Conjuncted: Bank Recapitalization – Some Scattered Thoughts on Efficacies.

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In response to this article by Joe.

Some scattered thoughts could be found here.

With demonetization, banks got a surplus liquidity to the tune of Rs. 4 trillion which was largely responsible for call rates becoming tepid. However, there was no commensurate demand for credit as most corporates with a good credit rating managed to raise funds in the bond market at much lower yields. The result was that banks ended up investing most of this liquidity in government securities resulting in the Statutory Liquidity Ratio (SLR) bond holdings of banks exceeding the minimum requirement by up to 700 basis points. This combination of a surfeit of liquidity and weak credit demand can be used to design a recapitalization bond to address the capital problem. Since the banks are anyways sitting on surplus liquidity and investing in G-Secs, recapitalization bonds can be used to convert the bank liquidity to actually recapitalize the banks. Firstly, the government of India, through the RBI, will issue Recapitalization Bonds. Banks, who are sitting on surplus liquidity, will use their resources to invest in these recapitalization bonds. With the funds raised by the government through the issue of recapitalization bonds, the government will infuse capital into the stressed banks. This way, the surplus liquidity of the banks will be used more effectively and in the process the banks will also be better capitalized and now become capable of expanding their asset books as well as negotiating with stressed clients for haircuts. Recapitalization bonds are nothing new and have been used by the RBI in the past. In fact, the former RBI governor, Dr. Y V Reddy, continues to be one of the major proponents of recapitalization bonds in the current juncture. More so, considering that the capital adequacy ratio of Indian banks could dip as low as 11% by March 2018 if the macroeconomic conditions worsen, the motivation for going in for recap bonds has no logical counters. As I have often said this in many a fora, when banks talk numbers, transparency and accountability the way it is perceived isn’t how it is perceived by them, and moreover this argument gets diluted a bit in the wake of demonetization, which has still been haunted by lack of credit demand. As far as the NPAs are concerned, these were lying dormant and thanks to RBI’s AQR, these would not even have surfaced if let be made decisions about by the banks’ free hands. So, RBI’s intervention was a must to recognize NPAs rather than the political will of merely considering them as stressed assets. The real problem with recap bonds lie in the fact that the earlier such exercise in the 90s has still resulted in bonds maturing, and unless, these bonds are made tradable, these would be confined to further immaturities.

Financial Fragility in the Margins. Thought of the Day 114.0

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If micro-economic crisis is caused by the draining of liquidity from an individual company (or household), macro-economic crisis or instability, in the sense of a reduction in the level of activity in the economy as a whole, is usually associated with an involuntary outflow of funds from companies (or households) as a whole. Macro-economic instability is a ‘real’ economic phenomenon, rather than a monetary contrivance, the sense in which it is used, for example, by the International Monetary Fund to mean price inflation in the non-financial economy. Neo-classical economics has a methodological predilection for attributing all changes in economic activity to relative price changes, specifically the price changes that undoubtedly accompany economic fluctuations. But there is sufficient evidence to indicate that falls in economic activity follow outflows of liquidity from the industrial and commercial company sector. Such outflows then lead to the deflation of economic activity that is the signal feature of economic recession and depression.

Let us start with a consideration of how vulnerable financial futures market themselves are to illiquidity, since this would indicate whether the firms operating in the market are ever likely to need to realize claims elsewhere in order to meet their liabilities to the market. Paradoxically, the very high level of intra-broker trading is a safety mechanism for the market, since it raises the velocity of circulation of whatever liquidity there is in the market: traders with liabilities outside the market are much more likely to have claims against other traders to set against those claims. This may be illustrated by considering the most extreme case of a futures market dominated by intra-broker trading, namely a market in which there are only two dealers who buy and sell financial futures contracts only between each other as rentiers, in other words for a profit which may include their premium or commission. On the expiry date of the contracts, conventionally set at three-monthly intervals in actual financial futures markets, some of these contracts will be profitable, some will be loss-making. Margin trading, however, requires all the profitable contracts to be fully paid up in order for their profit to be realized. The trader whose contracts are on balance profitable therefore cannot realize his profits until he has paid up his contracts with the other broker. The other broker will return the money in paying up his contracts, leaving only his losses to be raised by an inflow of money. Thus the only net inflow of money that is required is the amount of profit (or loss) made by the traders. However, an accommodating gross inflow is needed in the first instance in order to make the initial margin payments and settle contracts so that the net profit or loss may be realized.

The existence of more traders, and the system for avoiding counterparty risk commonly found in most futures market, whereby contracts are made with a central clearing house, introduce sequencing complications which may cause problems: having a central clearing house avoids the possibility that one trader’s default will cause other traders to default on their obligations. But it also denies traders the facility of giving each other credit, and thereby reduces the velocity of circulation of whatever liquidity is in the market. Having to pay all obligations in full to the central clearing house increases the money (or gross inflow) that broking firms and investors have to put into the market as margin payments or on settlement days. This increases the risk that a firm with large net liabilities in the financial futures market will be obliged to realize assets in other markets to meet those liabilities. In this way, the integrity of the market is protected by increasing the effective obligations of all traders, at the expense of potentially unsettling claims on other markets.

This risk is enhanced by the trading of rentiers, or banks and entrepreneurs operating as rentiers, hedging their futures contracts in other financial markets. However, while such incidents generate considerable excitement around the markets at the time of their occurrence, there is little evidence that they could cause involuntary outflows from the corporate sector on such a scale as to produce recession in the real economy. This is because financial futures are still used by few industrial and commercial companies, and their demand for financial derivatives instruments is limited by the relative expense of these instruments and their own exposure to changes in financial parameters (which may more easily be accommodated by holding appropriate stocks of liquid assets, i.e., liquidity preference). Therefore, the future of financial futures depends largely on the interest in them of the contemporary rentiers in pension, insurance and various other forms of investment funds. Their interest, in turn, depends on how those funds approach their ‘maturity’.

However, the decline of pension fund surpluses poses important problems for the main securities markets of the world where insurance and pension funds are now the dominant investors, as well as for more peripheral markets like emerging markets, venture capital and financial futures. A contraction in the net cash inflow of investment funds will be reflected in a reduction in the funds that they are investing, and a greater need to realize assets when a change in investment strategy is undertaken. In the main securities markets of the world, a reduction in the ‘new money’ that pension and insurance funds are putting into those securities markets will slow down the rate of growth of the prices in those markets. How such a fall in the institutions’ net cash inflow will affect the more marginal markets, such as emerging markets, venture capital and financial futures, depends on how institutional portfolios are managed in the period of declining net contributions inflows.

