Complexity, Risk, and Financial Markets (Wiley Investment)

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The Handbook of Commodity Investing. Frank J. Fabozzi , Roland Fuss , Dieter G. Filled with a comprehensive collection of information from experts in the commodity investment industry, this detailed guide shows readers how to successfully incorporate commodities into their portfolios. Created with both the professional and individual investor in mind, The Handbook of Commodity Investments covers a wide range of issues, including the risk and return of commodities, diversification benefits, risk management, macroeconomic determinants of commodity investments, and commodity trading advisors.

Starting with the basics of commodity investments and moving to more complex topics, such as performance measurement, asset pricing, and value at risk, The Handbook of Commodity Investments is a reliable resource for anyone who needs to understand this dynamic market. Level II — 1st Saturday in June. Two testing windows: March and September. Two testing windows: April and October Three testing windows: March, July and November Topic areas Level I focuses on investment tools and tests your basic knowledge and comprehension with some questions requiring analysis.

Part I is a question multiple-choice exam that focuses on the tools used to assess financial risk: quantitative analysis, fundamental risk management concepts, financial markets and products, and valuation and risk models. Level I focuses on definitions, concepts and terminology around technical analysis. Part II is an question multiple-choice exam emphasizing the application of the tools acquired in Part I: market, credit, operational, integrated risk, and investment management, as well as current market issues.

Level II asks you to apply your understanding gained in Level I within a portfolio-management context. Level II requires applications of theories and concepts explored in Level I. Level III looks more in-depth at certain areas such as ethics. Also it asks candidates to apply their practical skills by analyzing case studies and making recommendations. Level I: The exam is open to anyone with an interest in technical analysis. The 3-month CD investment was a zero-coupon investment over one quarter.

The 1-year rollover strategy consisted of four consecutive quarterly zero-coupon investments. All interest and principal were fully reinvested every quarter. Graphically, the simplest way to present a zero-coupon rate is as follows: PV0 0 0. This is very misleading, since all rates zero, coupon, and amortizing can be spot i.

They all can also be forward i. Zero-coupon interest can accrue on either an add-on or a discount basis. The distinction here is only of the form, not of substance. In the add-on case, the investment is purchased for a full face value, which is a multiple of some round number, and interest accrues based on the principal equal to the face value of the security.

In contrast, most short-term securities sell at a discount from face value, and the interest rate is only implied by the ratio of the round-numbered face value and the purchase price of the security. For example, the price of Treasury Bill is expressed as percent of par. Similarly, a buyer of a 6-month U. T-Bill paying Even though the two T-Bills have the same 3. Depending on at what rate we can reinvest the 3-month T-Bill, we could end up with more or less than the principal and interest on the six-month T-Bill in 6 months. Assuming no change in rates, we can use EAR as a comparison tool.

To obtain EARs, we compute the yields each investment earns if it is rolled over at its original yield for a total holding period of 1 year. Let us re-emphasize that, upfront, the realized yield on any rollover strategy is not known, as the reinvestment rate can change. The equivalent yield calculation relies on the unrealistic assumption that the reinvestment rate is known and will not change.

Today, only some Eurobonds come in bearer form with physical coupons. All U. But the way interest is paid on bonds is the same as ever. Let us consider an example. The maturity date is June 30, and coupons accrue from June 30 to June The actual CFs from the bond are portrayed in the following picture: Most commonly, CFs are represented as percentages of par, as in the following normalized picture:. Coupon yields and rates can be expressed on a variety of compounding and daycount bases, typically following a particular convention.

But the legal language can be far from plain English. The stated rate is always annualized and needs to be de-compounded. On any other basis, the numerator and the denominator of the day count may not result in an exact division by 4 for all periods. A 3-year 4. Notice that the 3-year 4. When computing the present value of the bond, it is convenient to break.

We used a yield of 5. If the bond sells in the market for Zero-coupon bonds are special cases of coupon bonds with the coupon rate equal to 0. That is, the knowns are:. Purchase price. Sale price. HPRs can be expressed on any compounding and day-count basis. A yield to maturity YTM is a holding period return over a holding period equal to the maturity of the instrument. The assumption that the instrument is held to maturity also ensures that the sale price is equal to the face value.

