# Introduction to Credit Default Swaps

The Credit Default Swap became a household name in 2008 when it became clear that its widespread use contributed to the financial crisis. While the CDS market has shrunk since its heyday and looks very different today, CDS contracts continue to be used by banks and hedge funds as a way of managing risk and speculating on debt.

What is a Credit Default Swap

Credit Default Swap contracts are simply insurance policies on bond investments. If I own a bond issued by Company A, and am concerned that Company A might go bankrupt and not be able to pay back the full principal amount, I can protect my investment against the potential default entering into a CDS contract.  I pay the insurance provider an annual fee in exchange for a promise to receive the full principal amount of my bond investment if Company A defaults.

There are few key attributes that specify a CDS contract:

Reference Entity: The bond issuer on which the insurance policy is written.
Notional Amount: The notional amount of bond that is being protected by the contract.
Tenor: The time length of the contract (e.g. 5 years, or as specified by a given maturity date).
Coupon Rate: The annual insurance premium as a fraction of the notional amount, usually expressed in basis points.

Here’s an example of a investor that purchased a bond issued by Netflix, and entered into a CDS contract to protect that investment.

Here we say that the Bond Investor has purchased protection on Netflix, and the Counterparty has sold protection.

Historically, the coupon rate on a CDS contract was set according to how risky the bonds on the Reference Entity were. For risky companies, the coupon rate could be as high as 1000 bps and for safer companies as low as 50 bps. However, to simplify the CDS market, ISDA (the International Swaps and Derivatives Association) standardized CDS contracts to only pay either 100bps or 500bps in coupon. Older contracts that were written before this standardization are called legacy CDS contracts.

The granularity once offered by the coupon rate is now available in an upfront payment made to either party. For example, if Netflix were a very risky company, protection sellers might require an additional upfront payment of 10% of notional to compensate for a 500bps coupon rate that they deem is too low. Conversely, if Netflix were a very safe company, protection buyers might demand an upfront payment to compensate for a coupon rate they think is too high.

In the event of default, the protection seller doesn’t actually exchange the full notional for the defaulted bond. Rather, the value of cheapest defaulted bond is decided by an auction process and deemed to be the Recovery Rate. The protection seller then pays the protection buyer $Notional * (1 - Recovery Rate)$ and does not receive the defaulted bond in exchange.

Here are the cashflows of the Netflix CDS described above where Netflix defaults a little after 3.5 years with a recovery rate of 40%.

Notice that a small upfront payment is made to the protection buyer. This is because the 500bps coupon is too high for Netflix’s level of risk. On default, the protection seller pays $\10\,000\,000 \times (1-40\%)=\ 6\,000\,000$ but also receives a tiny bit of Accrued Coupon, the portion of the next coupon payment that’s owed since Netflix defaulted in the middle of two coupon payment dates.

The CDS Market

You can find market data on some CDS contracts at the Wall Street Journal Market Data Center here. Notice that CDS market prices are generally quoted as either Price or a Spread.

The price of a CDS, (sometimes called the bond price) is equal to $1-Upfront$ where the convention is that a positive upfront indicates a payment to the protection seller, and a negative upfront indicates a payment to the protection buyer. For example, a CDS is trading at a price of 94 (points), then the upfront payment is 6pts made to the protection seller. This is a convenient way of thinking about CDS market price since it mimics the price of a hypothetical bond that the protection seller has purchased from the protection buyer.

The spread, or more accurately par spread of a CDS contract is the theoretical fair coupon rate that would require no upfront payment. Smaller par spreads indicate a safer reference entity, while larger par spreads indicate a riskier reference entity. Before the standardization of CDS contracts, the coupon on a CDS would be equal to the par spread at the time it was written. Par spreads are a convenient way of quoting CDS since you don’t need to know the coupon rate on the contract, to understand how the market is pricing the riskiness of the reference entity. Let’s take a look at the market spreads on some sovereign CDS.

Remember that these are the par spreads of 5 year CDS contracts written on the debt of each country. We can see from this that the market considers the debt of France, New Zealand and Canada very safe compared to Brazil, Serbia, and Italy.

While CDS contracts were originally invented as insurance products for bond investments, banks and hedge funds started using them to speculate on debt by entering into a CDS without actually owning a bond issued by the reference entity. CDS can be an attractive way to bet on a company because CDS trades don’t require a lot of cash to put on.

