Credit Default Swaps: hedging credit risk and valuing CDS (Excel) transcript

hello everyone and welcome again to nettle the best platform around for distance learning in business finance economics and much much more my name is Sarah and today we are investigating one really peculiar and slightly complicated type of derivatives credit default swaps they have originated in the 1990s and very soon they've become one of the most common derivative types and many analysts associate the Great Recession of 2007-2009 with the prevalence of credit default swaps and how they were used to hide credit risks of various companies so they're really catchy derivative instruments so it's really important to understand what they're all about how they are valued and how is the payoff of a credit default swap determined so in essence what a credit default swap is well it's again a contract that mostly occurs over-the-counter between a production buyer and a protection seller regarding the debt performance of third party basically there is a company somewhere out there that took out a loan or issued a bond so it has some debt and there are credit risks associated with the debt performance of this company and some other country parties presumably who hold the debt of this company or the speculator so arbitrage owners might want to enter derivatives contract to transform their credit risks they exposed to by holding the debt of the company or profit from obtaining the premium from taking extra risk so imagine there is a company that holds the debt of a relatively risky corporation and they want to hedge this risk so they want to buy protection so one of the ways to facilitate that would be for this company to enter a credit default swap agreement so what would they do well they would agree with another counterparty for a swap transaction every certain period of time if the company whose that performance is the underlying does not default or no credit events a care regarding this company's that performance and again here it's very important to stress that defaults and credit events are not synonyms defaults are narrower than credit events defaults are pretty much the worst squared event that could occur but there can be an instance where credit default spread is triggered if there is a credit event that is not seen in the most to default so you have to always look at the contract specifications to determine what the swap is all about but for simplicity let's now assume that we are only concerned with defaults as the the worst case scenario is pretty much that could happen to anyone holding someone else's debt right so if the company does not default then the protection buyer so the company who basically wanted to hatch out hedge away its exposure base the protection seller fixed premium that's determined by the terms of the contract if on the other hand the underlying company does default then the protection seller is obliged to repay the protection buyer the reasonable sum that they would have lost as their debt has basically been tremendously devalued so here in this case we can already figure out what are the most important characteristics that determine the variation and the performance of credit default swap we are obviously concerned with the probability of default how risky is the debt of the company we are concerned with in that case let's assume that the annual probability of default is 1.73 percent so every single year we roll the dice basically that's how we model risk events in finance all the time and if the probability one point 73% triggers then the company defaults and that is assumed to be the exactly the same probability every single yet in more advanced valuation models sometimes it's assumed that the probability of default is dependent on the year some companies might be experienced in liquidity crisis right now so they would be more likely to default right now or there might be some expectations of future economic growth or company specific performance that would change that's assumption but that's mostly a reasonable assumption that we can still stick with in this simple example so every single year our underlying company whose debt performance is the underline for our credit default swap has a one point seventy three percent probability of defaulting then we need to specify the recovery rate the recovery rate basically corresponds to the proportion of the par value of the debt that can be recovered by the debt holder if the company defaults most of the time the recovery rate is associated with collateral so for example if we loan or the bond that the company takes or issues respectively is collateralized then well the bank or the any other debt holder can just claim the collateral and recover part of the value that they've lost so why is it important for credit default swaps well the recovery rate is assessed by board of practitioners so the association of derivatives trading and they estimate what is the recovery rate if the company defaults and in case of that the protection seller so the protection seller so someone who takes all the risk instead of the protection buyer for the premium they would have to compensate for this loss so they'd have to provide the par value adjusted for the recovery rate so the higher is the recovery rate the less the production seller would have to provide in case of default and then as pretty much everywhere in finance we need to account for the cost of Finance the cost of capital for the company that enters into the swap because we obviously need to discount the cash flows by the cost of capital as that's the most relevant discount rate for obvious corporate finance reasons so we assume that the recovery rate for the debt of our underlying companies 25 percent and the cost of Finance for the counterparty that enters into the swap is six point eleven percent well then we need to also account for the fact how long is our credit default swap we can have long term credit default swaps which would be obviously more valuable to the production bar we could have shorter term credit default swaps but that would just protect them against short-term risks that would be less valuable in our case we are considering a six year credit default swap