Market-Based Estimation of Implied Corporate Credit Risk

Corporations that raise capital in the debt markets need to pay a spread over risk-free instruments to compensate the investor for the increased default risk that they assume when buying bonds issued by the said corporation. Thus the buyer of such "credit-risky" bonds need to be able to estimate the likelihood of default and compare this to the default rates implied by the current trading spread of the bond. Sure, credit rating agencies do dish out ratings such as AA, AAA, BBB etc but these are far too broad and not that useful here.

There are more exotics techniques but lets first examine a simplistic back-of-the-envelope approach to calculate implied credit risk. Assume p is the probability of default of the corporation, and r is the recovery rate if/when a default happens.

Assume a corporate bond is currently yielding 12% and government bonds with the same term are yielding 10%. Further suppose that if the corporate bond defaulted the recovery rate would be 50%.

By definition, the (US) government bond is default-free and thus returns $1.1 annually for every $1 of face value. During this same time period, an investor holding a corporate bond could have one of two things happen. The issuer could default, with probability p, or it might not with probability (1-p). The payouts associated with these events are, respectively, $1.12 and $0.5 per dollar of face value. Generalising this it it would look as follows:

The efficient-market hypothesis implies that all available information is used in pricing the instrument thus the corporate bond price/yield already has the credit risk probability embedded in it. Thus an investor should be indifferent to investing in the government bond or the corporate bond. We can therefore look at the expected value of both instruments after one period and solve for the implied credit risk.

1.1 = (1-p)1.12 + 0.5p
=> 1.1 = 1.12 - 1.12p + 0.5p
=> 0 = 0.02 - 0.62p
=> p = .02/.62 = 3.226%

In other words, this crude approach indicates that there is a 3.226% chance of default in the next 12 months for the said corporation.

As an aside, because only two events can happen, and we look only at distinct time epochs, this approach is effectively a one-period, discrete-time binomial tree model. Of course it is flawed in many ways (the real-world is continuous not discrete), but it's a good enough quick guess.


Fixed Income 101 - Pop Quiz

How is a zero-coupon bond different from an ordinary bond?
As the name implies, a zero-coupon bond pays no coupon throughout it's life. Instead all funds are returned at maturity. Because of this zero-coupon bonds are sold at a discount to their face value and return face value at maturity. Also, zero coupon bonds are free of reinvestment risk.

How could you manufacture a zero-coupon bond from a coupon-bearing bond?
A dealer holding coupon-bearing bonds can separate the coupons from the principal repayment creating two separate products that can be on-sold to investors with specific requirements. This practice is called stripping and it produces zero-coupon bonds.

Can you named a traded example of a zero-coupon bond?
US T-bills. And strip bonds.

What happens to the price of a bond as it gets closer to maturity?
The price should converge to the par value of the bond unless a default is imminent.

Fixed income is often considered less risky than equity investments. Can you name any fixed income products that are high risk?
The equity tranches of CDOs are high risk (remember the GFC?), so are "junk bonds" (remember Michael Milken?).

What are the risks associated with a bond?
Credit/default risk, basis risk, interest rate risk, liquidity risk and yield curve risk.

What is liquidity risk?
A common liquidity measure for trade-able instruments is the spread between bid and ask prices. Liquidity risk is the risk that arises from the difficulty of selling an asset and some more exotic fixed income products might be exposed to this.

Do you think Value-At-Risk is useful in the presence of black swans?
Highly subjective. I'll leave you to make up your own mind on that one!

What is the difference between an annuity and a perpetuity?
In an annuity the cashflows are constant. A perpetuity is also an annuity but there is an infinite amount of payments.

Can you name a perpetual bond?
Perpetual bonds have no set maturity. Consols issued by the UK Government are perpetual bonds.

What's the continuously compounding FV formula.
Pt = P0. ert

What does a bond's duration tell us?
Duration comes in two forms: modified duration and Macaulay duration. Macaulay duration is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows. It is an important measure for investors to consider as bonds with higher durations carry more risk and have higher price volatility than bonds with lower durations

Modified duration is the 1st derivative of how the price of a bond changes in response to interest rate changes. As interest rates change, the price is not likely to change linearly, but instead it would change over some curved function of interest rates. The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity.

What's the Macaulay duration of a zero-coupon bond?
In this case, the Macaulay duration is the bonds' time to maturity.

What is bond convexity?
Convexity is the 2nd derivative of price changes per unit of change in interest rate. It measures the sensitivity of the bond's modified duration (the 1st derivative) to changes in interest rates. In general, the higher the convexity the less sensitive the bond price is to interest rate shifts, the lower the convexity the more sensitive it is.

