Introduction to Bond Math: Bond Pricing
A bond is worth $80 and has a 10% coupon maturing in one year. What’s the YTM? What if it matures in two years, not one?
Before beginning to answer any question on bond math, you always want to make sure you have all the information you need.
For example, you havent been told here whether or not the coupon is paid semi-annually or annually. This makes a difference.
Note: The YTM will be lower if its paid twice a year.
For nearly all bond math questions – because they dont want you breaking out a calculator – theyll be annual calculations. However, you should have an intuition for how yields work nevertheless.
So, if we have a coupon being paid annually then we can use the simple, generalized formula C + (P-FV) / P where C is coupon and FV is face value.
Note: We werent told in the question what the face value was. However, you can assume its always $100 if the price is x < $100.
So we will have $10 of coupon payments, a spread between P and FV of $20 on maturity, and a current price of $80. Therefore, we have 30/80 = 37.5%, which is your answer (as can be verified with a YTM calculator).
Now what happens if the bond matures in two years, not one? Well were getting one more coupon payment ($10), which is good, but were now delaying getting the spread between FV and P for two years. So YTM is going to invariably be lower. Remember that YTM hinges on reinvesting proceeds at the same rate so cash flow today is always heavily preferred.
When dealing with maturities beyond just one year, we need to use the more formal estimated YTM formula. This wont get us the exact YTM – as that would involve a much more complicated set up – instead itll get us a reasonably close approximation. Its important to point that out in an interview.
The estimated YTM formula, for when you have maturities equal or greater to two years, is simply the following where n is the number of years until maturity:
Using this formula well get down to (10+10)/90, which is 22.22%. You can verify that this is the correct estimated YTM with the YTM calculator linked above (which provides both the exact YTM and the traditional estimated YTM as well).
Enterprise value (EV) is $200 and we have a TL of $100, Senior Secured Notes of $50, and Unsecured Notes of $100 and another tranche of Unsecured Notes (maturing two years after the first) of $ What are the recovery values throughout?
Here we have a slight spin on a typical waterfall question. First, we know that the TL and the Senior Secured Notes are going to be fully covered leaving $50 behind.
Now we have two groups of Notes (bonds) within the same class.
As an aside, which may be worth keeping in mind, in a Chapter 11 youll have a Plan of Reorganization (POR) that will need to be submitted by the debtor. One of the requirements of the POR is that it treats all claims within the same class equally, unless otherwise consented to by one of the claims in the class.
In other words, if youre in the same class you have to actively agree to take less than your proportional share (which, as you can imagine, is rare to find!).
So in this question we have two separate Notes (differing in maturity and size, but not in their seniority) and they must be treated equally.
So this class of claims should be thought of as being $150 ($100+$50) and the amount they can lay claim to is $50. So the recovery rate for both is 33.33%.
Its also important to have the right verbiage down in an interview. These Unsecured Notes represent an impaired class that in the event of a Chapter 11 would be the ones that would vote on a POR. So the recovery rate of this impaired class is 33.33%.
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Back of envelope approach:
$dP simeq frac{partial P}{partial y} times Delta y$
You know that when $y=3%$, $P=100$. So you can write
$P-100 simeq frac{partial P}{partial y} times (c-y)$
Price $simeq$ 100 + Duration x (3%-9%).
Guess a duration of around 7.0 for a 10 year bond (they would assume that you would have a feel for this number).
So I get 100 – 7 x 6 = 100 – 42 = $58.
If I do this carefully assuming annual compounding then I get $61.5 which is in the same ball park. You can refine this using a second order correction but this would be an acceptable first guess that you can do without calculators.
It might be more impressive to demonstrate that you have the tools and can use them. Go to the interview with a handheld calculator. The answer is a few keystrokes away.
consider your bond initially was at par (cpn=3%~=yld_0) and now answer the question what is the price change given new yld_1=9%. for a very dirty estimate use relationship between price change vs yield change and duration (~=10).for a less dirty estimate youll need some educated guess on the level of convexity. have a look at closed formula of convexity of par bond. hope this helps.
Use Taylor Expansion to approx price changes for some variations in Yield. Guess the Duration to be less than Full maturity since its Paying coupons and go from there. First price it at Par.
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Let’s say we have $100 in debt with 15% in PIK. How does this flow through the three statements? Let’s assume a 20% tax rate.
PIK accounting questions are very common in RX interviews because so many of the out-of-court restructurings done will involve some PIK.
Note: Why is this the case? The obvious answer is because the company likely doesnt have much cash on hand (negative FCF, limited liquidity) so PIK allows them to avoid imminent cash crunches. The less obvious answer, perhaps, is that PIK allows the company to offer a much higher interest rate (in this case 15%), which current holders who may be exchanging bonds into will find enticing.
So lets go through it. On the income statement (IS) you will have $15 in new interest expense in the form of newly issued debt. This creates a tax shield (another reason why you can have higher interest rate) of $3 ($15*20%). Therefore, net income is down by $12.
Moving to the top of the cash flow statement (CFS) you have $12, you then add back the $15 as its a non-cash expense (thats the primary reason to do PIK!), so you have cash flow from operations up by $3.
On the balance sheet (BS) you have assets (cash) up by $3, on the liabilities side you have debt up by $15, and within shareholders equity (retained earnings) you have a $12 decrease from net income. So both sides of the equation are up by $3.
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