Index-Linked Bonds

Anne-Sofie Reng Rasmussen, Market Operations

INTRODUCTION

In the late 1990s, Danish banks began to market a new financial product, index-linked bonds^[1]. This product is comprised of a conventional bond and an option linked to the development in e.g. equities, exchange rates or commodity prices.

Despite critical press coverage^[2], index-linked bonds have become a large sales success for Danish banks. At the beginning of May 2007, the nominal outstanding volume of index-linked bonds listed on the Copenhagen Stock Exchange was almost kr. 55 billion.

The volume of index-linked bonds issued has grown considerably in recent years. At the same time, the products have become increasingly complex, and issuers have been very innovative in their choice of underlying assets. This development makes it more difficult for investors to assess what they are actually buying, and what they are paying for it. This unavoidably increases the need for investor information and guidance in connection with such investments. It is not necessarily sufficient for the investor to rely on the information material received from the seller of the index-linked bond.

INDEX-LINKED BONDS

An index-linked bond of the type discussed in this article is a package comprised of a conventional bond and an option. The bonds are usually zero-coupon bonds. Unlike a conventional bond, which has a fixed coupon yield, an option is linked to the index-linked bond, a structure that entails some tax advantages, cf. Box 1.

The final payoff to the investor consists of a guaranteed payment stemming from the zero-coupon-bond element of the package, as well as an option payoff that is dependent on the development in a pre-selected underlying asset. The option payoff is not known until the bond matures, and in the worst case may be zero.

The underlying asset can be e.g. the development in a given exchange rate, a stock index, various commodities, interest rates or different macroeconomic indicators. The product enables the investor to speculate in these relatively high-risk assets, but offers protection against any substantial depreciation in the value of the assets via the guaranteed minimum payment at maturity. The price of this protecttion, or rather of the possible gain, is the loss of the yield that the investor would have received on a conventional instrument such as a government bond.

THE ISSUANCE PROCEDURE

A special feature of index-linked bonds is that the return is linked to the development in an underlying asset, while no fixed coupon payment is received. This section describes the issuance procedure for such bonds and the roles of the parties involved.

Typically, the first step before offering an index-linked bond for sale is to select the underlying asset and determine the link between the bond yield and the development in this asset. In many cases, the issuer wishes an index-linked bond to be based on a specific theme. This might be expectations of a strengthening of the Icelandic krona vis-à-vis the Danish krone, the development in a US stock index, or a general upswing in the Asian economies. The chosen scenario determines the option element of the index-linked bond and thus the bond's specific return profile. An important issue in this context is: what determines the choice of scenario on which the index-linked bond is based? Is there a financial rationale behind the structure?

The first index-linked bonds issued in the Danish market were often linked to a single stock index or exchange rate. Issues in recent years have, however, reflected a much wider range of investment strategies. Some are linked to baskets of currency pairs, while others depend on the development in portfolios of different assets such as equities, exchange rates and commodities. The product can be structured in countless different ways. Some of the most prevalent option types are described in Box 2.

Once the provider has determined a scenario, an issuer is contacted, typically an issuer with a sound credit rating, i.e. AAA to AA. The issuer lends its name and reputation to the index-linked bonds in order to minimise the credit risk on the principal guarantee. In most cases, however, the issuer does not wish to assume the risk in relation to the speculative option position in the index-linked bond. Normally, the issuer therefore fully hedges the option risk so as to eliminate all risk in relation to the development in the underlying asset. Part of the proceeds from the issue of the index-linked bonds is thus used to purchase options, while the remainder constitutes the price of the zero-coupon bonds in the structure.

In general, the provider of the index-linked bond arranges the hedging of the option for the bond issuer. The provider contacts a number of international investment banks in order to obtain bids for the option element of the structure. The provider and issuer of the index-linked bonds do not necessarily have the capacity to calculate a fair price for the option, but rely on the competition among the international investment banks to ensure a fair price. The seller of the option makes a bid for the option price, based on the prevailing market terms. The strike price of the option is determined on the date of issue of the index-linked bond, which is when the final conditions for the index-linked bond are laid down.

The above description of the issuance process makes it clear that there are three sources of costs behind the index-linked bond: the provider, the issuer, and the seller of the option. The costs to the investor of investing in the index-linked bond thus not only com-
prise the issuance costs and the sales and marketing costs, etc. payable to the provider – which are often stated in the bond prospectus – but also the proceeds to the issuer of the bond and the seller of the option.

THE SECONDARY MARKET

Index-linked bonds are " buy-and-hold" instruments. The bonds are typically purchased on issue and held until maturity, since there is little liquidity in the secondary market for this type of product. If the investor wishes to sell the bond prematurely, the market's illiquidity can make it difficult to obtain a good price^[3].

Normally, the maximum loss to the investor is the difference between the bond's market price and its nominal value if the bond is held until maturity. If the investor wishes to sell the bond prematurely, there is no guarantee that the market price will be equal to or higher than the nominal value of the bond.