If two bonds have the same duration and yield, but different convexity, which would you prefer?
Take the one with the higher convexity because it will be less sensitive to interest rate changes.

How can convexity be used for risk management purposes?
Convexity can be used similarly to the way 'gamma' is used in derivatives risks management; it is a number used to manage the market risk a bond portfolio is exposed to. If the combined convexity and duration of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur.

Name a bond type that inherently has high convexity and one that has low convexity?
In general, the higher the coupon rate the lower the convexity of a bond. Ergo, zero-coupon bonds have the highest convexity. At the other end of the scale, an adjustable-rate mortgage (ARM) have low convexity.

How would you explain "credit spread"?
The credit spread or yield spread is the difference in yield between different securities due to different credit quality. The credit spread reflects the additional net yield an investor can earn from a security with more credit risk relative to one with less credit risk. The credit spread of a particular security is often quoted in relation to the yield on a credit risk-free benchmark security or reference rate.

This is not to be confused with the vertical credit spread option trading strategy that involves the writes of a higher premium option with the purchase of a lower premium option with the same maturity.

What is the difference between "coupon" and "yield"?
Coupon is calculated based on face value of the bond. Yield is based on coupon / price of bond. Therefore yield is more meaningful to the investor.

How are a bond's price and yield related?
They are inversely correlated. As one goes up, the other goes down. Because of this bonds have interest rate risk. Note that the loss/gain on a bond will only be crystallized if bonds are sold before maturity. Of course, if you mark-to-market daily then you have a paper loss/gain immediately.

What is a discount bond and can you give an example of one?
A discount bond is one sold for less than it's face/par value. This happens when the coupon yield is not as attractive as other products with the same risk profile thus the investor pays a lesser amount to get an equivalent yield to other products. Zero coupons are always discount bonds since they have no interim payments.

How is yield-to-maturity (YTM) calculated?
YTM is the rate of return you receive if you hold a bond to maturity and reinvest all the interest payments at the YTM rate. It will produce a present value equivalent to the price of the security. It may also be called a redemption yield, and it is the internal rate of return (IRR) that an investor would get on an investment, such as a bond or other fixed-interest security, like gilts. This is assuming that the bond will be held until maturity, and all payments will be made on time.

How do you price a bond?
For an option-free coupon-bearing bond the price is the PV of the future cashflows and principal repayment. Things get much hairier when there is embedded optionality.

What's the difference between the clean and dirty price of a bond?
Clean price does not include accrued interest from the current period, whereas the dirty price does. Bond prices generally tend to be quoted as clean prices because they are more stable over time.

What factors affect the price of a bond?
Interest rates, credit quality of issuer, time to maturity, embedded optionality.

How does the issuing company's credit rating affect prices of its' corporate bonds?
If the company gets down-rated investors perceive more risk in lending them money so they therefore expect a higher yield.In other words it costs the company more to raise capital in the debt markets.

What is LIBOR and how is it calculated?
LIBOR, the London Interbank Offered Rate, is a well-used reference rate. LIBOR is the interest rate that banks with high credit ratings from ratings agencies (rating of AA- or above) charge each other for short-term financing. LIBOR is set daily by the British Banking Association (BBA) and is considered the benchmark for floating short-term interest rates. Note that no single bank sets this reference rate, it is compiled by independent authorities using a transparent algorithm based on information from numerous financial institutions.

Other than LIBOR what are the "local" reference rates in Australia?
In Australia, the 90-day Bank Bill Swap rate (BBSW) and the Australian Bank Bill Swap Bid Rate (BBSY) are heavily used reference rates. BBSW is compiled by the Australian Financial Markets Association (AFMA) at 10am every business day based on the average MID-price of market rates supplied by a select number of Australian bank bill market makers. BBSY is calculated similarly except that the average BID-price is used instead of the average MID-price. You can read more about their methodology here. Note that BBSW and BBSY are NOT to be confused with the RBA "cash rate".

Why do we need day count conventions?
In order to perform various calculations, such as accrued interest, we need to be able to count the number of days between two dates. There are established day count conventions to ensure this process is transparent and well-defined. Actual/Actual, Actual/360, 30/360, 30/365 are examples of day count conventions.

What are business day conventions?
Fixed-income products often involve payments to financial institutions. Contracts need to define exactly when these payments will happen and detail what adjustment happens if the scheduled payment date falls on a public holiday. Business day conventions specify such adjustments. Following, Modified Following, Preceding, Modified Preceding are examples of such adjustments.