Suppose Netflix has a 5% bond outstanding that will mature in 5 years and currently trades at 90. Furthermore, suppose that 5-year Netflix 500bp CDS trades at 91. As a speculator, if you wanted to put on a bet that Netflix will be a safer company in the future, you could either buy the bond, or buy the CDS (sell protection). If you bought the bond, you would have to put down $9,000,000 today on a$10,000,000 notional investment, but if you bought the CDS you would receive $900,000 today on the same$10,000,000 notional investment.

# Modigliani-Miller & Unlevered Betas

In 1958 Franco Modigliani and Merton Miller wrote a nobel-prize winning paper describing how leverage effects the value of companies. In this post we’ll discuss some of their results and apply it to the derivation of a well-known formula used to de-lever equity betas.

Suppose there exist two companies that own exactly the same assets and produce exactly the same stream of uncertain earnings. Let one company be completely unlevered, i.e. completely financed with equity, and the other company be partially financed by debt. Denote the annual earnings before interest and tax (or EBIT) of both companies by the random variables  $X_1, X_2, X_3, \ldots$ that are governed by some probability distributions. We assume that the value of each firm is only a function of its stream of after-tax EBIT (or EBIAT). However, since interest payments are tax-deductible, the value of each firm will depend on its leverage.

First suppose that the company is unlevered, i.e. has no debt and is fully financed with equity. In this case, on year $i$, the company produces after-tax EBIT of $X_i(1-\tau)$ where $\tau$ is the corporate tax rate for this company. The value of the company can be written as

$\displaystyle V_U = \mathbb{E}\left[\frac{X_1(1-\tau)}{(1+r_U)} + \frac{X_2(1-\tau)}{(1+r_U)^2} + \cdots \right]$

where $r_U$ is some discount factor appropriate for the risk associated with the earnings stream.

Note that we are discounting earnings and not cash. We assume that earnings convert into cash quickly and that management acts in the best interest of the shareholders when reinvesting earnings instead of paying cash dividends to equity holders.

Now consider the case where the company is partially financed with debt. Let $D$ be the market value of the debt and $r_d$ is the interest paid on debt or the cost of debt. On year $i$, this company will produce earnings after tax and before interest of $\displaystyle (X_i-r_dD)(1-\tau)+r_dD=X_i(1-\tau)+r_d\tau D$

Notice that we can separate these earnings into earnings identical to the unlevered firm $X_i(1-\tau)$, and fixed earnings $r_d\tau D$ from the tax benefit of debt financing. In fact, we can replicate these earnings exactly by holding a portfolio of the unlevered company and $\tau D$ amount of the debt of the company. Therefore, the value of the levered company must be exactly equal to the value of the portfolio containing the unlevered company and $\tau D$ debt.

$V_L = V_U+\tau D$

### Unlevered Betas

Now we will use the relationship derived above to show how the beta of a company’s equity to the market changes with its leverage. Let $E$ represent the value of the equity of a levered company (share price or market cap), and let $M$ be the level of a market index. By CAPM, we know that

$\displaystyle \frac{\Delta E}{E}=\beta_L \frac{\Delta M}{M} + \epsilon$

Where $\epsilon$ is a a random with $\epsilon \sim \mathcal{N}(0,\sigma^2)$. Here, $\beta_L$ is said to be the levered beta of the equity. Now suppose that the company had no leverage and denote the hypothetical value of the unlevered firm as $V_U$. Then we can write

$\displaystyle \frac{\Delta V_U}{V_U}=\beta_U \frac{\Delta M}{M} + \epsilon$

where $\beta_U$ is the unlevered beta of the company. Solving for $\frac{\Delta M}{M}$ in each and setting those equal to each other, we get

$\displaystyle \frac{1}{\beta_L}\left(\frac{\Delta E}{E} -\epsilon\right)=\frac{1}{\beta_U}\left(\frac{\Delta V_U}{V_U}-\epsilon\right)$

From above, we know that $V_L =D+E=V_U+\tau D$ so we can replace $V_U$ with $E+(1-\tau)D$. If we assume $D$ is constant and does not fluctuate with the market, this also implies that $\Delta E = \Delta V_U$.

$\displaystyle \frac{1}{\beta_L}\left(\frac{\Delta E}{E} -\epsilon\right)=\frac{1}{\beta_U}\left(\frac{\Delta E}{E+(1-\tau)D}-\epsilon\right)$

Taking expectations

$\displaystyle \frac{1}{\beta_L}\left(\frac{\mathbb{E}[\Delta E]}{E} \right)=\frac{1}{\beta_U}\left(\frac{\mathbb{E}[\Delta E]}{E+(1-\tau)D}\right)$

$\displaystyle \beta_U=\frac{E}{E+(1-\tau)D}\beta_L=\frac{\beta_L}{1+(1-\tau)\frac{D}{E}}$

# Bond Yield Explained

Suppose you are considering investing in one one of two bonds.

