## YTM

Finance Junkie
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### YTM

Pls explain:

A risk analyst seeks to find out the credit linked yield spread on a BB rated 2 yr zero coupon bond issued by a petroleum company. If the prevailing annual risk free rate is 3%, the default rate for BB-rated bonds is 7% per yr, and the loss given default is 60%. then the yield to maturity of the bond is:
a. 2.57%
b. 5.90%
c. 7.45%
d. 7.52%

Can you pls explain the working as this qn is seeming totally new to me.

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Finance Junkie
Posts: 99
Joined: Sat Apr 07, 2012 10:24 am

### Re: YTM

Pristine team,

will you reply on this??
to see no revert sinec last 2 days

content.pristine
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### Re: YTM

Hi Suresh, your query has come on Saturday night. We try to reply to the queries within two business days. Please let me know what is the exact doubt in the question. We should be able to answer it by tomorrow.

content.pristine
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Joined: Wed Apr 11, 2012 11:26 am

### Re: YTM

Suresh,

I've just looked this up. This question looks like its from the book "Financial Institutions Management" by Anthony Saunders. This reading is no longer in your FRM syllabus. Don't worry, you wont find a question like this in your exam this year

Finance Junkie
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### Re: YTM

Thanks for that Jyothi:)

content.pristine
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### Re: YTM

One of our instructor, Kiran Prasad, had something to add to the discussion:

"There are two ways to approach the question above. The hard way of remembering a (or more than one!) formula and application of it. Or the more sophisticated way of combining some of the fundamental concepts along with a logic to arrive at the answer. I recommend the second one as it is not only intuitive, but more often that not, this critical analysis is the one that proves to be the difference in the final exam!

Here is the approach:

We need to ask ourselves the following questions to solve this problem:

Does a BB rated bond(of any coupon rate and any maturity!) yield more than the Risk free rate of return of comparable maturity?
- Yes (this rules out the option 1 above)

Why does an investor of BB rated bond demand more yield?
- Because there is an element of riskiness (in crude terms) associated with the bond. To talk in more sophisticated terms, besides sharing all other risks, a BB rated fixed income instrument possesses an additional risk called Default Risk. Hence the investor demands a premium for absorbing that risk.

Now, here is the most important step:

Having understood that there is an element of default risk, the next question is "how does it matter to me and where is it captured?"
- The default risk/ chance of not generating the perceived cash flow is captured in the higher yield to maturity.

Concept:

Consider a Zero coupon risk free fixed income security of a Face Value of FV, same as the BB rated compnay in question. Maturity be 2 years. Its YTM is same as Rf, the risk free rate of return. Suppose it trades in the market at PV1.

PV1 = FV / (1+Rf)^2

Consider a BB rated 2 yr zero coupon bond issued by the petroleum company in question of comparable maturity. Lets assume its YTM and PV as Rc and PV2.

PV2 = FV / (1+Rc)^2

From these two equations above, we can say that,

PV1 *(1+Rf)^2 = FV = PV2*(1+Rc)^2

or

PV1 *(1+Rf)^2 = PV2*(1+Rc)^2 ---> 1

Now the trick lies in establishing a relationship between PV1 and PV2 here.

PV2 is nothing but PV1 but with the default risk embedded in it.

PV2 = PV 1 * A factor capturing the element of Riskiness

PV2 = PV 1 *{ Case 1 + Case 2}

Case 1: Is the scenario where there is no default in the two year maturity and the investor is able to retain full perceived/ promised face value)

Case 2: Is the scenario where the company defaults in the period of two years and the investor recovers only the LGD % of the face value.

We add these two because they are mutually exclusive events.

Case 1: PV1 * ( Probability of not defaulting in the two years * Recovery Rate in that scenario) = PV1 * (PD of not defaulting in two years * 100%) = PV1 * (PD of not defaulting in two years)

Case 2: PV1 * (Probability of defaulting in the two years (a.k.a. Cumulative Prob of default) * Corresponding Recovery Rate) = PV 1* ( PD of default in two years * 40%)

Cumulative Probability of default = 1- (Probability of not defaulting in year 1 * Probability of not defaulting in year 2) = 1- {(1- 0.07)*(1-0.07)} = .1351

Probability of not defaulting in two years = 0.8649

Equation 1 becomes

PV1 * (1+ Rf)^2 = PV1*{ (0.8649*1 + 0.1351*0.4)} * (1+ Rc)^2

or

(1+Rf)^2 = 0.9189 * (1+Rc)^2

(1+0.03)^2 = 0.9189*(1+Rc)^2

Solving this we get Rc as 0.07449 or 7.449% approximating to 7.45% being the closest option."

jaspreet.frm
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### Re: YTM

Solution: The generalized relationship between risk free rate and risky corporate bond yield given its probability of default and recovery rate is
(1+y)^T=(1+Y*)^T{(1-p)^T + f[1-(1-p)^T]}
Where
y=risk free rate
y*= risky corporate bond yield
T=Time Intervals
p=Probability of default
f=Recovery Rate

In the given question
y=3%
y*= To be found
T=2
p=7%
f=1-Loss given default=1-60%=40%
Putting the values in the equation
(1+0.03)^2= (1+y*)^2 {(1-0.07)^2+0.40[1-(1-0.07)^2]}
1.0609=(1+y*)^2{(0.93)^2+0.40[1-(0.93)^2]}
1.0609=(1+y*)^2{(0.8649)+0.40[1-(0.8649)]}
1.0609=(1+y*)^2{(0.8649)+0.40[0.1351]}
1.0609=(1+y*)^2{0.9189}
(1+y*)^2=1.0609/0.9189
(1+y*)^2=1.1545
(1+y*)=√ 1.1545=1.0745
y*=0.745=7.45%