*After a hiatus resulting from various disturbances, I am back with another book manuscript section. I just reworked this section, and hopefully did not introduce major issues into it. However, I wanted to get this out before next week. Right now, my main concern in life is getting my kitchen sink back.*This section introduces credit spreads from a bond pricing perspective. Looking at bonds is not completely inappropriate, as banks do hold bonds in their liquidity portfolio. The illiquid loans on bank balance sheets can be analysed in the same way, albeit bankers might use different terminology.A credit spread is the excess interest income an instrument earns when compared to an idealised benchmark security that has no perceived default risk. The “idealised” refers to fact that there may not be a useful benchmark bond to compare to, and so one needs to calculate the yield on a matching bond based on a fitted curve based on traded securities. (For example, a corporate bond may have a maturity of 11 years, and the closest government bond maturities might be at 10 and 12 years, forcing the analyst to interpolate an implied 11-year government bond yield.) The curve used would normally be the central government bonds in sensible developed countries, or possibly a swap curve. Banks would probably use some internal funding cost benchmark for anything other than its bond portfolio. The choice of benchmark is not of critical importance, as the spread is relative to that curve, and we can easily convert to a spread versus another curve.Standard practice in fixed income markets is to quote spreads in basis points (abbreviated “bps.”), where 100 basis points is 1%. For example, if a 5-year corporate bond yields 5.5% and the 5-year (central) government bond yields 5%, the credit spread is 50 basis points. Although the use of basis points seems like pointless jargon, it makes sense when you are dealing with minute changes in spreads (or yields) all day. Using a different unit also makes it clear that a spread is not the same thing as the outright yield on a security.

Yield Up, Price Down (and Vice-Versa)The most important part of fixed income analysis is being able to convert the price of an instrument to a yield to maturity. A fixed income instrument promises future payments based on rules set in the instrument legal documentation. Although those payments are often fixed amounts, the payments can be determined by future reference interest rates (“floating payments”).Since the future payments are contractually fixed, the less you pay for the instrument, the higher the return for the investor. Hence, the rule is “yield up, price down” (and the converse).Bond yields are equivalent to internal rates of return but are expressed in market conventions that are most convenient to bond traders. A detailed explanation is not needed for this text. The easiest way to visualise how bond yields change in practice is to experiment with bond price/yield functions in spreadsheet software.

Discount CurveDiscount curves are a key tool for fixed income pricing. The discount curve can be equivalently expressed in two ways:

The discounted value of receiving $1 at a future date *T*; or

the discount rate used on a cash flow received at future date *T*.

If we are using a simple annual interest rate convention, if the discount rate is 5%, the discounted value for one year in the future is $0.9524 (=1/1.05), and $0.9070 (=1/(1.05×1.05)) for two years in the future. (The calculations can be understood as $1 in the present will compound by 5% per year – a factor of 1.05 – and so take the inverse of the future compounded value to determine how much $1 in the future is worth now.) However, the discount rate does not have to be constant across maturities – it will be determined by a fitting function.**Example. **Assume that the benchmark curve has a 1-year discount rate is 4%, and the 2-year rate is 5% (implying a positive slope to the benchmark curve – longer maturities have a higher yield). The fair value of a 6% annual coupon bond when discounted by the benchmark rate is equal to $101.914 (all figured rounded), which is the sum of the following discounted cash flows.

The first coupon $6 coupon paid in one year is worth $5.769 (=$6/1.04).

The second $6 coupon plus $100 principal is worth $96.145 (=$106/(1.05×1.05)).

If a bond is trading at fair value versus the benchmark discount curve, its market price will equal the sum of the discounted value of its future cash flows. If its price is lower (higher) than that, we can then create new discount curve where the discount rates are higher (lower) than the benchmark curve by a fixed amount – that is, we add (subtract) a fixed spread to the benchmark curve’s discount rates. Determining the spread shift to the benchmark curve is the generic method to calculate a credit spread.If we are using (central) government bonds to determine the benchmark discount curve, almost all bonds will be trading cheap to the fair value implied by the discount cash flows – we need to add a positive credit spread to create a shifted discount curve that matches the market price of the bond. It is possible that some bonds will trade with a negative spread to the (central) government fitted curve – some government bonds will trade at expensive levels versus others, and we might encounter bonds with tax treatment (e.g., American municipal bonds) that leads to them trading with yields below bonds that do not get such treatment.**Example continued.** If the previous bond was observed to be trading at $100.054, we would find that the credit spread was 100 basis points. That is determined by noting that the market price is equal to the sum of the discounted values if we add 100 basis points to the benchmark discount curve.