In general, investment managers in their own firms, or as employees of merchant or investment banks, compete to manage institutions’ funds. Such competition is likely to increase as investment funds approach ‘maturity’, i.e., as their cash outflows to investors, pensioners or insurance policyholders, rises faster than their cash inflow from contributions and premiums, so that there are less additional funds to be managed. In principle, this should not affect financial futures markets, in the first instance, since, as argued above, the short-term nature of their instruments and the large proportion in their business of intra-market trade makes them much less dependent on institutional cash inflows. However, this does not mean that they would be unaffected by changes in the portfolio preferences of investment funds in response to lower returns from the main securities markets. Such lower returns make financial investments like financial futures, venture capital and emerging markets, which are more marginal because they are so hazardous, more attractive to normally conservative fund managers. Investment funds typically put out sections of portfolios to specialist fund managers who are awarded contracts to manage a section according to the soundness of their reputation and the returns that they have made hitherto in portfolios under their management. A specialist fund manager reporting high, but not abnormal, profits in a fund devoted to financial futures, is likely to attract correspondingly more funds to manage when returns are lower in the main markets’ securities, even if other investors in financial futures experienced large losses. In this way, the maturing of investment funds could cause an increased inflow of rentier funds into financial futures markets.

An inflow of funds into a financial market entails an increase in liabilities to the rentiers outside the market supplying those funds. Even if profits made in the market as a whole also increase, so too will losses. While brokers commonly seek to hedge their positions within the futures market, rentiers have much greater possibilities of hedging their contracts in another market, where they have assets. An inflow into futures markets means that on any settlement day there will therefore be larger net outstanding claims against individual banks or investment funds in respect of their financial derivatives contracts. With margin trading, much larger gross financial inflows into financial futures markets will be required to settle maturing contracts. Some proportion of this will require the sale of securities in other markets. But if liquidity in integrated cash markets for securities is reduced by declining net inflows into pension funds, a failure to meet settlement obligations in futures markets is the alternative to forced liquidation of other assets. In this way futures markets will become more fragile.

Moreover, because of the hazardous nature of financial futures, high returns for an individual firm are difficult to sustain. Disappointment is more likely to be followed by the transfer of funds to management in some other peripheral market that shows a temporary high profit. While this should not affect capacity utilization in the futures market, because of intra-market trade, it is likely to cause much more volatile trading, and an increase in the pace at which new instruments are introduced (to attract investors) and fall into disuse. Pension funds whose returns fall below those required to meet future liabilities because of such instability would normally be required to obtain additional contributions from employers and employees. The resulting drain on the liquidity of the companies affected would cause a reduction in their fixed capital investment. This would be a plausible mechanism for transmitting fragility in the financial system into full-scale decline in the real economy.

The proliferation of financial futures markets has only had been marginally successful in substituting futures contracts for Keynesian liquidity preference as a means of accommodating uncertainty. A closer look at the agents in those markets and their market mechanisms indicates that the price system in them is flawed and trading hazardous risks in them adds to uncertainty rather than reducing it. The hedging of financial futures contracts in other financial markets means that the resulting forced liquidations elsewhere in the financial system are a real source of financial instability that is likely to worsen as slower growth in stock markets makes speculative financial investments appear more attractive. Capital-adequacy regulations are unlikely to reduce such instability, and may even increase it by increasing the capital committed to trading in financial futures. Such regulations can also create an atmosphere of financial security around these markets that may increase unstable speculative flows of liquidity into the markets. For the economy as a whole, the real problems are posed by the involvement of non-financial companies in financial futures markets. With the exception of a few spectacular scandals, non-financial companies have been wary of using financial futures, and it is important that they should continue to limit their interest in financial futures markets. Industrial and commercial companies, which generate their own liquidity through trade and production and hence have more limited financial assets to realize in order to meet financial futures liabilities in times of distress, are more vulnerable to unexpected outflows of liquidity in proportion to their increased exposure to financial markets. The liquidity which they need to set aside to meet such unexpected liabilities inevitably means a reduced commitment to investment in fixed capital and new technology.

Banking and Lending/Investment. How Monetary Policy Becomes Decisive? Some Branching Rumination.

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Among the most notoriously pernicious effects of asset price inflation is that it offers speculators the prospect of gain in excess of the costs of borrowing the money to buy the asset whose price is being inflated. This is how many unstable Ponzi financing structures begin. There are usually strict regulations to prevent or limit banks’ direct investment in financial instruments without any assured residual liquidity, such as equity or common stocks. However, it is less easy to prevent banks from lending to speculative investors, who then use the proceeds of their loans to buy securities or to limit lending secured on financial assets. As long as asset markets are being inflated, such credit expansions also conceal from banks, their shareholders and their regulators the disintermediation that occurs when the banks’ best borrowers, governments and large companies, use bills and company paper instead of bank loans for their short-term financing. As long as the boom proceeds, banks can enjoy the delusion that they can replace the business of governments and large companies with good lending secured on stocks.

In addition to undermining the solvency of the banking system, and distracting commerce and industry with the possibilities of lucrative corporate restructuring, capital market inflation also tends to make monetary policy ineffective. Monetary policy is principally the fixing of reserve requirements, buying and selling short-term paper or bills in the money or inter-bank markets, buying and selling government bonds and fixing short-term interest rates. As noted in the previous section, with capital market inflation there has been a proliferation of short-term financial assets traded in the money markets, as large companies and banks find it cheaper to issue their own paper than to borrow for banks. This disintermediation has extended the range of short-term liquid assets which banks may hold. As a result of this it is no longer possible for central banks, in countries experiencing capital market inflation, to control the overall amount of credit available in the economy: attempts to squeeze the liquidity of banks in order to limit their credit advances by, say, open market operations (selling government bonds) are frustrated by the ease with which banks may restore their liquidity by selling bonds or their holdings of short-term paper or bills. In this situation central banks have been forced to reduce the scope of their monetary policy to the setting of short-term interest rates.

Economists have long believed that monetary policy is effective in controlling price inflation in the economy at large, as opposed to inflation of securities prices. Various rationalizations have been advanced for this efficacy of monetary policy. For the most part they suppose some automatic causal connection between changes in the quantity of money in circulation and changes in prices, although the Austrian School of Economists (here, here, here, and here) tended on occasion to see the connection as being between changes in the rate of interest and changes in prices.