YTMs can be expressed on any compounding and day-count basis. Most dealers convert non-native YTMs to a bond-equivalent basis which, in the U. We illustrate the concepts of a holding period return and a YTM through some examples. Today it matures and you receive What semi-annual YTM have you earned? Now let us look at the coupon bond. This follows the logic of how we solve for the YTM. And 3 All of our examples assume that the investment is purchased a moment after a coupon has been paid. That is, we do not need to make here, and in fact throughout the entire book, a distinction between the dirty price and the clean price.

When a coupon bond is purchased between coupon payment dates, the buyer pays the seller the so-called dirty price, which includes an allowance for the interest that has accrued between the last coupon date and the purchase date. The seller is entitled to that interest, but by surrendering the bond, he has no possibility to collect it. The buyer will receive it included in the whole coupon payment on the next coupon date. The so-called clean price is equal to the dirty price minus the accrued interest. We can show that if the interest rate the money-market account paid on any balances left in it was equal to the computed YTM of 4.

There is another interpretation of YTM. In reality, both the reinvestment rates and the term rates are not likely to be constant or equal to each other. It is a mathematical construct. It is the average rate earned over the life. Amortizing rates Another way of earning interest, in addition to zero and coupon rates, is through amortizing interest rates. Some bonds and most mortgage loans follow this arrangement. Instead, he repays it piece by piece with each periodic payment. To distinguish the amortizing loan from a coupon loan, the latter is often referred to as a balloon loan and the principal repayment at maturity as the balloon payment.

The mortgage borrower obtains the full amount of the loan upfront with which he pays for a piece of real estate. The monthly payments cover both the interest on the loan and the repayment of the principal. They can also help construct the so-called amortization table which breaks down each payment into its interest and principal components. What is immediately clear from such a table is that over time the interest portion of the level payment decreases while the principal portion increases as the loan is paid down.

This is obvious as each month interest is paid on the decreasing outstanding principal i. Interest and principal portions balance each other in such a way that the total payment remains constant. An amortization schedule for a mortgage looks quite complicated. We can easily lose sight of the simple nature of the amortizing loan.

It is nothing but an ordinary textbook annuity. Let us illustrate this point on an example. Consider a 2-year, 4. The repayment consists of four equal payments in 6, 12, 18, and 24 months such that the PV of those payments is equal to the face value of the loan. These can be portrayed as: CF. Let us also illustrate the logic of interest and principal component calculation. The loan balances are summarized in Table 2. Table 2. Floating-rate bonds So far, we have considered only bonds with interest rates that are known in advance.

Suppose ABC Inc. The second coupon rate will be set 1 year from today and paid 2 years from today. It will be set equal to the then prevailing 1-year interest rate. The third coupon will be set 2 years from today and paid 3 years from today, and so on. How much will investors pay for such a bond?

In order to answer that question, we make a few observations. The timing of the coupon-setting process corresponds to a sequence of new 1-year loans. On such loans, the rate would be set at the beginning of the year and paid at the end i. Consider owning the 5-year bond 4 years from today, just after a coupon payment. The recursive argument continues all the way to year 1 for which the interest rate is known and equal to the 1-year discount rate.

That price will stay close to par throughout the life of the bond deviating only slightly inside the interest periods and returning to par right after each coupon payment. At any given moment, zero rates, coupon rates, and amortizing rates are not equal to each other. A graph of market interest yields against their maturities is called the term structure of interest rates, or the yield curve.

The graph is a snapshot of the market YTMs for a given moment in time. The x-axis contains maturities relative to the date of the graph e. As there are many types of interest rates, there are many term structures. This assumes that the credit quality of the issuer remains the same throughout the life of the bond. For example, U. Abstracting from all these credit issues, generally, we look at two types of term structures:. The term structure of discount rates i. The term structure of par rates i. These are newly issued T-Bills, notes, and bonds with a few standard maturities.