What is an IR swap?
A plain-vanilla interest rate (IR) swap is a bilateral OTC contractual arrangement where the counterparties agree to swap streams of future interest payments based on a specified "notional" principal for a pre-defined period of time. In the "plain vanilla" case this is done using a common currency on a fixed-for-floating basis whereby one party exchanges a stream of fixed interest payments for a stream of floating interest payments that are based on the future movement of a specified floating reference rate like BBSW or LIBOR. Of course, the other party does the reciprocal operation contemporaneously. The interest payments of the swap are based on a notional principal meaning there is no upfront exchange of principal - it wouldn't make any sense since the net exchange would be zero. The start and end dates, the coupon frequency, the notional principal, the day count convention to be used, along with a number of other input parameters are all negotiated up-front.

I've written more about IR swaps here.

How would you bootstrap a yield curve?
This is a big topic. I'll cover this separately.

What part of the yield curve is the most risky?
Trick question. For a normal yield curve the longer tenors are where the risk is. For an inverted yield curve shorter maturities carry more risk.

Name some bonds with embedded optionality?
Callable and puttable bonds.

What is a callable bond? When would you use one?
A callable or redeemable bond is a bond that can, but doesn't have to be, redeemed by the issuer (borrower) prior to its maturity. It effectively lets the issuers pay out the loan early. Because this is beneficial to the issuer, the holder (lender) needs to be compensated for the additional re-investment risk that they are exposed to. The issuer is only likely to call/redeem the bond when interest rates drop thus the holder would face the prospect of having to reinvest their money at the prevailing lower rate. Thus a premium, above and beyond non-callable coupon rates, is built into the coupon of callable bonds.

Since the issuer has the right but not the obligation to call the bond on certain call dates, we say that the bond has an embedded call option. At this point the simple discounted cash flow calculations get intertwined with stochastic differential equations and assumptions about Gaussian behaviour (more on that another day).

How does the price of a callable bond react to interest rate declines?
Whilst regular bond prices are inversely correlated to interest rate moves, callable bonds exhibit a phenomenon called price compression. Callable bonds also rise in price, but price compression means that they do not rise as much because of the risk that the issuer will call the bond and simply repay the principal, depriving the bondholder of future interest.

What is an arithmetic progression? A geometric progression? Can you sum them easily?
I've covered arithmetic progressions already and how easy it is to sum them. Geometric progressions(GPs) are very similar but each term in the series is a constant multiple of previous term. Both of these are extremely useful in discounted cash flow analysis and PVing. To sum the first n terms in a GP:

How would you price a callable bond?
Generally we value a bond by discounting each of its cash flows at its own zero-coupon ("spot") rate. However, when a bond has one of more embedded options its' cash flow is uncertain. If a callable bond is called by the issuer its cash flow will be truncated. To value such a bond, one must consider the volatility of interest rates as their volatility will affect the possibility of the call option being exercised. One can do so by constructing a binomial interest rate tree that models the random evolution of future interest rates. The volatility-dependent one-period forward rates produced by this tree can be used to discount the cash flows of any bond in order to arrive at a bond value.

Alternately, you can think of a callable bond as a regular bond coupled with a call option and value them separately using PV'ing and Black-Scholes, then add the two prices together.

As a bond investor you expect fiscal policy to incorporate quantitative easing and therefore inflation to increase over the next 2 years. Does this change your approach?
Inflation reduces the real return on all investments including coupon-bearing bonds. There are inflation-linked bonds that can be sought to mitigate this risk. In these bonds either the capital or the coupon is indexed according to some agreed-upon, published measure of inflation.

In the context of bonds what is the Fischer Equation used for?
This equation helps us convert between nominal returns and real rates of return given estimates of inflation. It has important implications in the trading of inflation-indexed bonds, where changes in coupon payments are a result of changes in break-even inflation (BEI), real interest rates and nominal interest rates.

What is Break Even Inflation (BEI)?
Break-even inflation is the difference between the nominal yield on a fixed-rate investment and the real yield (fixed spread) on an inflation-linked investment of similar maturity and credit quality. If inflation averages more than the break-even, the inflation-linked investment will outperform the fixed-rate. Conversely, if inflation averages below the break-even, the fixed-rate will outperform the inflation-linked.