 Bond A Bond B Coupon Rate 5% 6% Maturity (years) 10 7 Price 92 97

Both bonds pay coupons semi-annually starting six months from today, and ending in 10 and 7 years. If neither bond defaults, which will be the more profitable investment? Since these two bonds have different coupon rates, maturities and trade at different prices, the answer to this question is not immediately clear. To answer this question, it is important to look at the yield of the bond.

The Yield to Maturity (YTM) of a bond is the discount rate at which the present value of the bond’s cashflows is equal to its market price. It is also the annual rate of return on your bond investment if the bond doesn’t default and all coupon payments are reinvested. (We will revisit this later.) Concretely, the YTM is the rate $y$ that satisfies

$\displaystyle P=\frac{c/2}{(1+y/2)^{.5\times 2}}+\frac{c/2}{(1+y/2)^{1\times 2}} + \cdots + \frac{100+c/2}{(1+y/2)^{T\times 2}}$

where $c$ is the annual coupon payment on $100 of notional,$T$is the number of years to maturity, and $P$ is the market price of the bond. Note that bond yields are almost always communicated as a semi-annually compounded rate since the vast majority of U.S. corporate bonds pay coupons semi-annually. The easiest way to compute the yield of a bond is by using the YIELD function in Excel. From here we can see that Bond A has a yield of 6.08% and Bond B has a yield of 6.54%. We can check these numbers by confirming that they satisfy the equation above. $\displaystyle \frac{\2.50}{1+3.04\%}+\frac{\2.50}{(1+3.04\%)^2}+\cdots+\frac{\102.50}{(1+3.04\%)^{20}}=92.00$ $\displaystyle \frac{\3.00}{1+3.27\%}+\frac{\3.00}{(1+3.27\%)^2}+\cdots+\frac{\103.00}{(1+3.27\%)^{14}}=97.00$ Since Bond B has the higher yield, Bond B is the more profitable investment (assuming your earnings are always reinvested). The fact that Bond B trades at a higher yield than Bond A also implies that the market considers Bond B to have greater default risk and thereby demands a higher yield. ## Important Properties of Yield Yield is a very important part of bond analysis and investing. Here are a few properties of yield that are important to understand. Yield is another, better way of saying price There is a one-to-one relationship between the price and the yield of the bond: any price implies exactly one yield, and any yield implies exactly one bond price. They are simply two different ways of communicating the same information. In practice, yield is usually the better indicator of how risky and profitable the bond is. Yield and price are inversely related At any given point in time, the lower the price of the bond, the higher the yield, and the higher the price the lower the yield. Below is a plot of the yield vs. price of a 10-year bond with a 5%, 10% and 15% coupon rate. Notice that at par, the yield of each bond is exactly equal to its coupon. This, it turns out, is always true for all bonds. Bond prices pull to par over time The conversion between yield and price depends on the time to maturity of the bond. As time passes and the bond approaches its maturity date, either the yield or the price must change to reflect the shortened bond. If the yield stays constant throughout the life of the bond, the price of the bond must drift to par over time. This effect is called pulling to par. Consider a bond that pays a 9% coupon and matures in 10 years. Under various fixed yields, the price of this bond over time looks like this: Yield is an annual return Suppose you invest D dollars in a bond that always trades at a fixed yield y (expressed as a semi-annually compounded rate as usual). Additionally, when the bond makes coupon payments, you reinvest the payments by buying more notional of the bond. After t years, the market value of your investment will be exactly $D(1+y/2)^{t\times 2}$. I.e. your annualized rate of return will be exactly y. This is the basis for our earlier assertion that the yield of a bond like an annual rate of return on the bond. A proof of this is provided at the end of this article. ## The Nitty Gritty of Yield Computation Let’s consider this bond issued by Rent-A-Center Inc. On 03/08/2017, this bond traded at 85.00, which implied a yield of 11.765%. Let’s see if we can verify that number by discounting the bond’s cashflows by its yield and arriving at its price. In order to do that, however, we first need to discuss a few more nuances in bond yield calculation. Settlement Date vs. Trade Date When a bond is traded between two parties, it doesn’t officially exchange hands typically until three business days later. This is becuase it takes time to verify that the seller actually owns the bond and to update central records to reflect the new owner of the bond. Yields are generally computed by discounting cashflows to the settlement date, not the trade date, which in this case is 03/13/17. Discounting and Day Counts This bond makes coupon payments on the 15th of May and November every year until its maturity in May 2023. The next coupon will be paid on 05/15/17, which is sooner than six months from the settlement date of 03/13/17. Therefore we can no longer assume that the next coupon will be paid exactly six months from now, and must discount each cashflow $CF_i$ by $(1+y/2)^{2\times t_i}$ where $t_i$ is the remaining time to the next coupon date (as a fraction of a year). Specifically, $t_1$ is the number of years between 03/13/17 and 05/15/17, $t_2$ is the years between 03/13/17 and 11/15/17, and so on. Computing each $t_i$ exactly can be complicated, especially when dealing with leap years, so the calculation is conventionally simplified using a Day Count Convention. The vast majority of all U.S. corporate bonds use a “30/360” day count convention, which assumes that every year is comprised of twelve equal 30-day months. In this case, the number of days between 03/13/17 and 5/15/17 is $(5-3)\times 30 + (15-13)=62$, and $t_1=\frac{63}{360}=0.173$. The 30/360 approximation does produce an error in the yield computation, but it’s fairly small as you’ll see below.  Date Payment Actual Days Actual Years Discounted Value 30/360 Days 30/360 Years Discounted Value 05/15/17 3.3125 63 0.173 3.25 62 0.172 3.25 11/15/17 3.3125 247 0.677 3.07 242 0.672 3.07 05/15/18 3.3125 428 1.173 2.90 422 1.172 2.90 11/15/18 3.3125 612 1.677 2.73 602 1.672 2.74 05/15/19 3.3125 793 2.173 2.58 782 2.172 2.58 11/15/19 3.3125 977 2.677 2.44 962 2.672 2.44 05/15/20 3.3125 1159 3.174 2.30 1142 3.172 2.30 11/15/20 103.3125 1343 3.677 67.86 1322 3.672 67.89 Total 87.13 87.17 Summing up the 30/360 discounted cashflows, we get 87.17 as the price of the bond, which doesn’t match 85.00. That’s because we’re missing one final piece of the story. Accrued Interest, Clean Price and Dirty Price The seller of this bond received the last coupon payment on 11/15/16, and has already waited almost three months without any coupon payment. Therefore, the seller should receive around half of the coupon that will be paid out three months from now. To adjust for this, the seller simply charges the buyer of the bond the fractional coupon that is owed in addition to the$85 price. On a 30/360 basis, $\frac{118}{360}$ will have passed between the last coupon date and the settlement date, so the seller is owed $\frac{118}{360}\times \6.625=\2.17$. Therefore, even though the bond traded at a price of $85, the buyer actually paid$87.17 for the bond, which matches our result above. In general, the actual amount that is paid for the bond ($87.17) is called the dirty price while the paid amount less accrued interest ($85.00) is called the clean price.