The first coupon is worth $5.714 (=$6/1.05).

The final coupon and principal is worth $94.340 (=$106/(1.06×1.06)).

The price of the bond is close to par since the final cash flow represents the bulk of the value of the bond, and the coupon rate is close to the 2-year risky discount rate (6%), but the first coupon is discounted at a lower rate (5%) since the benchmark curve is not flat.There are bonds without cash flows before maturity (bills, zero coupon bonds) whose price is directly comparable to the discount (price) curve. However, most bond trading involves bonds that have payments before maturity, and so we need to look at how different types of bonds behave in a more qualitative fashion.

Fixed Coupon BondsThe most common form of bond are ones with fixed coupons. The bonds pay a fixed amount of interest per year, plus they repay the principal amount at maturity. The usual pricing convention is to use $100 as the par price. For example, a 5% coupon bond would pay $5 per year in interest, plus $100 at maturity (along with the final interest payment). Coupon payments are typically made either annually, semi-annually (twice a year) or quarterly. A 5% semi-annual bond will pay $2.50 every six months until its final payment of $102.50 at maturity.We can now give the simplest example of a fixed coupon bond spread. We have a 10-year corporate bond with a 7% coupon trading at $100 (par), which implies a yield-to-maturity of 7%. We then compare it to the fitted government bond yield curve. If the fitted 10-year government bond yield is 5%, the corporate bond has a spread of 200 basis points over the government curve. (The fitted 10-year government bond is assumed to also trade at par.) If we compare cash flows, every year the corporate bond pays $2 more in interest ($7 versus $5) and they both pay $100 at maturity.What makes fixed income pricing exciting is that bonds do not always trade at par*. (OK, somewhat exciting.) *Imagine an old 30-year bond issued with a 10% coupon that is now 5 years to maturity with a yield of 5%. In order to have so low a yield relative to its coupon, it needs a high price (about $121.65 for an annual coupon bond). If we compare thar bond to a 5-year 5% coupon bond trading at par, we see that the old bond’s cash flows are weighted more towards the coupon payments at times before maturity. Since the discount rates tend to differ based on maturity – there is a “slope” to the yield curve – this weighting mismatch should be taken into account when calculating the spread.

Floating BondsSome bonds feature interest payments that are based on a fixed spread over a reference interest rate, and then pay the principal amount at maturity. If the bond is bought at par ($100), the spread on the bond over the reference rate is the contractually fixed spread. If the bond is purchased at a price away from par, the premium/discount would be amortised over the life of the bond.Floating rate bonds are not a significant portion of the bond universe, but floating bank loans (which are analysed in the same way) are more common.

Amortising BondsMany bonds – particularly securitisations of residential mortgages – have principal payments that are repaid ahead of maturity. To compare the amortising structure to standard (“bullet”) bonds that just repay principal at maturity, one needs to decompose the payments and see how much of a spread they have versus the benchmark discount curve.**Example.** Let us take a non-standard 2-year amortising bond with an annual coupon rate of 6% and repays 50% of its principal in the first year. The cash flows on the bond are:

$6 coupon (on original principal) plus $50 principal for a total payment of $56.

A final payment of $53, which is the $50 principal plus $3 interest (=6% of $50).