Whatever effect changes in the rate of interest may have on the aggregate of money circulating in the economy, the effect of such changes on prices has to be through the way in which an increase or decrease in the rate of interest causes alterations in expenditure in the economy. Businesses and households are usually hard-headed enough to decide their expenditure and financial commitments in the light of their nominal revenues and cash outflows, which may form their expectations, rather than in accordance with their expectations or optimizing calculations. If the same amount of money continues to be spent in the economy, then there is no effective reason for the business-people setting prices to vary prices. Only if expenditure in markets is rising or falling would retailers and industrialists consider increasing or decreasing prices. Because price expectations are observable directly with difficulty, they may explain everything in general and therefore lack precision in explaining anything in particular. Notwithstanding their effects on all sorts of expectations, interest rate changes affect inflation directly through their effects on expenditure.

The principal expenditure effects of changes in interest rates occur among net debtors in the economy, i.e., economic units whose financial liabilities exceed their financial assets. This is in contrast to net creditors, whose financial assets exceed their liabilities, and who are usually wealthy enough not to have their spending influenced by changes in interest rates. If they do not have sufficient liquid savings out of which to pay the increase in their debt service payments, then net debtors have their expenditure squeezed by having to devote more of their income to debt service payments. The principal net debtors are governments, households with mortgages and companies with large bank loans.

With or without capital market inflation, higher interest rates have never constrained government spending because of the ease with which governments may issue debt. In the case of indebted companies, the degree to which their expenditure is constrained by higher interest rates depends on their degree of indebtedness, the available facilities for additional financing and the liquidity of their assets. As a consequence of capital market inflation, larger companies reduce their borrowing from banks because it becomes cheaper and more convenient to raise even short- term finance in the booming securities markets. This then makes the expenditure of even indebted companies less immediately affected by changes in bank interest rates, because general changes in interest rates cannot affect the rate of discount or interest paid on securities already issued. Increases in short-term interest rates to reduce general price inflation can then be easily evaded by companies financing themselves by issuing longer-term securities, whose interest rates tend to be more stable. Furthermore, with capital market inflation, companies are more likely to be over-capitalized and have excessive financial liabilities, against which companies tend to hold a larger stock of more liquid assets. As inflated financial markets have become more unstable, this has further increased the liquidity preference of large companies. This excess liquidity enables the companies enjoying it to gain higher interest income to offset the higher cost of their borrowing and to maintain their planned spending. Larger companies, with access to capital markets, can afford to issue securities to replenish their liquid reserves.

If capital market inflation reduces the effectiveness of monetary policy against product price inflation, because of the reduced borrowing of companies and the ability of booming asset markets to absorb large quantities of bank credit, interest rate increases have appeared effective in puncturing asset market bubbles in general and capital market inflations in particular. Whether interest rate rises actually can effect an end to capital market inflation depends on how such rises actually affect the capital market. In asset markets, as with anti-inflationary policy in the rest of the economy, such increases are effective when they squeeze the liquidity of indebted economic units by increasing the outflow of cash needed to service debt payments and by discouraging further speculative borrowing. However, they can only be effective in this way if the credit being used to inflate the capital market is short term or is at variable rates of interest determined by the short-term rate.

Keynes’s speculative demand for money is the liquidity preference or demand for short-term securities of rentiers in relation to the yield on long-term securities. Keynes’s speculative motive is ‘a continuous response to gradual changes in the rate of interest’ in which, as interest rates along the whole maturity spectrum decline, there is a shift in rentiers’ portfolio preference toward more liquid assets. Keynes clearly equated a rise in equity (common stock) prices with just such a fall in interest rates. With falling interest rates, the increasing preference of rentiers for short-term financial assets could keep the capital market from excessive inflation.

But the relationship between rates of interest, capital market inflation and liquidity preference is somewhat more complicated. In reality, investors hold liquid assets not only for liquidity, which gives them the option to buy higher-yielding longer-term stocks when their prices fall, but also for yield. This marginalizes Keynes’s speculative motive for liquidity. The motive was based on Keynes’s distinction between what he called ‘speculation’ (investment for capital gain) and ‘enterprise’ (investment long term for income). In our times, the modern rentier is the fund manager investing long term on behalf of pension and insurance funds and competing for returns against other funds managers. An inflow into the capital markets in excess of the financing requirements of firms and governments results in rising prices and turnover of stock. This higher turnover means greater liquidity so that, as long as the capital market is being inflated, the speculative motive for liquidity is more easily satisfied in the market for long-term securities.

Furthermore, capital market inflation adds a premium of expected inflation, or prospective capital gain, to the yield on long-term financial instruments. Hence when the yield decreases, due to an increase in the securities’ market or actual price, the prospective capital gain will not fall in the face of this capital appreciation, but may even increase if it is large or abrupt. Rising short-term interest rates will therefore fail to induce a shift in the liquidity preference of rentiers towards short-term instruments until the central bank pushes these rates of interest above the sum of the prospective capital gain and the market yield on long-term stocks. Only at this point will there be a shift in investors’ preferences, causing capital market inflation to cease, or bursting an asset bubble.

This suggests a new financial instability hypothesis, albeit one that is more modest and more limited in scope and consequence than Minsky’s Financial Instability Hypothesis. During an economic boom, capital market inflation adds a premium of expected capital gain to the market yield on long-term stocks. As long as this yield plus the expected capital gain exceed the rate of interest on short-term securities set by the central bank’s monetary policy, rising short-term interest rates will have no effect on the inflow of funds into the capital market and, if this inflow is greater than the financing requirements of firms and governments, the resulting capital market inflation. Only when the short-term rate of interest exceeds the threshold set by the sum of the prospective capital gain and the yield on long-term stocks will there be a shift in rentiers’ preferences. The increase in liquidity preference will reduce the inflow of funds into the capital market. As the rise in stock prices moderates, the prospective capital gain gets smaller, and may even become negative. The rentiers’ liquidity preference increases further and eventually the stock market crashes, or ceases to be active in stocks of longer maturities.

At this point, the minimal or negative prospective capital gain makes equity or common stocks unattractive to rentiers at any positive yield, until the rate of interest on short-term securities falls below the sum of the prospective capital gain and the market yield on those stocks. When the short-term rate of interest does fall below this threshold, the resulting reduction in rentiers’ liquidity preference revives the capital market. Thus, in between the bursting of speculative bubbles and the resurrection of a dormant capital market, monetary policy has little effect on capital market inflation. Hence it is a poor regulator for ‘squeezing out inflationary expectations’ in the capital market.