Dirty Price = Clean Price + Accrued Interest

Why is it called dirty price? Since Accrued Interest drops down to zero right after coupon payments are made, the dirty price also suffers from the same sudden jumps. If the yield on this bond never changed, the clean and dirty price of the bond would look like this:

To avoid these jumps, bonds are always quoted with their clean price, and it is on the buyer to know that he or she will pay accrued interest on top of the quoted price.

## Proofs on Yield

Here we will provide rough proofs for two of the assertions made above. For each we assume coupons are paid annually, and therefore yield will be expressed as an annually compounded rate. Generalizing to the k-compounded case requires minimal extensions.

The yield of a bond trading at par is equal to the coupon rate.
If a bond with an annual coupon rate of $c$, maturing in $T$ years is trading at a yield of $c$, its price (per $1 notional) must be: $\displaystyle P = c\sum_{k=1}^T\frac{1}{(1+c)^k}+\frac{1}{(1+c)^T}$ The sum here is simply a geometric sum which can be simplified as follows. $\displaystyle P = c\times \frac{1}{1+c}\times \frac{1-\left(\frac{1}{1+c}\right)^T}{1-\frac{1}{1+c}}+\frac{1}{(1+c)^T} = 1$ Thus, the price of this bond is exactly par, independent of $c$ and $T$. An investment in a bond with a fixed yield y will produce annual returns of y. Consider a bond that pays a coupon c with payments made annually, and matures in T years. Additionally, assume that the bond trades at a fixed yield y at all times. The price of$1 of notional of the bond at the end of year $t$ must then be