so without further ado let's estimate what the payoffs of both counterparties gonna be in case of the swap and how to assess using only these variables the fair value of the credit default swap premium of the swap rate of the credit default swap that would keep the expected payoff of both counterparties at zero so that would basically prevail at the market at equilibrium regarding the probability of default at year one it's obviously just equal to the annual probability of default the probability of survival is obviously by definition just one minus the respective probability of default but later down the line he has two three five five yes two three four five and six we need to adjust for the fact that here the company might have already defaulted in year one so everything else already is around to adjust for that we need to multiply this probability of survival by the our natural annual of default because we would need to calculate the total unconditional probability of default of this yeah one point seventy three percent would be the probability of default in year two given the company has survived me year one but as the company doesn't always survive at year one we need to adjust for this fact so just multiply the two probabilities and obviously here we assume that those events are relatively independent so if the company survived last year its probability of default next year doesn't change so here we can see that the total probability is decreasing because well the probability of survival is less than one then how we might adjust these figures the probability of survival in two years is less than the probability of survival in one year obviously because more things can happen in two years than canon one year so we just need to subtract the probability of default in here - from the probability of survival at the end of year one and given this formulas we can just drag the formulas down until the end of the lifetime of our credit default swap and see that at the very end of our credit default swap period the probability of company's survival is slightly higher than 90% and it decreases year by year which is something that we would expect and the probability of survival is key to our calculations because the probability of survival reflects how likely it is that the protection seller would receive the premium from the protection buyer if everything goes smoothly and the company doesn't default then the protection buyer is obliged to pay the protection seller the premium at the end of each year so we need to discount it using the relevant discount rate so the cost of Finance of the counterparties entering into the swap agreement and then we can figure out this discounted cash flows so the discount factor would just be one divided by one plus the annual cost of finance and we need to raise it to the power power of the number of the yeah as our discount factor slowly diminishes with time to reflect time value of money so we can enforce this formula all the way down and see how our discount factor indeed slowly diminishes then to discount those cash flows we just need to multiply the probability of survival times the discount factor and those would be the sums that the protection buyer pays the protection seller that would be proportional to the premium obviously we still don't know what the colibri premium is going to be but given that this premium is fixed and certain the contractual terms we can state that it would be proportional to these numbers times the swap rate that's going to be measured in basis points but we'll return to that a little bit later so here next as we assume that the defaults occur media we need to account for the fact that if defaults occur media then the protection buyer still needs to pay the protection seller the swap rate for the period of time within this year when the company hasn't yet defaulted basically to do that we just need to consider what is the probability of this event and adjust for the fact that we are paying as the production buyer the production seller not the premium for the whole year but just for half of this period as of the media the company has already defaulted and we are not obliged to pay after that so our probability of default year by year is that and as defaults occur mid year we just need to divide that by tail so those would be again this sums that protection seller receives proportional to the credit default swap premium to the swap rate that we still don't know as we want to figure out what the fair value of this premium would be to basically equilibrating expected payoffs of both counterparties to 0 then discount factors here are a little bit complicated as we are dealing with swap rates that are being paid and received at the middle of the year we would need to discount them by media discount factors to account for that we need to start as usual 1/1 plus and you'll cost of finance that should be locked and then we need to raise it to the power of the number of the year minus 0.5 because here we want to discount by a semi-annual discount factor here we want to discount by a discount factor that represents one half years two and a half years and so on and so forth as we want to bring the cash flows from meadia periods to the present day having applied that we can figure out those discount factors that would be slightly greater than those as here we are concerned with time periods that a half year closer to the present day always so all of these discount factors are smaller than these and now we can figure out the discounted cash flows of those partial credit default swaps premia that the protection buyer has to pay even though the company would be defaulting at the mid-year so multiplying this cash flow by this discount factor would get this discounted cash flow and we can already figure out what is the expected premium that the a protection seller would receive so first of all we can sum up those discounted cash flows if all goes smoothly and the underlying company does not default and those tiny premia that we need to adjust for given that we assume that defaults occur media and the expected received premium neglecting the swap rate as for now would be this part of the payoff receivable by the production