Because they are newly issued, they trade very close to par as the coupon rate is set close to the market yield, and they are very liquid as there is great interest in them among investors. On September 18, the Treasury yields in Table 2. From this information we can produce the term structures for the four dates on one graph as in Figure 2. On any given date the slope of the curve can be smooth, jagged, humped, etc.

Market analysts and economists often aver that the shape of the yield curve predicts the economic cycle to follow. For example, upward sloping is purported to signal expansion; inverted, or downward, sloping a recession; and inverted with a high short rate perhaps a currency crisis. We show some examples in Figure 2. The Treasury yield curve blends discount i. This is done for completeness as there are no 3-month coupon-paying Treasuries.

For consistency, all rates are expressed as semiannual bond equivalents. Although most yield curves found in the press are those for coupon par instruments, the most useful one is the term structure of zero-coupon interest rates, commonly referred to as the discount curve, or the zero curve. We explain in the next few sections why that is the correct procedure. We also explain how to construct the discount curve from observed market yields. Their maturities and yields are summarized in Table 2.

How much would we be willing to pay for the coupon bond? For the 1-year zero. That is, they do not represent the amounts we would have to pay for real securities. We had obtained those by discounting at the zero rates of the four real discount bonds. In reality, many issuers tend to only issue coupon securities i.

What can we do when this is the case? Constructing the zero curve by bootstrapping The discount curve can be obtained by a process known as a zero bootstrap. This takes as given the par coupon rates observed in the market and sequentially produces the zero rates one by one from the shortest to the longest maturity, just like lacing boots. A bootstrap looks messy on paper, but setting it up in a spreadsheet requires no skill. The only confusion usually has to do with day-count conventions. We will describe the process for annual rates. These are newly issued bonds and some old bonds that originally had much longer maturities.

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This can be depicted as: The coupon bond is just labeled. Given that the coupon bond and the zero-coupon bond are perfect substitutes for each other, they must have the same yield. Solving for the 2-year zero rate Let us examine the 2-year coupon bond. It has a coupon rate of 5. The arbitrage argument But suppose that the 2-year coupon bond cannot be separated and sold as two strips. Can we still claim that the 2-year discount rate should be 5. This is actually the crux of the argument. She could do it by entering into the following two trades simultaneously:. This matches exactly the 2-year strip.

Thus, a 2-year zero investment can be synthetically replicated by going long a 2-year coupon bond and short a 1-year zero-coupon bond. All investors would simply enter into synthetic replicating strategies. The strategy could be summarized as the following three trades:. Her future obligations would be matched at every point in time as shown below: Long 2-year zero. All she would have to do in the future is to collect the receipts on her 1-year and 2-year zero investments and use them to satisfy her coupon obligations to the lender of the bond. She would not be the only one pursuing this arbitrage strategy.

All investors would immediately pursue strategies like this one, driving the price of the newly issued 2-year zero up and its rate down to the 5. It is also possible that the 2-year coupon rate might be driven up and the 1-year zero rate down at the same time. One lesson from all of this is that the no-arbitrage principle requires that all instruments be freely traded and both longs and shorts allowed.

Our example also presumes that the bid—ask spread i. Illiquid markets with wide bid—ask spreads also follow arbitrage rules, but the spread has to be explicitly taken into account to compute arbitrage bands around mid-market prices and rates. In liquid markets, the standard procedure is to use mid-market rates to compute the implied mid-market zero rate and then adjust it to bid or ask.

Solving for the 3-year zero rate Let us examine the 3-year coupon bond. It has a coupon rate of 6. It is a pure coincidence that the 3-year zero rate, or the 3-year zero yield, is. The 3-year coupon yield is not equal to 6. If XYX GmbH were to issue another 3-year coupon bond and wanted to sell it at par, then its coupon would have to be set below 6. We will come back to this point after the completion of the bootstrap. The arbitrage argument Let us again show that the 3-year zero of 6. We assume that the 1- and 2-year zero-coupon bonds issued by XYZ GmbH are actively traded in the market or they can be created synthetically by taking simultaneous positions in existing coupon bonds and zero bonds.