What is a FRA?
A forward rate agreement (FRA) is an over-the-counter forward contract in which one party pays a fixed interest rate, and receives a floating interest rate equal to a reference rate (the underlying rate). The payer of the fixed interest rate is also known as the borrower or the buyer, whilst the receiver of the fixed interest rate is the lender or the seller. An IR swap is clearly a combination of two FRAs. FRAs are often a hedge against interest rate changes. The buyer of the contract locks in the interest rate in an effort to protect against an interest rate increase, while the seller protects against a possible interest rate decline

What is the risk premium?
Risk premium refers to the amount by which an asset's expected rate of return exceeds the risk-free interest rate. When measuring risk, a common approach is to compare the risk-free return on T-bills and the risky return on other investments (using the ex post return as a proxy for the ex ante expected return). The difference between these two returns can be interpreted as a measure of the excess expected return on the risky asset. This excess expected return is known as the risk premium. In the context of bonds, the term "risk premium" is often used imprecisely to refer to the credit spread (the difference between the bond interest rate and the risk-free rate).

The current price of 5 different bonds are: 18, 22, 17, 19, 20. Assuming zero transaction costs, if you know that tomorrow 4 of these bonds will go to 0 and one of them, 100, how would you arbitrage?
Buy them all. You'll pay 94 now for all 4 and get a total of 100 back tomorrow.

You expect the yield curve to steepen. How could you profit from it?
Short answer: Buy a yield curve call option.

If the yield curve steepens, this means that the spread between long- and short-term interest rates increases. Therefore, long-term bond prices will decrease relative to short-term bonds. A yield curve option is an option on the spread between two rates at different maturities/tenors on the same yield curve. They are usually struck on the yield of a longer maturity bond/index less the yield of a shorter maturity bond/index. Thus yield curve calls profit if the spread widens (yield curve steepens), and yield curve puts profit if it narrows (yield curve flattens). These options allow investors to take a view on the shape of the yield curve without taking a directional view on the underlying bond market. A yield curve option costs less than the series of calls and puts on the underlying securities/indices used to construct the yield curves because it only pays off on the change in spread whereas one of a pair of separate options might be in-the-money as the result of a parallel shift in the yield curve.

You expect the yield curve to flatten. How could you profit from it?
Short answer: Buy a yield curve put option. To understand why see the answer above.

You have purchased a 5 yr zero coupon bond. How much do you make or lose if the continuously compounded zero curve shifts upwards by 1 basis point ?
The modified duration of the bond tells you the approximate percentage change in the bond price for a 100 basis point change. Since 1 basis point is a small change then modified duration should be a reasonable estimate. If there was a larger change duration would likely be less accuracy and you'd need to grad the 2nd derivative of price wrt interest rate and do a convexity adjustment.

How are interest rates and FX rates related?
Give todays exchange rate for a currency pair, and the interests rates for each currency investors should be indifferent to: investing one currency today for a period of time and then converting to the other currency at the end of the investment period; or doing a currency conversion today and investing in the other currency for a period of time. Because we expect arbitrage-free conditions this arrangement indicates that the unknown forward FX rate is a function of today's FX rate and the interest rates of the two currencies. This is called interest rate parity and it clearly indicates that FX and IR markets are linked.

If the yen/dollar exchange rate is 100yen/$ today and the one year forward rate is 105yen/$, what does this imply?
That the current interest rate for the yen is 5% pa (see interest rate parity discussion above).


Classical Test Functions for Genetic Algorithms

Recently, I was talking to a rather intelligent chap who was quizzing me on some research I did many years ago. I gave him a simplistic explanation of discrete optimization and genetic algorithms and he fired back a very good question... how do you measure how good a genetic optimization algorithm is? Clearly, you test it on something that you know the global optimum of and see if the algorithm can find it. But there is more to it than that. We also need to use a problem that makes it hard for the algo to get to the "global optima". There are 2 such functions that come to mind that are often used precisely for this purpose...

The Rastrigin Function

where x can range from -5.12 to +5.12.

But it's best visualised as a plot:

As you can see, the external variable A controls the amplitude of the peaks. It should also be clear that this is a non-linear, multimodal function that is not convex. In other words, this function has been custom made to contain many "local minima". A local minimum is a value that is the smallest value in a given area but is not the global minimum. These local minima are the blue-shaded "valleys" in the plot. The global minimum is obviously the smallest value of the function over the entire area. In the case of the Rastrigin function the global minimum is 0 at (0,0) in the two-dimension version of this function.

At any local minimum other than (0,0) the value of Rastrigin's function is greater than 0. In fact, the farther the local minimum is from the origin the larger the value of the function is at that point. The presence of all of these local minima is precisely why this function is often used to test genetic algorithms, because the numerous local minima make it difficult for standard, gradient-based methods to find the global minimum.

Rosenbrock Function

You can read more about it at Wikipedia but again it's best illustrated with a plot:

From Wikipedia, "The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial, however to converge to the global minimum is difficult."

There are others, but these 2 functions are a damn good test for any genetic / evolutionary algo you need to shake out.

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