seller plus that now we need to figure out what is going to be the amount payable by the protection seller in case there is a default or more broadly a credit event of any case so what should be payable by the production seller if default occurs well first of all we need to account for the likelihood of this event so probability of default at each of those years and then we need to remember that the protection seller is not obliged to compensate the whole par value of the loan or a bond or whatever is the debt that the company has defaulted on they need to only compensate for the non recovered part of the par value so we need to multiply that by one minus the recovery rate and that is where the recovery rate comes into the equation if the protection buyer can recover large proportion of the debt from collateral that the company has pledged to guarantee their loans or bonds or whatever then the amount that they would have to be compensated for is lower and the protection seller is less unhappy about having to do sell so we have to account for this logic and we can enforce this formula all the way through and then we also need to discount that using the relevant discount factor again we should remember that the defaults acquire media so the discount factor should be exactly the same as over here so we can just basically copy them across and then we can multiply those to figure out the present value of some payable by the production seller and those sums are not dependent on the swap rate the credit default swap premium because they reflect the spected proportion of the par value that the production seller would have to pay back in case the underlined company defaults then we can bottom allocate all the way around and sum that all up to get the expected payoff for the protection prior something that they would receive from the protection seller in case the company whose that performance is the underlying default so here we just can copy that across and we see that the expected payable by the production seller is zero point zero six to nine so it's six point twenty nine percent of the par value of debt and that is the premium that they receive unaccounted for the swap rate so to figure out the swap rate naturally what we have to assume is that the expected inflow for the production seller should be equal to the expected outflow of the production seller so both counterparties are indifferent in terms of whether to enter into this credit default swap in present value terms they still could benefit from hedging but if the present values are positive for one side of the transaction then they are negative for the other side of the transaction and that's the logic that persists throughout the whole topic of swaps swaps generally are zero-sum game on paper so they can be mutually beneficial in terms of present value at least when the fair value for both counterparties is zero and that's the logic that's constantly used to value not only credit default swaps but other types of swaps for example interest rate swaps and if you would like to know a little bit more about these please leave comments below and I'll record a video about that in the near future but without further ado without digress in any further how can we calculate the premium of the series in terms of the swap rate well what we can do is we can divide the expected payable in terms of the fraction of the notional amount of the that we can divide that by this and because this times the swap rate should equal that if we do it the other way around dividing we would get the equilibrium swap rate that would equilibrates the positive and negative payoffs the influence and they are close and that would bring the expected payoffs of both counterparties to exactly zero and as we need to express it in basis points that's the well-known convention for credit default swap rates we just need to multiply it by 10,000 and when we will if we force the formula we see that the room swap rate for that credit default swap is however 35 basis points so one point 35 percent per annum one point 35 percent per annum of the par value of the loan of bond or any other debt the production buyer has to pay to the protection seller for them to be interested in taking on that credit risk from the protection buyer and just to verify that we can see that if we multiply the received premium by the equilibrium swap rate divided by 10,000 and subtract the payable that is expected to be an outflow of the production seller we get exactly zero which verifies our claim that both parties do not gain or lose on average in terms of present value from this credit default swap but that's not the main way credit default swaps are used because how would you know that how on earth would you figure out what is the probability of default of a particular company well you could obviously use some credit risk techniques some sophisticated models for example augment that score or you could estimate a locket or a probate in terms of known past default events but those estimations would necessarily be backward-looking because they use past data and euro metric estimation would come out with some logic in terms of what factors influence likely defaults of companies credit default swaps can be used to estimate what is the market implied forward-looking probability of default for a wide range of companies that have CTS traded to do that well what we know do we obviously know costs of finance for pretty much all companies they report that we can roughly estimate recovery rates because that is what various professional bodies do and we obviously know the duration for which there is credit default swaps contracted but we don't know that in the real life but in the real life this swap rate is not big blue bream swap rate but that's the market swap rate that's written in the contract so if we know the equilibrium swap rate and we know those characteristics we could arrive at the implied probability of default that the market expects from the dead performance of a particular company so to do that what we needed to do we need to assume that we see a certain market formed swap rate CES premium and using this very logic of the calculations we can then assess what is the probability