If we had a flat 6% credit curve for discounting, the bond would trade at par. If it trades at a 100 basis point spread to the previously used benchmark curve, the price is equal to $100.503, since the large payment in the first year is discounted at 5%, not 6%. The yield on the bond at that price is equal to 5.64%, which is below the coupon rate, and lies between the risky interest rates at the 1-year and 2-year maturities.The previous example highlights that an amortising instrument will have a yield that reflects an average of the discount rates across the lifetime of the bond, which can be quite different that the discount rate at maturity. This makes the yield on a amortising security not directly comparable to yields on conventional (bullet) bonds that only repay principal at maturity. In particular, yields on American 30-year conventional mortgages can decouple from 30-year Treasury bonds, which have a much longer weighted average maturity (or duration). Prepayment OptionThe most complex spread calculations involve bonds where there is optionality in the payments. The most important case is the one where the borrower can pre-pay principal amounts (typically with some “fees” or premium associated with the payment). Residential mortgages – particularly in the United States – typically give households the right to prepay the mortgage.The disadvantage of owning a bond that can be prepaid is that the prepayments are typically done when interest rates have fallen. Imagine a 10-year bond that is issued at par with a 5% coupon and can be prepaid at par after five years have passed since issuance. We then assume that the interest rate the borrower sees at five years is 4%. The issuer will then have an incentive to issue a new bond yielding 4%, and repay the original bond at par. For the holder of the bond, this means that the value of the bond is at most $100 (what they will be repaid), and thus will not increase due to the fall in interest rates. However, if the prevailing interest rate rose to 6%, and the issuer will likely not finance. The value of the original bond will fall to be in line with the 6% yield of comparable bonds. As such, the owner of the bond faces the prospect of future capital losses if interest rates rise and limited capital gains (if any) if yields fall.Such a bond is analysed on the basis that it consists of two underlying instruments: a standard bullet bond plus selling a call option (a short position) on the bond. *(For readers unfamiliar with options terminology, owning a call option gives the holder the option – and not the obligation – to buy a security at a fixed price at a future date. Being short the call option means that you have sold the option to someone else – in this case, the issuer of the bond.)* In order to determine the value of such a bond versus benchmark bonds – which have no embedded options – one needs to calculate the value of the option. To the extent that fixed income options are traded, it is possible to come up with the price of the embedded option. We can then use that estimated option price to determine the value of the theoretical bond that does not feature a prepayment option, and thus be able to determine the spread. This process gives us the option-adjusted spread (OAS). What Should the Spread Be?A risky bond spread provides compensation for holding that bond instead of a high-quality benchmark bond. Why own the bond of some dodgy company when you can buy the bond of the government that controls the central bank (which ensures that it can make payments)?Although bond spreads are effectively set based on what the market is pricing for comparably risk bonds (e.g., bonds with the same rating), we can come up with a fair value estimate for what the spread should be. There are two risk factors that the spread compensates for. (This discussion assumes that other technical factors like tax effects and embedded options are already considered.) When I discuss “fair value” for a price, it refers to the fair value for an investor who is concerned about the expected return on the investment, and not the riskiness of outcomes away from that expected outcome (known as a risk neutral investor).

There is an illiquidity premium. Bonds that are less liquid than actively traded benchmarks need a spread advantage in order to be equally attractive. Less liquid bonds typically have a wider spread between the bid and offer prices (the bid/offer spread, the bid/ask spread is also used), which means that it is more costly to enter and exit a position holding the bonds. Since bond portfolios are often used for liquidity management, transaction costs matter. On top of the bid/offer spread, illiquid bonds are less likely to attract bids when liquidity dries up in a financial crisis (which is when investors want to be able to rotate out of bonds to buy risk assets at distressed prices). The liquidity premium is normally not very large (e.g., typically 20 basis points for high quality bankruptcy-remote issuers), but it creates a floor for observed credit spreads.

There is a default risk premium, which is much larger than illiquidity premium for anything other than the top-rated issuers. The spread associated with the default risk premium should compensate for credit losses caused by credit events. We can back out a theoretical probability of default based on assuming that the observed spread is at the level that matches this expected loss, as well as having an assumed recovery rate upon default. For example, a 100 basis point credit spread corresponds to an expected credit loss of 1% per year. This could result from a 1% annual chance of default with no recovery, or a 2% annual chance of default with a 50% recovery rate. (One either can have a credit analyst come up with a recovery rate assumption or use historical recovery rate averages that are provided by credit rating agencies.) The usual approach when presenting these implied default probabilities is to ignore the existence of the illiquidity premium.