Convertible Arbitrage. Thought of the Day 108.0

A convertible bond can be thought of as a fixed income security that has an embedded equity call option. The convertible investor has the right, but not the obligation, to convert (exchange) the bond into a predetermined number of common shares. The investor will presumably convert sometime at or before the maturity of the bond if the value of the common shares exceeds the cash redemption value of the bond. The convertible therefore has both debt and equity characteristics and, as a result, provides an asymmetrical risk and return profile. Until the investor converts the bond into common shares of the issuer, the issuer is obligated to pay a fixed coupon to the investor and repay the bond at maturity if conversion never occurs. A convertible’s price is sensitive to, among other things, changes in market interest rates, credit risk of the issuer, and the issuer’s common share price and share price volatility.

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Analysis of convertible bond prices factors in three different sources of value: investment value, conversion value, and option value. The investment value is the theoretical value at which the bond would trade if it were not convertible. This represents the security’s floor value, or minimum price at which it should trade as a nonconvertible bond. The conversion value represents the value of the common stock into which the bond can be converted. If, for example, these shares are trading at $30 and the bond can convert into 100 shares, the conversion value is $3,000. The investment value and conversion value can be considered, at maturity, the low and high price boundaries for the convertible bond. The option value represents the theoretical value of having the right, but not the obligation, to convert the bond into common shares. Until maturity, a convertible trades at a price between the investment value and the option value.

A Black-Scholes option pricing model, in combination with a bond valuation model, can be used to price a convertible security. However, a binomial option model, with some adjustments, is the best method for determining the value of a convertible security. Convertible arbitrage is a market-neutral investment strategy that involves the simultaneous purchase of convertible securities and the short sale of common shares (selling borrowed stock) that underlie the convertible. An investor attempts to exploit inefficiencies in the pricing of the convertible in relation to the security’s embedded call option on the convertible issuer’s common stock. In addition, there are cash flows associated with the arbitrage position that combine with the security’s inefficient pricing to create favorable returns to an investor who is able to properly manage a hedge position through a dynamic hedging process. The hedge involves selling short a percentage of the shares that the convertible can convert into based on the change in the convertible’s price with respect to the change in the underlying common stock price (delta) and the change in delta with respect to the change in the underlying common stock (gamma). The short position must be adjusted frequently in an attempt to neutralize the impact of changing common share prices during the life of the convertible security. This process of managing the short position in the issuer’s stock is called “delta hedging.”

If hedging is done properly, whenever the convertible issuer’s common share price decreases, the gain from the short stock position should exceed the loss from the convertible holding. Equally, whenever the issuer’s common share price increases, the gain from the convertible holding should exceed the loss from the short stock position. In addition to the returns produced by delta hedging, the investor will receive returns from the convertible’s coupon payment and interest income associated with the short stock sale. However, this cash flow is reduced by paying a cash amount to stock lenders equal to the dividend the lenders would have received if the stock were not loaned to the convertible investor, and further reduced by stock borrow costs paid to a prime broker. In addition, if the investor leverages the investment by borrowing cash from a prime broker, there will be interest expense on the loan. Finally, if an investor chooses to hedge credit risk of the issuer, or interest rate risk, there will be additional costs associated with credit default swaps and a short Treasury position. This strategy attempts to create returns that exceed the returns that would be available from purchasing a nonconverting bond with the same maturity issued by the same issuer, without being exposed to common share price risk. Most convertible arbitrageurs attempt to achieve double-digit annual returns from convertible arbitrage.

Delta Hedging.

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The principal investors in most convertible securities are hedge funds that engage in convertible arbitrage strategies. These investors typically purchase the convertible and simultaneously sell short a certain number of the issuer’s common shares that underlie the convertible. The number of shares they sell short as a percent of the shares underlying the convertible is approximately equal to the risk-neutral probability at that point in time (as determined by a convertible pricing model that uses binomial option pricing as its foundation) that the investor will eventually convert the security into common shares. This probability is then applied to the number of common shares the convertible security could convert into to determine the number of shares the hedge fund investor should sell short (the “hedge ratio”).

As an example, assume a company’s share price is $10 at the time of its convertible issuance. A hedge fund purchases a portion of the convertible, which gives the right to convert into 100 common shares of the issuer. If the hedge ratio is 65%, the hedge fund may sell short 65 shares of the issuer’s stock on the same date as the convertible purchase. During the life span of the convertible, the hedge fund investor may sell more shares short or buy shares, based on the changing hedge ratio. To illustrate, if one month after purchasing the convertible (and establishing a 65-share short position) the issuer’s share price decreases to $9, the hedge ratio may drop from 65 to 60%. To align the hedge ratio with the shares sold short as a percent of shares the investor has the right to convert the security into, the hedge fund investor will need to buy five shares in the open market from other shareholders and deliver those shares to the parties who had lent the shares originally. “Covering” five shares of their short position leaves the hedge fund with a new short position of 60 shares. If the issuer’s share price two months after issuance increases to $11, the hedge ratio may increase to 70%. In this case, the hedge fund investor may want to be short 70 shares. The investor achieves this position by borrowing 10 more shares and selling them short, which increases the short position from 60 to 70 shares. This process of buying low and selling high continues until the convertible either converts or matures.

The end result is that the hedge fund investor is generating trading profits throughout the life of the convertible by buying stock to reduce the short position when the issuer’s share price drops, and borrowing and selling shares short when the issuer’s share price increases. This dynamic trading process is called “delta hedging,” which is a well-known and consistently practiced strategy by hedge funds. Since hedge funds typically purchase between 60% and 80% of most convertible securities in the public markets, a significant amount of trading in the issuer’s stock takes place throughout the life of a convertible security. The purpose of all this trading in the convertible issuer’s common stock is to hedge share price risk embedded in the convertible and create trading profits that offset the opportunity cost of purchasing a convertible that has a coupon that is substantially lower than a straight bond from the same issuer with the same maturity.

In order for hedge funds to invest in convertible securities, there needs to be a substantial amount of the issuer’s common shares available for hedge funds to borrow, and adequate liquidity in the issuer’s stock for hedge funds to buy and sell shares in relation to their delta hedging activity. If there are insufficient shares available to be borrowed or inadequate trading volume in the issuer’s stock, a prospective issuer is generally discouraged from issuing a convertible security in the public markets, or is required to issue a smaller convertible, because hedge funds may not be able to participate. Alternatively, an issuer could attempt to privately place a convertible with a single non-hedge fund investor. However, it may be impossible to find such an investor, and even if found, the required pricing for the convertible is likely to be disadvantageous for the issuer.