of default that justifies such a premium so for example let's assume that the swap rate would be not hundred and thirty five basis points that justifies the probability of default of one point seventy three percent per annum but rather something like two hundred fifty two basis points so slightly higher here we can see that given this swap rate the protection seller would not benefit from entering a swap so it would mean that the probability of default is actually higher than that and what does the market believe this value is to do that we would need to use the solver function in Excel that basically tries to match values in certain cells so that the values of certain functions satisfy predetermined conditions for example we could use solver to figure out which probability of default would justify an equilibrium swap rate of two thousand fifty two basis points of two point fifty two percent per annum so to do that we first need to enable solver in our excellence so we can click file options add-ins and manage Excel add-ins click go and here we will see which atoms are currently enabled in your Excel we can here see that in our case the solver add-in is not being enabled yet so we can just check here and click OK sometimes you will see that the solve is already enabled in your Excel so you don't have to worry about it at all but in our case we just click here we just tick solver and press ok and now solver as an Excel atom tool would appear under data over here so we just need to click solver and certain interface would appear so we would need to set the objective well what is the objective that we want to achieve well we want to achieve the objective of the expected payoff of both to parties was the protection southern production buyer to be equal to zero so we want to set this cell to be equal to zero what do we want to change well we want to change the probability of default we want to figure out what is the probability of default that justifies that premium and also you could put constraints on your optimization problem but here we have a very easy task so we don't need to be concerned with any other functions of this wonderful solver edit we just can click solve and see that given the equilibrium operate of 250 to the annual default rate that would justify such a high premium would be 3 point 21 percent per annum and here you can be slightly worried that this number is not exactly zero but that's just the Excel way of representing very small numbers here it basically says it's roughly minus 3 times 10 to the power of minus 8 so for all purposes that would be equivalent to 0 roughly so what we can do now is we can do some comparative statics using credit default swaps and reason how do these factors contribute to the equilibrium swap rate equilibrium c.d.s premium so we just need to return to the initial conditions so type 1.73 here and here we need to type this divided by that times 10,000 so we are back to square one what we want to figure out here is how do these characteristics affect equilibrium swap rate so our credit default swaps more or less valuable if the company is more likely to default well intuitively it would seem that the come that are more risky would require higher credit default swap premium to buy protection against their default let's see if it's true let's increase this probability of default to some very high value like 10% per annum what we would see is that the equilibrium swap rate would skyrocket to 812 latest point so eight point 12% brand which is a very high value and very small amount of companies have such high annual probabilities of default what happens if the probability of default drops to a very low value like 0.1 percent per annum then the credit defaults for premium the swap rate would consequently drop to only eight basis points of 0.08 percent per annum which is very negligible it means that the more sound the company is the less credit risk it manifests the less would be the premium that the protection seller would require from the production buyer so enter into credit default swap well back to square one one point 73% what would happen if the recovery rate changes well even theoretically even the recovery rate increases it would mean that the debt of this company even given the fact that the probability of default doesn't change is safer so the swap would be less valuable alternatively you can think about it in terms of the protection seller if the recover rate is higher then in case of default the protection star would have to compensate less of a fraction of the par value of debt so they would be happy to enter into a default swap even if they are compensated with a lower premium so let's figure out if that's true if the recovery rate increases to 50% the equilibrium swap swap rate drops to just 90 basis points so our intuition was correct but if the recovery rate is lower for example it's only 5% now then the equilibrium swap rate would increase to 171 basis points it means that there is a very predictable monogamous relationship between both probability of default and recovery rate in terms of the equilibrium swap rate the city has premium that would justify such a condition back to square one twenty-five percent hundred thirty-five BPS equilibrium swap rate now this would be harder to simulate but we already can kind of assess it and intuitively grasp it theoretically if the credit default swap is of higher duration high maturity then it would mean that there are more events that gonna happen and that are likely to happen within a longer time frame so buying protection against the company defaulting in a longer time period would be more valuable so the equilibrium swap rate would be higher if on the other hand it is a shorter term credit default swap you could protect yourself against credit risk with a lower premium that you have to pay to your production seller and that's it regarding credit default swaps hope I managed to explain this kind of sophisticated topic in credit derivatives in a relatively understandable manner leave a like under this video if you found it helpful in the comments below I would be glad to see any of your feedback or suggestions for future videos as for now stay tuned thank you very much