$\displaystyle P_t = \sum_{p = 1}^{T-t}\frac{c}{(1+y)^p} + \frac{1}{(1+y)^T}$

Note that we can write this as the recursive formula

$\displaystyle P_t = P_{t-1}(1+y)-c$

Now, suppose you spend $D$ dollars and buy $\frac{D}{P_0}$ in notional of this bond. As we recieve coupon payements on this investment, we will reinvest those payments back into the bond so the notional of our investment will change over time. Let $B_t$ represent the notional balance of our investment in the bond at the end of year $t$, where $B_0=\frac{D}{P_0}$. We can write $B_t$ recursively in the following way:

$\displaystyle B_t = B_{t-1}+\frac{B_{t-1}c}{P_t} = B_{t-1}\left(\frac{P_t+c}{P_t}\right)=B_{t-1}\frac{P_{t-1}(1+y)}{P_t}$

Expanding the recursion, we get:

$\displaystyle B_t = \frac{D}{P_0}\times \frac{P_0(1+y)}{P_1}\times \frac{P_1(1+y)}{P_2}\times \cdots \times \frac{P_{t-1}(1+y)}{P_t} = \frac{D}{P_t}(1+y)^t$

The market value of our investment at time $t$ is simply the notional balance times the price at time $t$. Therefore

$\displaystyle MV_t = B_tP_t=D(1+y)^t$

# Introduction to Bonds

When companies or governments want to raise money, one way they can do so is by borrowing money from the public by issuing bonds. A bond is simply a contract whereby the issuer borrows money today and promises to pay back that money with interest on a fixed schedule. While there’s a lot of variety in how bonds are structured in the real world, most bonds have the same basic structure.

The total dollar amount borrowed is called the Face Value or Notional of the bond. A typical bond will make semi-annual payments as a percentage of its notional called coupon payments until the bond matures. On the maturity date, the bond will make a final coupon payment and will pay back the entire notional of the bond, called the principal payment.

For example, consider a fictitious company Coyote Inc. that needs to raise $100,000,000 to expand its business. It issues$100,000,000 of bonds to investors that pay a 10% coupon semi-annually and mature 10 years from now. You as an investor decide to buy $1,000 in notional of this bond from Coyote. You should expect to receive (10%)($1,000) = $100 every year paid out in two$50 payments every six months and the full $1,000 with the final coupon payment in the end of ten years. Note that if you simply discount the bond cash flows by the risk-free rate, say 2% semi-annually compounded, you would find that the their present value is actually$1,721.82, much higher than the $1,000.00 that you paid. $\displaystyle \frac{\ 50}{\left(1+\frac{02}{2}\right)^{.05\times 2}}+ \cdots + \frac{\ 1,050}{\left(1+\frac{.02}{2}\right)^{10\times 2}} = \ 1{,}721.82.$ If this were a risk-less bond, like a U.S. government bond, then that would be an appropriate way of computing a fair value for the bond. However, Coyote Inc. is not the U.S. government and could go bankrupt before the bond matures in ten years. If that happens, you may end up receiving only a small fraction of the principal payment you are owed. Therefore, investors would only be willing to buy this bond if they could buy it for much cheaper than$1,721.82, in this case $1,000.00. Once bonds have been issued, investors are free to sell their bonds to other investors willing to buy them. However, as the bond markets move up and down, and as the financial health of Coyote Inc. changes, so will the amount investors are willing to pay for this bond. Let’s leave Coyote Inc. now and look at a real-life examples. Coca-Cola and Netflix We can find up-to-date market data as well as historical trading data on most bonds for free at the FINRA website.1 If we search for Coca Cola, we’ll find that it issued a bonds over the last few years. Here is one of those bonds: In this entry, you’ll see that this bond was traded on 03/03/2017 at$93.42. Bond prices are generally quoted as a price per $100 of notional, so$1,000 of notional of this bond would cost you only $934.20.2 Since this bond is trading below$100 (trading below par), this bond is said to be trading at a discount. This means that investors are no longer willing to earn only 2.5% on their investment in Coca Cola because of changes in Coca Colas financial health or overall market conditions. Now let’s take a look at a bond issued by Netflix.

This bond traded at \$108.00 on 03/03/2017, well above par, and is therefore trading at a premium. This indicates that investors are attracted to the bonds high coupon and are willing to pay even more for that coupon given changes in Netflix’s financial health or market conditions.

1. For more information on TRACE market data from FINRA, visit www.finra.org/industry/trace
2. It’s also common to quote bond prices as cents per dollar of notional, so you might hear that this bond is trading at 93 cents.