She could do it by entering into the following three trades simultaneously:. This matches exactly the 3-year strip. All investors would simply enter into this synthetic replicating strategy with existing bonds. Again, we can also argue the yield on a new 3-year zero could not be higher than 6. The strategy could be summarized as the following four trades:. The last three trades—. Her future obligations would be matched at every point in time and can be depicted as: Long 3-year zero.

In the future, she would simply collect the receipts from her 1-, 2-, and 3-year zero investments and use them to satisfy her coupon obligations to the lender of the 3-year coupon bond. Again, she would not be the only one pursuing this arbitrage strategy. All investors would immediately do the same, driving the price of the new 3-year zero up and its rate down to the 6.

They might also force the yields on the other bonds to adjust to a level at which arbitrage would not be possible. A vast majority of coupon bonds are issued at par or at a price close to par. Practically, what that implies is that at the time of issue the coupon rate is set close to the prevailing market yield. This phenomenon is normally portrayed as a downward-sloping concave relationship between the price of a bond and the YTM on the bond.

This can also be seen through a recursive argument. Six months prior to maturity i. Note that there is a maximum price for the bond. Most of the time, we are somewhere in the middle. The graph is a convex bowed to below curve due to the nature of compound interest. This can also be seen from the. What does the magnitude of that risk depend on?

Three factors come to mind:. Time to maturity—other things being equal, the longer the maturity of the bond, the larger the price swings are for the same change in interest rates. Coupon rate—other things being equal, the lower the coupon rate of the bond, the larger the price swings are for the same change in interest rates. The intuition is similar to the time to maturity argument.

Coupon frequency—other things being equal, the less frequently the coupons on the bond are paid, the larger the price swings are for the same change in interest rates. Monthly bonds bring the coupons closer to today than annual coupon bonds and, hence, are less sensitive to the discount rate change. Floating-rate bonds have virtually no interest rate sensitivity as their prices always return to par after coupon payments.

The most commonly used interest rate risk metrics are duration and convexity. Both of these measures are local in nature. Duration The universe of bonds, even for the same issuer, can be enormous. How do we choose which bond to invest in or which one to sell out of a portfolio? But the bond may also provide substantial coupons prior to maturity. We can say that a bond has, for example, a duration of 3. As we will show, this is a very intuitive measure. But it is important to understand the intuitive Macaulay meaning of duration.

The essence of the duration computation is columns 5 and 6. Figure 2. The height of the block is taken from the appropriate row of column 5. The Macaulay concept is extremely intuitive. With a little experience, we can guess bond durations fairly accurately. Here are some heuristics:. All other things being equal, the longer the maturity, the longer the duration i. The larger the coupon, the shorter the duration i. The greater the frequency of the coupons, the shorter the duration i.

All three of these correspond closely to the heuristics behind the interest rate risk of bonds. Here are two more observations:. The duration of a zero-coupon bond is equal to its maturity. Floating rate bonds have very short durations equal to the next coupon date. Let us now examine the more practical meaning of duration, that of the interest rate sensitivity applied to our example bond.

We computed the Macauley duration to be 4. If the yield on the bond were to increase from 8. Based on the starting value of In the case of a small change in the yield, duration was a very good approximation to the change in the value of the bond. This would not be so if the yield change considered were large.

In the extreme case of the yield going down to 0 i. What went wrong? Duration is a local measure. Thus, it is a linear approximation based on a line that is tangent to a polynomial curve, the true price—yield relationship. This is represented in Figure 2. Later we show how the duration-based linear approximation can be improved with the use of convexity. Most computer applications do not compute duration the way we presented it in Table 2.

That is, they compute the sensitivity of price to yield directly. For example, using the yield of 8. Of course, we could use a smaller yield change or adjust the centering in the numerator. The smaller the blip used, the smaller the error. We encourage the reader to repeat the calculation by using a 1 basis point or a 0. Portfolio duration Duration is very popular with managers of large bond portfolios. This is due to its one very attractive property: the duration of a portfolio is equal to the weighted average of the durations of individual bonds.