When a new convertible security is priced in the public capital markets, it is generally the case that the terms of the security imply a theoretical value of between 102% and 105% of face value, based on a convertible pricing model. The convertible is usually sold at a price of 100% to investors, and is therefore underpriced compared to its theoretical value. This practice provides an incentive for hedge funds to purchase the security, knowing that, by delta hedging their investment, they should be able to extract trading profits at least equal to the difference between the theoretical value and “par” (100%). For a public market convertible with atypical characteristics (e.g., an oversized issuance relative to market capitalization, an issuer with limited stock trading volume, or an issuer with limited stock borrow availability), hedge fund investors normally require an even higher theoretical value (relative to par) as an inducement to invest.

Convertible pricing models incorporate binomial trees to determine the theoretical value of convertible securities. These models consider the following factors that influence the theoretical value: current common stock price; anticipated volatility of the common stock return during the life of the convertible security; risk-free interest rate; the company’s stock borrow cost and common stock dividend yield; the company’s credit risk; maturity of the convertible security; and the convertible security’s coupon or dividend rate and payment frequency, conversion premium, and length of call protection.

Synthetic Structured Financial Instruments. Note Quote.

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An option is common form of a derivative. It’s a contract, or a provision of a contract, that gives one party (the option holder) the right, but not the obligation to perform a specified transaction with another party (the option issuer or option writer) according to specified terms. Options can be embedded into many kinds of contracts. For example, a corporation might issue a bond with an option that will allow the company to buy the bonds back in ten years at a set price. Standalone options trade on exchanges or Over The Counter (OTC). They are linked to a variety of underlying assets. Most exchange-traded options have stocks as their underlying asset but OTC-traded options have a huge variety of underlying assets (bonds, currencies, commodities, swaps, or baskets of assets). There are two main types of options: calls and puts:

  • Call options provide the holder the right (but not the obligation) to purchase an underlying asset at a specified price (the strike price), for a certain period of time. If the stock fails to meet the strike price before the expiration date, the option expires and becomes worthless. Investors buy calls when they think the share price of the underlying security will rise or sell a call if they think it will fall. Selling an option is also referred to as ”writing” an option.
  • Put options give the holder the right to sell an underlying asset at a specified price (the strike price). The seller (or writer) of the put option is obligated to buy the stock at the strike price. Put options can be exercised at any time before the option expires. Investors buy puts if they think the share price of the underlying stock will fall, or sell one if they think it will rise. Put buyers – those who hold a “long” – put are either speculative buyers looking for leverage or “insurance” buyers who want to protect their long positions in a stock for the period of time covered by the option. Put sellers hold a “short” expecting the market to move upward (or at least stay stable) A worst-case scenario for a put seller is a downward market turn. The maximum profit is limited to the put premium received and is achieved when the price of the underlyer is at or above the option’s strike price at expiration. The maximum loss is unlimited for an uncovered put writer.

Coupon is the annual interest rate paid on a bond, expressed as percentage of the face value.

Coupon rate or nominal yield = annual payments ÷ face value of the bond

Current yield = annual payments ÷ market value of the bond

The reason for these terms to be briefed here through their definitions from investopedia lies in the fact that these happen to be pillars of synthetic financial instruments, to which we now take a detour.

According to the International Financial Reporting Standards (IFRS), a synthetic instrument is a financial product designed, acquired, and held to emulate the characteristics of another instrument. For example, such is the case of a floating-rate long-term debt combined with an interest rate swap. This involves

  • Receiving floating payments
  • Making fixed payments, thereby synthesizing a fixed-rate long-term debt

Another example of a synthetic is the output of an option strategy followed by dealers who are selling synthetic futures for a commodity that they hold by using a combination of put and call options. By simultaneously buying a put option in a given commodity, say, gold, and selling the corresponding call option, a trader can construct a position analogous to a short sale in the commodity’s futures market.

Because the synthetic short sale seeks to take advantage of price disparities between call and put options, it tends to be more profitable when call premiums are greater than comparable put premiums. For example, the holder of a synthetic short future will profit if gold prices decrease and incur losses if gold prices increase.

By analogy, a long position in a given commodity’s call option combined with a short sale of the same commodity’s futures creates price protection that is similar to that gained through purchasing put options. A synthetic put seeks to capitalize on disparities between call and put premiums.

Basically, synthetic products are covered options and certificates characterized by identical or similar profit and loss structures when compared with traditional financial instruments, such as equities or bonds. Basket certificates in equities are based on a specific number of selected stocks.

A covered option involves the purchase of an underlying asset, such as equity, bond, currency, or other commodity, and the writing of a call option on that same asset. The writer is paid a premium, which limits his or her loss in the event of a fall in the market value of the underlying. However, his or her potential return from any increase in the asset’s market value is conditioned by gains limited by the option’s strike price.

The concept underpinning synthetic covered options is that of duplicating traditional covered options, which can be achieved by both purchase of the underlying asset and writing of the call option. The purchase price of such a product is that of the underlying, less the premium received for the sale of the call option.

Moreover, synthetic covered options do not contain a hedge against losses in market value of the underlying. A hedge might be emulated by writing a call option or by calculating the return from the sale of a call option into the product price. The option premium, however, tends to limit possible losses in the market value of the underlying.

Alternatively, a synthetic financial instrument is done through a certificate that accords a right, based on either a number of underlyings or on having a value derived from several indicators. This presents a sense of diversification over a range of risk factors. The main types are

  • Index certificates
  • Region certificates
  • Basket certificates

By being based on an official index, index certificates reflect a given market’s behavior. Region certificates are derived from a number of indexes or companies from a given region, usually involving developing countries. Basket certificates are derived from a selection of companies active in a certain industry sector.

An investment in index, region, or basket certificates fundamentally involves the same level of potential loss as a direct investment in the corresponding assets themselves. Their relative advantage is diversification within a given specified range; but risk is not eliminated. Moreover, certificates also carry credit risk associated to the issuer.

Also available in the market are compound financial instruments, a frequently encountered form being that of a debt product with an embedded conversion option. An example of a compound financial instrument is a bond that is convertible into ordinary shares of the issuer. As an accounting standard, the IFRS requires the issuer of such a financial instrument to present separately on the balance sheet the

  • Equity component
  • Liability component

On initial recognition, the fair value of the liability component is the present value of the contractually determined stream of future cash flows, discounted at the rate of interest applied at that time by the market to substantially similar cash flows. These should be characterized by practically the same terms, albeit without a conversion option. The fair value of the option comprises its

  • Time value
  • Intrinsic value (if any)

The IFRS requires that on conversion of a convertible instrument at maturity, the reporting company derecognizes the liability component and recognizes it as equity. Embedded derivatives are an interesting issue inasmuch as some contracts that themselves are not financial instruments may have financial instruments embedded in them. This is the case of a contract to purchase a commodity at a fixed price for delivery at a future date.