ISSUANCE VOLUME

The outstanding volume of index-linked bonds listed on the Copenhagen Stock Exchange has increased considerably in recent years, cf. Chart 1.

At the beginning of May 2007, the outstanding volume was almost kr. 55 billion. Index-linked bonds have also gained considerable ground in the rest of Europe. Particularly Germany, Italy, Spain and Belgium have seen extensive issue of structured bonds to private investors. The aggregate outstanding bond volume in these four markets was estimated at approximately kr. 3,000 billion at the beginning of 2007^[4].

THE INVESTOR'S POSITION

On the face of it, an index-linked bond seems an attractive package for investors who want some risk exposure, combined with a limited risk of loss. Via the index-linked bond, the investor in fact buys a zero-coupon bond and an option on an underlying asset. In principle, the private investor could design an equivalent product simply by buying zero-coupon bonds and options directly. However, the private investor has limited opportunity to trade in the derivatives market. Many of the related options have longer maturities and are more complex than the options available to private investors. The index-linked bonds thus give private investors access to an option market to which they have no direct access. In addition, the providers of index-linked bonds achieve certain economies of scale by buying options at a level that is not possible for individual investors.

There are, however, a number of product-specific circumstances to be taken into account. In order to take a closer look at the index-linked bonds and their payment flows, three examples of investment portfolios are presented:

• Portfolio 1: Buy an index-linked bond. Assume that the bond is issued at a premium of 4 and has a guaranteed minimum payment of its nominal value on maturity. If the nominal value of the bond is kr. 1,000, the bond costs kr. 1,040, and the guaranteed payment at maturity is kr. 1,000.
• Portfolio 2: Deposit kr. 1,000 in a non-interest-bearing bank account^[5] and invest kr. 40 in an equity portfolio. Again, the total amount invested is kr. 1,040, and the guaranteed final payment is kr. 1,000.
• Portfolio 3: Deposit kr. 1,000 in a non-interest-bearing bank account and spend kr. 40 on a lotto coupon. Once again, the total investment is kr. 1,040, and the guaranteed payment is kr. 1,000.

All three portfolios thus require an initial investment of kr. 1,040, offer a guaranteed payment of kr. 1,000 at expiry, and may yield a positive return.

It is nonetheless evident that the three investments are not identical. Chart 2 illustrates the payment profile at expiry for portfolios 2 and 3, as well as the expected underlying probability distribution.

Chart 2 (left) shows that for portfolio 2 a very high positive return is unlikely, but there is a high probability that the final payment will exceed kr. 1,000. Chart 2 (right) shows that for portfolio 3 there is very low probability of a positive return, but that an extremely high return is possible. As long as these probability distributions are transparent, the individual investor can make an informed choice based on his preferences. Things become more complicated if there is no such transparency, i.e. if the investor cannot see whether he is investing in a lotto coupon, a diversified equity portfolio, or an entirely different asset.

The main problem in relation to index-linked bonds is this lack of transparency – in relation to both the actual return profile of the index-linked bond, and the probability that the development scenario described in the prospectus and sales brochure will materialise. This is illustrated in Box 3 on p.63. For an investor to be able to decide whether the index-linked bond is a good investment, he must be able to understand the underlying expected return distribution. It would be misleading to sell the product as a sound investment with low risk if it is in fact a lotto coupon. Likewise, if the investment is merely placed in a non-interest-bearing bank account and a relatively small equity portfolio, the consumer might as well invest the funds himself and avoid the extra costs of the structured bond. The more complex and exotic the option linked to the bond is, the more difficult it becomes to understand the underlying distributions.

Another problem for the private investor is that it can be difficult to assess whether the price paid for the index-linked bond is fair in relation to the value of the underlying components. In many cases, the built-in options are not traded in an open market, and the complexity of the products makes it very difficult to calculate their theoretical value.

The non-existence of a secondary market for index-linked bonds prevents the investor from obtaining a real indicative market price for the product purchased. An investor who buys a stock or a simple coupon bond may not necessarily have insight into all the mechanisms that influence the market price. However, the investor knows that the instrument is traded in a liquid market where the market participants have this insight. The market price is thus very likely to be a good reflection of the value of the asset. The purchaser of an index-linked bond has to take the provider's word that the sales price is the right price.

It could be argued that the investor can simply contact 4-5 different providers, compare the prices of the index-linked bonds, and then choose the cheapest offer. However, the low degree of standardisation of the index-linked bonds and their prospectuses, as well as the close link between the investor's bank and the provider of the index-linked bond, who are more often than not identical, make this approach unrealistic.

ISSUANCE DOCUMENTS

The three investment portfolios presented above have a common overall structure. They are all packages comprising a relatively secure product (the zero-coupon bond in portfolio 1, and the non-interest-bearing bank accounts in portfolios 2 and 3) and a high-risk investment (the option in portfolio 1, the equity investment in portfolio 2, and the lotto coupon in portfolio 3). The difference is that in portfolios 2 and 3 it is clearly evident how large a share of the invested sum is placed in, respectively, low-risk and high-risk assets, while this is not the case for portfolio 1. Here the two products are bundled, and in principle the investor can only see the aggregate price.