One standard analysis approach for corporate bonds is to compare average bond spreads for a credit rating tier versus the historical default/recovery rates for that tier. A typical situation is that the implied default probability is higher than the historical default rate average. To the extent that we believe that the implied default probability is too high, there is an additional risk premium embedded in the corporate bond spread. (Alternatively, we can argue that investors are not risk neutral – they weigh default loss risk more than the profitable non-default events.) The tendency in modern financial academia is to take such a sub-division of risk premia seriously, but we need to accept that there is considerable uncertainty in the expected default probability.In the previous discussion, I assumed that the credit rating reflects the credit risk associated with a bond. Given the difficulties encountered during the 2008 Financial Crisis, that might sound complacent. However, it is defensible if “credit rating” refers to the rating given to the issuer internally by the investment firm’s credit analysts. A standard approach is to have internal ratings for bonds, which can depart from those provided by credit rating agencies – who disagree amongst themselves. A more serious problem is finding the average spread based on a credit rating tier – since most sources are forced to stick with the public information of the ratings provided by agencies. If large issuers suddenly find themselves in trouble, they may trade at an effective rating that is much lower than the average rating provided by rating agencies, since they take time to adjust ratings. A few large issuers in the same rating bracket cratering at the same time can cause the spread on that bracket to jump while other ratings brackets are untouched.

BankingThe psychology of bank loan portfolios is different from bond portfolio investing. There are no *direct* additional costs for holding a corporate bond than a government bond, and so bond portfolio managers face a straightforward question: do we expect this corporate bond outperform a government bond once we incorporate potential credit losses*? There is an indirect management cost associated with adding new classes of securities to a portfolio – a sensible bond manager wants to ensure that they have analysts and traders that understand that type of bond. (Not all portfolio managers follow that rule of thumb, which shows up when a financial crisis hits.) But once the new team has been added, there is no additional cost to allocating portfolio assets to them.*Bank loan portfolios are not interchangeable with government bonds. Different classes of loans have different management costs associated with them. The perceived riskiness of the loans shows up in bank capital calculations, and if more bank capital needs to be held against the loan portfolio, the weighted cost of capital will go up. Increasing the riskiness of a bank’s assets will eventually show up in the form of credit rating downgrades, which will presumably increase the bank’s financing costs relative to other banks. (The funding costs banks in aggregate face in the market is determined by the state of the corporate bond market.) The bank Treasury team will raise the cost of funding for teams based on these factors, leaving the operating units to decide whether the book of potential loans will generate a profit versus their internal funding cost after taking into account operating expenses and credit losses.Bank loan spreads are determined by a negotiation between the bank and the borrower, and to the extent the borrowers cannot access the bond market, the pricing can be insulated from what is happening to corporate bond spreads. However, if loans are securitised, pricing on new loans is going to follow what is happening to the securitisations in the bond market. In cases where there is less linkage to the bond markets, spreads over the bank’s cost of funds will be determined by the bank’s corporate strategy and historical experience with that class of loans.

Concluding RemarksPutting aside the arcana of the illiquidity premium and if we account for other known factors (like prepayment optionality), the fair value of the spread on a risky fixed income is equal to its expected annual credit loss. Of course, we would like some extra spread over that fair value, but we cannot always get what we want. We must be careful about what the previous statement implies.

It does not mean that future default losses will match what is currently priced into the market.

Even though bond investors can agree that the spread is the implied default loss expectation, it does not mean that this their personal forecast for those losses. It is entirely possible that no major investor has a default loss forecast that aligns with market spreads.

References and Further ReadingIt is easy to find introductory references discussing how a bond yield is calculated, or how to discount future cash flows. The unfortunate problem with credit spreads is that for anything other than the spread between two conventional bonds, the calculations spiral down the rabbit hole of fixed income mathematics. That level of detail is overkill considering that the only insight one needs is that the credit spread is there to cover default losses (as well as the opportunity cost of holding an illiquid, risky instrument).