Contracts of this type have embedded in them a derivative that is indexed to the price of the commodity, which is essentially a derivative feature within a contract that is not a financial derivative. International Accounting Standard 39 (IAS 39) of the IFRS requires that under certain conditions an embedded derivative is separated from its host contract and treated as a derivative instrument. For instance, the IFRS specifies that each of the individual derivative instruments that together constitute a synthetic financial product represents a contractual right or obligation with its own terms and conditions. Under this perspective,

  • Each is exposed to risks that may differ from the risks to which other financial products are exposed.
  • Each may be transferred or settled separately.

Therefore, when one financial product in a synthetic instrument is an asset and another is a liability, these two do not offset each other. Consequently, they should be presented on an entity’s balance sheet on a net basis, unless they meet specific criteria outlined by the aforementioned accounting standards.

Like synthetics, structured financial products are derivatives. Many are custom-designed bonds, some of which (over the years) have presented a number of problems to their buyers and holders. This is particularly true for those investors who are not so versatile in modern complex instruments and their further-out impact.

Typically, instead of receiving a fixed coupon or principal, a person or company holding a structured note will receive an amount adjusted according to a fairly sophisticated formula. Structured instruments lack transparency; the market, however, seems to like them, the proof being that the amount of money invested in structured notes continues to increase. One of many examples of structured products is the principal exchange-rate-linked security (PERLS). These derivative instruments target changes in currency rates. They are disguised to look like bonds, by structuring them as if they were debt instruments, making it feasible for investors who are not permitted to play in currencies to place bets on the direction of exchange rates.

For instance, instead of just repaying principal, a PERLS may multiply such principal by the change in the value of the dollar against the euro; or twice the change in the value of the dollar against the Swiss franc or the British pound. The fact that this repayment is linked to the foreign exchange rate of different currencies sees to it that the investor might be receiving a lot more than an interest rate on the principal alone – but also a lot less, all the way to capital attrition. (Even capital protection notes involve capital attrition since, in certain cases, no interest is paid over their, say, five-year life cycle.)

Structured note trading is a concept that has been subject to several interpretations, depending on the time frame within which the product has been brought to the market. Many traders tend to distinguish between three different generations of structured notes. The elder, or first generation, usually consists of structured instruments based on just one index, including

  • Bull market vehicles, such as inverse floaters and cap floaters
  • Bear market instruments, which are characteristically more leveraged, an example being the superfloaters

Bear market products became popular in 1993 and 1994. A typical superfloater might pay twice the London Interbank Offered Rate (LIBOR) minus 7 percent for two years. At currently prevailing rates, this means that the superfloater has a small coupon at the beginning that improves only if the LIBOR rises. Theoretically, a coupon that is below current market levels until the LIBOR goes higher is much harder to sell than a big coupon that gets bigger every time rates drop. Still, bear plays find customers.

Second-generation structured notes are different types of exotic options; or, more precisely, they are yet more exotic than superfloaters, which are exotic enough in themselves. There exist serious risks embedded in these instruments, as such risks have never been fully appreciated. Second-generation examples are

  • Range notes, with embedded binary or digital options
  • Quanto notes, which allow investors to take a bet on, say, sterling London Interbank Offered Rates, but get paid in dollar.

There are different versions of such instruments, like you-choose range notes for a bear market. Every quarter the investor has to choose the “range,” a job that requires considerable market knowledge and skill. For instance, if the range width is set to 100 basis points, the investor has to determine at the start of the period the high and low limits within that range, which is far from being a straight job.

Surprisingly enough, there are investors who like this because sometimes they are given an option to change their mind; and they also figure their risk period is really only one quarter. In this, they are badly mistaken. In reality even for banks you-choose notes are much more difficult to hedge than regular range notes because, as very few people appreciate, the hedges are both

  • Dynamic
  • Imperfect

There are as well third-generation notes offering investors exposure to commodity or equity prices in a cross-category sense. Such notes usually appeal to a different class than fixed-income investors. For instance, third-generation notes are sometimes purchased by fund managers who are in the fixed-income market but want to diversify their exposure. In spite of the fact that the increasing sophistication and lack of transparency of structured financial instruments sees to it that they are too often misunderstood, and they are highly risky, a horde of equity-linked and commodity-linked notes are being structured and sold to investors. Examples are LIBOR floaters designed so that the coupon is “LIBOR plus”:

The pros say that flexibly structured options can be useful to sophisticated investors seeking to manage particular portfolio and trading risks. However, as a result of exposure being assumed, and also because of the likelihood that there is no secondary market, transactions in flexibly structured options are not suitable for investors who are not

  • In a position to understand the behavior of their intrinsic value
  • Financially able to bear the risks embedded in them when worst comes to worst

It is the price of novelty, customization, and flexibility offered by synthetic and structured financial instruments that can be expressed in one four-letter word: risk. Risk taking is welcome when we know how to manage our exposure, but it can be a disaster when we don’t – hence, the wisdom of learning ahead of investing the challenges posed by derivatives and how to be in charge of risk control.

Open Market Operations. Thought of the Day 93.0

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It can be argued that it would be much more democratic if the Treasuries were allowed to borrow directly from their central bank. By electing a government on a program, we would know what deficit it intends to run and thus how much it will be willing to print, which in the long run is a debate about the possible level of inflation. Instead, it has been argued that decisions made on democratic grounds might be unstable as they are affected by elections. However, the independence of central banks is also serving the interest of commercial bankers as we argue now.

In practice, the central bank buys and sells bonds in open market operations. At least it is always doing so with short term T-bonds as part of the conventional monetary policy, and it might decide sometimes to do it as well with longer maturity T-bonds as part of the unconventional monetary policy. This blurs the lines between a model where the central bank directly finances the Treasury, and a model where this is done by commercial banks since they result in the same final situation. Indeed, before an open market operation the Treasury owes central bank money to a commercial bank, and in the final situation it owes it to the central bank itself, and the central bank money held by the commercial bank has been increased accordingly.

The commercial bank has accepted to get rid of an IOU which bears interest, in exchange of a central bank IOU which bears no interest. However the Treasury will never default on its debt, because the state also runs the central bank which can buy an infinite amount of T-bonds. Said differently, if the interest rates for short term T-bonds start to increase as the commercial banks become more and more reluctant to buy these, the central bank needs to buy as many short-term bonds as necessary to ensure the short term interest rates on T-bonds remain at the targeted level. By using these open market operations a sovereign state running a sovereign currency has the means to ensure that the banks are always willing to buy T-bonds, whatever the deficit is.