The weight for each bond is simply the proportion of the portfolio invested in that bond. Let us look at an example. In the case of the portfolio represented in the above table, this could translate into the following statement: if the YTM on each bond in the portfolio decreased by 7 basis points, then the value of the portfolio would increase by 6. In dollars, that is equal to an increase of 0. By knowing one statistic about the portfolio—its duration—the manager can predict the value change for the entire portfolio very accurately for small changes in yields!

Often, bond managers engage in what is called duration matching, or portfolio immunization. These terms refer to a conscious selection of bonds to be added to the portfolio in order to reduce the duration of the portfolio to 0 i. This is done by selecting the right amount of bonds to be shorted or by buying bonds with negative duration. Many managers of corporate bond portfolios short government bonds with the same duration to eliminate exposure to interest rates, leaving themselves with pure credit spread exposure.

Note, in the above example, that if the YTMs do not change in parallel e. However, an estimate obtained by summing the products of the changes in yields for all bond times will have individual durations that are still very accurate. This is still much easier than revaluing all bonds. Convexity Convexity is often used to improve the accuracy of the duration approximation to the change in value of the bond. It is important to include it in the approximation for:. Large changes in YTM.

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Bonds whose price—yield relationship is highly non-linear e. Convexity is equal to half the second derivative of bond price with respect to the yield, and as such it measures the average rate of change in the slope of the tangent duration line. What we are mostly interested in is in improving that approximation.

In order to do that, we need to multiply the convexity by the relevant yield change Dy to obtain the change in the duration over that entire yield change. Recall the true value of the bond at a yield of 8. Convexity is widely used as a summary statistic to describe large bond portfolios. It is important to remember, however, that convexities, unlike durations, are not additive and are computed by blipping entire portfolios and revaluing all the bonds in them. The multiplication of the second derivative by 12 averages the point estimates of convexity per unit of yield over the entire range of Dy.

We should, however, bear in mind that those values are not computed through simple discounting, but, rather, with the use of an option-pricing model. As such they take into account other inputs, the most important of which is the volatility of the yield. Imagine a portfolio of two corporate bonds both with the same maturity and both trading at par. The volatility of the yield refers precisely to that concept.

Computing portfolio durations may be of little help in this case. Rather, we may prefer to compute individual durations and scale the assumed yield movements by the respective yield volatilities to arrive at portfolio value change approximations for more realistic yield movements. For most bonds, coupons and principal repayments are guaranteed by the issuer.

This is not the case with stocks. A company is under no obligation to pay dividends. If and when it returns cash to its shareholders, whether in the form of dividends or capital gains, is not known in advance. Even if we assume that we do, then what rate of discount should we apply? Equity holders get paid only after debt holders do.

So the discount rate must be higher than that applied to bonds. Given all these uncertainties, the math applied will be much simpler than that for bonds, and, paradoxically, complicating it will not make things more accurate. There is one more point of view to bear in mind. There are two related ways to answer that question. The accounting book value is of very little help; rather, it is more realistic to view the value of the debt as known. After all, we can discount the value of all future debt obligations. The discount rates can be easily gleaned from bond yields and rates quoted on bank debt.

Thus, although for risk-free government debt all the discounting machinery is well established, for risky corporate debt it is not so simple.


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This point becomes very clear when we consider hybrid securities like convertible debt. It becomes even clearer when we consider companies close to bankruptcy, where the possibility that debt holders will soon become equity holders is very real. Add to that the uncertainty in the value of the assets how much is the Starbucks brand worth? Perhaps we used the standard capital asset-pricing model CAPM 7 which related the discount rate for the stock to the rate on risk-free government obligations and a market risk premium. In any case, we have determined the appropriate rate r, which we will assume to remain constant over time.

We want to buy one share of ABC Corp. ABC will pay a dividend of D1 dollars a year from today and, at that point, we will be able to sell the stock for P1 dollars. D1 could be 0 here. Also note that we may not know the value P1 that we will be able to sell the share for 1 year from today. We would have gotten the same result if we had assumed that we did not intend to sell the stock after 1 year, but that we were going to hold it for n years, collect all intervening dividends, if any, and then sell only at the end of year n.

