp(-4%*0.5) + $3.00*exp(-4%*1.0) + $3.00*exp(-4%*1.5) + $103.00*exp(-4%*2.0) = $103.73. The bond's same market price of $104.73 can also be derived by discounting the same cash flows according to the continuous discount rates given by the the steep blue line. The lower steep line, which shows a rate of 0.40% at six months, is actually two nearby curves: a swap rate curve and nearby spot rate curve. Both start at 0.40% but, as the spot rate curve is slightly steeper, by year 2.0, the spot rate is 1.61% while the swap rate is 1.60%. For this purpose, we assume both the spot and are risk-free curves; e.g., US Treasury.

Each of the following is true about this bond EXCEPT which is false?

a. The bond's yield-to-maturity is 4.0%
b. The yield spread, represented by the solid red vertical arrow, is the difference between 4.0% (yellow line) and 0.40% (spot rate at six months)
c. If the price of the bond decreased due solely to perceived credit risk of bond (without any change in market risk), the upper curves (yellow and blue) would shift up
d. The z-spread, represented by the dashed red vertical arrow, is the difference between the (upper steep) blue line and the (lower steep) spot rate; e.g., 2.42% = 4.03% - 1.61%


306.2. The risk-free spot rate curve is (unrealistically) steep and given by the following: 1.0% at 0.5 years, 2.0% at 1.0 year, and 3.0% at 1.5 years, with continuously compounded rates (this question being sourced in Malz). (Source: Allan Malz, Financial Risk Management: Models, History, and Institutions (Hoboken, NJ: John Wiley & Sons, 2011))



A 1.5 year bond that pays a 10.0% semi-annual coupon has a price of $105.62 such that its z-spread happens be a round 3.00%. Specifically, $105.62 = $5.00*exp[-(1.0%+3.0%)*0.5] + $5.00*exp[-(2.0%+3.0%)*1.0] + $105.00*exp[-(3.0%+3.0%)*1.5].

Which is nearest to the bond's Spread '01 (aka, DVCS) per $1,000,000 of par value?

a. $0.14
b. $36.09
c. $151.16
d. $2,836.24

306.3. In regard to Malz's credit spreads, each of the following is accurate EXCEPT which is false?

a. A risky bond has a different z-spread at each coupon date; e.g., a semi-annual coupon bond with five years to maturity has 10 different z-spreads
b. The z-spread is the spread that must be added to the risk-free spot rate curve in order to arrive at the market price of the bond
c. The i- (or interpolated) spread is the difference between the bond's risky yield (YTM) and the linearly interpolated yield between the two swap rates with maturities flanking that of the risky bond
d. The spread '01 (DVCS) is generally a decreasing function of the z-spread level

(Source: Allan Malz, Financial Risk Management: Models, History, and Institutions (Hoboken, NJ: John Wiley & Sons, 2011))

Answers:
 
#2
The answers are (if I am right?):
306.1 c
306.2 c
306.3 a

By the way could you be so kind to explain why the spread '01 (DVCS) is generally a decreasing function of the z-spread level?
Is it due to convexity? The larger the z-spread - the lower the price of the bond compared to the initial z-spread level and the less it is flexible in response to changes for 0,00005 bp in both directions?
Thank you for you kind feedback!
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#4
Hi @[email protected]
  • I have a different answer for 306.1 (mine do match yours for .2 and .3). In regard to (C): "without any change in market risk" is meant to assume the riskfree rate curves do not change (in basic models we proxy market risk by changes to the riskfree rate; and credit risk manifests as spread changes). Then, I think, if the bond price decreases (presumably due to credit risk), the spread must increase. So I have (c) as true ..
  • In regard to "why is the spread '01 (DVCS) is generally a decreasing function of the z-spread level?," here is Malz:
Malz 7.1: "At very low or high spread levels, however, as seen in Figure 7.2, the spread01 can fall well above or below$400. The intuition is that, as the spread increases and the bond price decreases, the discount factor applied to cash flows that are further in the future declines. The spread-price relationship exhibits convexity; any increase or decrease in spread has a smaller impact on the bond’s value when spreads are higher and discount factor is lower. The extent to which the impact of a spread changeis attenuated by the high level of the spread depends primarily on the bond maturity and the level and shape of the swap or risk-free curve."
 
#5
Great questions, especially .1 and .2. Thanks for posting.

However, there are typos in the descriptions of both questions:
In .1, $103.00*exp(-4%*0.5) should read as $103.00*exp(-4%*2.0)
In .2, $5.00*exp[-(2.0%+3.0%)*1.5] should instead be $5.00*exp[-(2.0%+3.0%)*1.0]
 

Pflik

Active Member
#7
mistake is still in the pdf. Also i'm noticing that the calculation doesn't seem to be correct either. (i.e. i'm getting 150.31) just a small difference.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#8
Hi @Pflik thanks, we missed the revision (this thread is not the source).
Source is here @ https://www.bionicturtle.com/forum/...ads-and-spread-01-dvcs-malz-section-7-1.6921/

Re: Also i'm noticing that the calculation doesn't seem to be correct either. (i.e. i'm getting 150.31) just a small difference.
I don't see how you get 150.31. Depending on the shock method, the exact answer still looks to me to be $151.151, with a range from $151.140 to $151.162, here is my solution @ https://www.dropbox.com/s/hw854ix1j36hr18/T6.306.2_spread01.xlsx
... see rows 29 to 31, with row 30 replicating Malz exact approach to shock the spread up and down by 0.5 bps
 

Pflik

Active Member
#9
seems to be a rounding error in excel (or actually lack of rounding)

the value after shock is 105.635031133000 minus 105.62 (as per given in the question) will give you 0.015031133. However in the excel sheet the 105.620 was calculated not set as a constant (and thus there was more precision after the decimal sign i.e. 105,619914943). Hence the different answer. Minor difference only, but enough to make me wonder if the calculation was correct :)
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#10
@Pflik oh, great point. Agreed, I can get $150.31 by shocking down 1 bps and subtracting rounded $105.62; it occurs to me this is avoided by shock up + 0.5 and down - 0.5, as the 105.62 isn't used in that approach. I think it's a good illustration of why the question needs to ask for "nearest to ..." (and further, I'm glad my answers have big gaps). Thanks!
 
#11
I am struggling with a very basic concept related to 306.1:

The bond's YTM is 4%. Ok sounds good, but if a check it with my calculator:

PV = -100 FV= 103,75 PMT = 3 N =4 this is: YTM = 3,88 which is different from 4

where is my big mistake? thanks
 

QuantMan2318

Active Member
Subscriber
#12
I am struggling with a very basic concept related to 306.1:

The bond's YTM is 4%. Ok sounds good, but if a check it with my calculator:

PV = -100 FV= 103,75 PMT = 3 N =4 this is: YTM = 3,88 which is different from 4

where is my big mistake? thanks
Hi @Carlos Madrid

The Market value of the Bond is higher than the Face value which suggests that the Coupon (6%) should be greater than the YTM. Here the FV should be 100 as you are redeeming it at par. And the PV should be -103.75. So, PV = -103.75; FV = 100; PMT = 3; N = 4
which gives the YTM as 4% if you use semi annual compounding on your calculator


P.S And if you use semi annual compounding in your calculator, you should directly get the YTM which is 7.76% in your example which is a way of verifying your calculations

Hope this helps

Thanks
 
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