However, this system has a drawback. First when the commercial bank bought the T-bond, it had to pretend that it was worried the state might never reimburse, so as to ask for interests rates which are at least slightly higher than the interest rate at which they can borrow from the central bank, and make a profit on the difference. Of course the banks knew they would always be reimbursed, because the central bank always stands ready to buy bonds. As the interest rates departed from the target chosen by the central bank, the latter bought short term bonds to prevent the short term rate from increasing. In order to convince a commercial bank to get rid of a financial instrument which is not risky and which bears interest, the only solution is to pay more than the current value of the bond, which amounts to a decrease of the interest rate on those bonds. The bank thus makes an immediate profit instead of a larger profit later. This difference goes directly into the net worth of the banker and amounts to money creation.

To conclude, we reach the same stage as if the Treasury had sold directly its bond to the central bank, except that now we have increased by a small amount the net worth of the bankers. By first selling the bonds to the commercial banks, instead of selling directly to the central bank, the bankers were able to realize a small profit. But this profit is an immediate and easy one. So they have on one side to pretend they do not like when the Treasury goes into debt, so as to be able to ask for the highest possible interest rate, and secretly enjoy it since either they make a profit when it falls due, or even better immediately if the central bank buys the bonds to control the interest rates.

The commercial banks will always end up with a part of their assets denominated directly in central bank money, which bears no interest, and T-bonds, which bear interest. If we adopt a consolidated state point of view, where we merge the Treasury and the central bank, then the commercial banks have two types of accounts. Deposits which bear no interests, and saving accounts which generate interests, just like everybody. In order to control the interest rate, the consolidated state shifts the amounts from the interest-less to the interest-bearing account and vice-versa.

Asset Backed Securities. Drunken Risibility.

Asset Backed Securities (ABS) are freely traded financial instruments that represent packages of loans issued by the commercial banks. The majority are created from mortgages, but credit card debt, commercial real estate loans, student loans, and hedge fund loans are also known to have been securitized. The earliest form of ABS within the American banking system appears to stem from the creation of the Federal National Mortgage Association (Fannie Mae) in 1938 as part of amendments to the US National Housing Act, a Great Depression measure aimed at creating loan liquidity. Fannie Mae, and the other Government Sponsored Enterprises buy loans from approved mortgage sellers, typically banks, and create guaranteed financial debt instruments from them, to be sold on the credit markets. The resulting bonds, backed as they are by loan insurance, are widely used in pension funds and insurance companies, as a secure, financial instrument providing a predictable, low risk return.

The creation of a more general form of Mortgage Backed Security is credited to Bob Dall and the trading desk of Salmon brothers in 1977 by Michael Lewis (Liar’s Poker Rising Through the Wreckage on Wall Street). Lewis also describes a rapid expansion in their sale beginning in 1981 as a side effect of the United States savings and loans crisis. The concept was extended in 1987 by bankers at Drexel Burnham Lambert Inc. to corporate bonds and loans in the form of Collateralized Debt Obligations (CDOs), which eventually came to include mortgage backed securities, and in the form of CDO-Squared instruments, pools of CDO.

Analysis of the systemic effects of Asset Backed Security has concentrated chiefly on their ability to improve the quantity of loans, or loan liquidity, which has been treated as a positive feature by Greenspan. It has also been noted that securitization allowed banks to increase their return on capital by transforming their operations into a credit generating pipeline process. Hyun Song Shin has also analysed their effect on bank leverage and the stability of the larger financial system within an accounting framework. He highlights the significance of loan supply factors in causing the sub-prime crisis. Although his model appears not to completely incorporate the full implications of the process operating within the capital reserve regulated banking system, it presents an alternate, matrix based analysis of the uncontrolled debt expansion that these instruments permit.

The systemic problem introduced by asset backed securities, or any form of sale that transfers loans made by commercial banking institutions outside the regulatory framework is that they allow banks to escape the explicit reserve and regulatory capital based regulation on the total amount of loans being issued against customer deposits. This has the effect of steadily increasing the ratio of bank originated loans to money on deposit within the banking system.

The following example demonstrates the problem using two banks, A and B. For simplicity fees related to loans and ABS sales are excluded. It is assumed that the deposit accounts are Net Transaction accounts carry a 10% reserve requirement, and that both banks are ”well capitalised” and that the risk weighted multiplier for the capital reserve for these loans is also 10.

The example proceeds as a series of interactions as money flows between the two banks. The liabilities (deposits) and assets (loans) are shown, with loans being separated into bank loans, and Mortgage Backed Securities (MBS), depending on their state.

Initial Conditions: To simplify Bank B is shown as having made no loans, and has excess reserves at the central bank to maintain the balance sheet. The normal inter-bank and central bank lending mechanisms would enable the bank to compensate for temporary imbalances during the process under normal conditions. All deposit money used within the example remains on deposit at either Bank A or Bank B. On the right hand side of the table the total amount of deposits and loans for both banks is shown.

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Step 1: Bank A creates a $1000 Mortgage Backed Security from the loan on its balance sheet.

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Step 2: The securitized loan is sold to the depositor at Bank B. $1000 is paid to Bank A from that depositor in payment for the loan. Bank A now has no loans outstanding against its deposits, and the securitized loan has been moved outside of banking system regulation. Note that total deposits at the two banks have temporarily shrunk due to the repayment of the loan capital at A. The actual transfer of the deposits between the banks is facilitated through the reserve holdings which also function as clearing funds.

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Step 3: As Bank A now has no loans against its deposits, and is within its regulatory capital ratios, it can make a new $1000 loan. The funds from this loan are deposited at Bank B. The sum of the deposits rises as a result as does the quantity of loans. Note that the transfer of the loan money from Bank A to Bank B again goes through the reserve holdings in the clearing system and restores the original balance at Bank B.

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Step 4: Bank A securitizes the loan made in Step 3 repeating Step 1.

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Step 5: Bank A sells the MBS to the depositor at Bank B, repeating Step 2.

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Step 6: Bank A makes a new loan which is deposited at Bank B, repeating Step 3.

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Step 7: Bank A securitizes the loan made in Step 6, repeating Step 4.

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Since the Deposit and Loan positions of the two banks are identical in all respects in Steps (1,4), (2,5), (3,6) and (4,7) the process can continue indefinitely, resulting in expansion of the total commercial bank originated loan supply independent of central bank control.

This is a simplified version of the flows between loans, deposits, and asset backed securities that occur daily in the banking system. At no point has either bank needed recourse to central bank funds, or broken any of their statutory requirements with respect to capitalisation or reserve requirements where they apply.