It does not depend on who will hold the stock over the next n years. It also does not depend on whether the company pays any dividends at all i. For example, see Richard A. Brealey, Stewart C. Myers and Alan J. The owners could always force liquidation after n years to get Pn. Let us look at some numerical examples. Implicit in these price calculations is the assumption that each company will be able to maintain a particular stream of dividends. The rate used is the average for the economy as a whole i. Let us think of our ABC companies as manufacturing operations.

It is hard to imagine that ABC-NoGrowth would be able to maintain a steady dividend without replacing and modernizing its equipment. Its earnings before depreciation must grow just to keep the dividend constant. Its earnings net of depreciation can be constant as long as it pays out all of its earnings as dividends.

Suppose, instead of paying a dividend next year, ABC-NoGrowth decides that it will reinvest the dividend in existing operations. What must its return on investment ROI be just to keep investors equally happy? We apply the perpetuity formula 1 year from today and then discount the fair value 1 year from today back to today. It can also be easily shown that it does not matter for how long the company decides to reinvest earnings, instead of paying them out as dividends. Next we show that the same logic applies to ABC-ConstGrowth, except that it must ensure that its reinvested earnings not only return the discount yield, but also that the return on that return grows at rate g!

Let us consider a scenario where ABC-ConstGrowth decides that it will not pay out its earnings E1 in the form of dividends D1 1 year from now. Instead, it will reinvest those earnings in projects yielding the return on investment equal to ROI. We would like to use the same valuation trick as before by applying a growing perpetuity formula 1 year from today and then discount it by one period back to today. This implies that the production base that yielded year 1 earnings E1 must grow at rate r and it must sustain the growth of those earnings E1 at the rate g.

But it must also. The growing perpetuity formula assumes that the entire numerator grows at the rate g for ever. In general, if a company decides not to distribute earnings in the form of dividends, the projects that it invests these retained earnings in i. If the company always reinvests its earnings, then all future reinvestments must guarantee that growth rate.

Observation A growth company is not simply one whose earnings grow, but one for which any new investment, whether in the form of retained earnings or additional capital, is expected to produce new earnings that will grow at a rate at least equal to the return on its current earnings. That is a tall order especially if the current market price already assumes that the growth will continue for ever.

Consider two more ABC sisters. ABC-Value Corp. ABC-Growth Corp. These could be interpreted to mean that investors pay a lot more per dollar of earnings for the ABC-Growth shares. It is easy to imagine ABCValue to be a mature company with stable earnings, but no growth prospects, and ABC-Growth as a young entrant into a new, rapidly expanding industry. Why would one not invest in the latter? After all, its growth rate may even exceed that imputed in its share price, while ABC-Value may fail to deliver the steady earnings it has produced so far. What determines the value of a stock is not only its current earnings, but also the earnings growth.

None of these factors remain constant and there are many less tangible ones management, state of industry, economy, etc. Currencies are commodities like gold, oil, or wheat. Normally, the price of commodities is quoted in terms of a monetary unit per a commonly used quantity of commodity, like bushel, barrel, or metric ton.

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The other feature distinguishing currencies from other commodities is that we often want to know the cross-ratio. We are rarely interested in how many barrels of oil 20 bushels of wheat can buy.


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However, when returning to the U. Let us review quote conventions and some potential issues. In what follows, we use the easily recognizable three-letter currency codes as adopted by the payment clearing system SWIFT.

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Most currencies around the world are quoted with respect to a vehicle currency, which is one of the major hard currencies e. A currency can be quoted in European terms i. Most others e. Remembering this is only important when observing quotes or percentage appreciation rates that are not labeled; this text follows the notation in which a spot foreign exchange rate X is labeled with the terms in square brackets or as a superscript.

In the same way, the price of 1. For any two currencies, the FX rate in currency 1 per unit of currency 2 uniquely determines the FX rate in currency 2 per unit of currency 1 i. Often, the quotation units follow a convention and are dropped.

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