The problem is the implicit assumption with reserve based banking systems that bank originated loans remain within the banking system. Allowing the sale of loans to holders outside of the regulated banking system (i.e. to entities other than regulated banks) removes these loans from that control and thus creates a systemic loophole in the regulation of the commercial bank loan supply.

The introduction of loans sales has consequently created a novel situation in those modern economies that allow them, not only in causing a significant expansion in total lending from the banking sector, but also in changing the systemic relationship between the quantity of money in the system to the quantity of bank originated debt, and thereby considerably diluting the influence the central bank can exert over the loan supply. The requirement that no individual bank should lend more than their deposits has been enforced by required reserves at the central bank since the 19th century in Europe, and the early 20th century in the USA. Serendipitously, this also created a systemic limit on the ratio of money to bank originated lending within the monetary system. While the sale of Asset Backed Securities does not allow any individual bank to exceed this ratio at any given point in time, as the process evolves the banking system itself exceeds it as loans are moved outside the constraints provided by regulatory capital or reserve regulation, thereby creating a mechanism for unconstrained growth in commercial bank originated lending.

While the asset backed security problem explains the dramatic growth in banking sector debt that has occurred over the last three decades, it does not explain the accompanying growth in the money supply. Somewhat uniquely of the many regulatory challenges that the banking system has created down the centuries, the asset backed security problem, in and of itself does not cause the banks, or the banking system to ”print money”.

The question of what exactly constitutes money in modern banking systems is a non-trivial one. As the examples above show, bank loans create money in the form of bank deposits, and bank deposits can be used directly for monetary purposes either through cheques or more usually now direct electronic transfer. For economic purposes then, bank deposits can be regarded as directly equivalent to physical money. The reality within the banking system however is somewhat more complex, in that transfers between bank deposits must be performed using deposits in the clearing mechanisms, either through the reserves at the central bank, or the bank’s own asset deposits at other banks. Nominally limits on the total quantity of central bank reserves should in turn limit the growth in bank deposits from bank lending, but it is clear from the monetary statistics that this is not the case.

Individually commercial banks are limited in the amount of money they can lend. They are limited by any reserve requirements for their deposits, by the accounting framework that surrounds the precise classification of assets and liabilities within their locale, and by the ratio of their equity or regulatory capital to their outstanding, risk weighted loans as recommended by the Basel Accords. However none of these limits is sufficient to prevent uncontrolled expansion.

Reserve requirements at the central bank can only effectively limit bank deposits if they apply to all accounts in the system, and the central bank maintains control over any mechanisms that allow individual banks to increase their reserve holdings. This appears not to be the case. Basel capital restrictions can also limit bank lending. Assets (loans) held by banks are classified by type, and have accordingly different percentage capital requirements. Regulatory capital requirements are divided into two tiers of capital with different provisions and risk categorisation applying to instruments held in them. To be adequately capitalised under the Basel accords, a bank must maintain a ratio of at least 8% between its Tier 1 and Tier 2 capital reserves, and its loans. Equity capital reserves are provided by a bank’s owners and shareholders when the bank is created, and exist to provide a buffer protecting the bank’s depositors against loan defaults.

Under Basel regulation, regulatory capital can be held in a variety of instruments, depending on Tier 1 or Tier 2 status. It appears that some of those instruments, in particular subordinated debt and hybrid debt capital instruments, can represent debt issued from within the commercial banking system.

Annex A – Basel Capital Accords, Capital Elements Tier 1

(a) Paid-up share capital/common stock

(b) Disclosed reserves

Tier 2

(a) Undisclosed reserves

(b) Asset revaluation reserves

(c) General provisions/general loan-loss reserves

(d) Hybrid (debt/equity) capital instruments

(e) Subordinated debt

Subordinated debt is defined in Annex A of the Basel treaty as:

(e) Subordinated term debt: includes conventional unsecured subordinated debt capital instruments with a minimum original fixed term to maturity of over five years and limited life redeemable preference shares. During the last five years to maturity, a cumulative discount (or amortisation) factor of 20% per year will be applied to reflect the diminishing value of these instruments as a continuing source of strength. Unlike instruments included in item (d), these instruments are not normally available to participate in the losses of a bank which continues trading. For this reason these instruments will be limited to a maximum of 50% of tier 1.

This is debt issued by the bank, in various forms, but of guaranteed long duration, and controlled repayment. In effect, it allows Banks to hold borrowed money in regulatory capital. It is subordinate to the claims of depositors in the event of Bank failure. The inclusion of subordinated debt in Tier 2 allows financial instruments created from lending to become part of the regulatory control on further lending, creating a classic feedback loop. This can also occur as a second order effect if any form of regulatory capital can be purchased with money borrowed from within the banking system

Stock Hedging Loss and Risk

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A stock is supposed to be bought at time zero with price S0, and to be sold at time T with uncertain price ST. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices Ki (i = 1,2,··· ,n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget. The chosen put option is supposed to finish in-the-money at maturity, and the constraint of hedging expenditure is supposed to be binding.

Suppose the market price of the stock is S0 at time zero, the hedge ratio is h, the price of the put option is P0, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S0erT + hP0erT —– (1)

and the market price of the portfolio is

ST + h(K − Sτ)+ er(T − τ) —— (2)

therefore the loss of the portfolio is

L = S0erT + hP0erT − (ST +h(K − Sτ)+ er(T − τ)—– (3)

where x+ = max(x, 0), which is the payoff function of put option at maturity. For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures.

The mathematical model of stock price is chosen to be a geometric Brownian motion

dSt/St = μdt + σdBt —– (5)

where St is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and Bt is a standard Brownian motion. The solution of the stochastic differential equation is

St = S0 eσBt + (μ − 1/2σ2)t —– (6)

where B0 = 0, and St is lognormally distributed.

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I{X≤c1}FY(g(X) − X)] + E [I{X≥c1}FY (c2 − X)] —– (7)

where E[X] is the expectation of random variable X. I{X<c} is the index function of X such that I{X<c} = 1 when {X < c} is true, otherwise I{X<c} = 0. FY(y) is the cumulative distribution function of random variable Y, and

c1 = 1/σ [ln(k/S0) – (μ – 1/2σ2)τ]

g(X) = 1/σ [ln((S0 + hP0)erT − h(K − f(X))er(T − τ) − v)/S0 – (μ – 1/2σ2)T]

f(X) = S0 eσX + (μ−1σ2

c2 = 1/σ [ln((S0 + hP0)erT − v)/S0 – (μ – 1/2σ2)T]

X and Y are both normally distributed, where X ∼ N(0, √τ), Y ∼ N(0, √(T−τ)).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from (8) by numerical searching methods.