One way to increase transparency could be for the index-linked-bond prospectus to state how much of the issue price is related to, respectively, the bond and the option. This would allow investors to see " how much option" they are in fact buying. In recent years, one Danish provider has begun to include this calculation in the prospectuses for the products it offers. However, this has by no means been standardised across providers. It should be noted that this approach tells the investor what the provider has paid for the option, but not whether the price is fair.

To gain an impression of the value of the option in the structure, it is necessary to consider the characteristics of the underlying asset. The information material on index-linked bonds typically includes a graph of the historical development in the underlying asset. However, this description gives only little information to the investor in terms of the valuation of the related option. As such, the historical development in the asset price has no influence on the value of an option on the asset.

The decisive factor is the volatility of the asset price. The more volatile the asset, the higher the value of the option, all other things being equal. This is because the greater volatility increases the probability of the price of the underlying asset reaching a level where the option contributes a large payoff. One possibility might therefore be to state the historical volatility of the asset compared with the implied volatility on which the price of the purchased option is based. If the implied volatility is far higher than the historical volatility, the price of the option is too high.

Comparison of the implied and historical volatilities is difficult, however, when it comes to the very complex products issued in recent years. Here, the return on the bond is often linked to one or more baskets of underlying indices, so we are no longer talking about options on single underlying assets. In this case, the decisive factor is no longer the variation in the individual asset, but just as much the covariation between the assets.

INDEX-LINKED BONDS AS INVESTMENT OBJECTS

There are two approaches to assessing index-linked bonds as investment objects. One relates to the price the investor pays for the product compared to the value of the underlying components. Is the price fair? The other relates to the quality of the product in relation to the investor's expectations of the potential return of the index-linked bonds. Below, the value of the underlying components for a range of index-linked bonds is calculated. Then the actual returns on a number of index-linked bonds are analysed in order to assess whether the bonds in question have historically been sound investments.

Is the price fair?
On the basis of a selection of the simplest products, i.e. currency-indexed bonds linked to a single exchange rate, the theoretical price of the option can be calculated. The price of the index-linked bond less the theoretical price of the option gives an implied price for the zero-coupon bond. If the investment entails no extra costs for the investor, the implied price of the zero-coupon bond will be the same as the price of an equivalent zero-coupon bond without an option. Any difference in price can be taken to be the implied premium on the theoretical value of the product paid by the investor for the index-linked bond.

Calculations based on 10 currency-indexed bonds show that the implied zero-coupon yield varies between ‑ 0.2 per cent p.a. and 1.8 per cent p.a.^[6] There is thus some variation in the implied costs related to the individual index-linked bond. On average, the implied placement yield is 2.2 percentage points lower than the annual yield to maturity on a government bond with equivalent maturity. In other words, the investor pays a substantial premium – in excess of the theoretical price of the option – for access to the option in the index-linked bond.

A survey by Stoimenov and Wilkens (2005) of equity-linked bonds issued in the German market yields similar results. In addition, it is seen that the implied premium increases with the degree of complexity in the option bundled with the index-linked bond.

The historical return
Below, the historical return on this type of product is investigated. Since the products are classified as buy-and-hold instruments, the return at redemption is considered.

Index-linked bonds were not introduced until the late 1990s and are thus relatively new in the Danish investment universe. Due to the limited issuance volume in the early years, combined with a typical life of 2-5 years, it has only been possible to perform systematic analysis of the historical return on this type of investment within the last couple of years.

A review of stock-exchange releases, prospectuses and providers' websites has identified a total of 67 index-linked bonds listed on the Copenhagen Stock Exchange that have been redeemed.^[7] The 67 bonds represent an aggregate nominal value of approximately kr. 18 billion. The average annual return weighted by the nominal value of the bonds has been 3.37 per cent. If the annual returns are also weighted by the maturities of the bonds, the average annual return is 2.47 per cent.^[8] Chart 3 presents the distribution of the annual returns.

Of the 67 bonds, 31 were redeemed with a zero or negative return. Looking at the excess return on the index-linked bonds in relation to government bonds with the same maturity, we find an average value-weighted annual excess return of 0.32 percentage point. The maturity-weighted excess return is 0.11 percentage point.

The historical excess return has thus been modest, particularly if a number of other factors are taken into account. Firstly, the credit risk on index-linked bonds is greater than on government bonds. Secondly, government bonds are far more liquid than index-linked bonds. Thirdly, a normally risk-averse investor is willing to pay a higher price for a secure return than for a very volatile return.

CONCLUDING REMARKS

Index-linked bonds give private investors access to an investment universe, the option market, that has not previously been so accessible, and in which direct investments often entail relatively high costs. Indexation of the bond payoff gives the investor access to a return profile that would otherwise be difficult to achieve. For example, inflation-indexed bonds offer investors protection against inflation.^[9]

As with all other investments, a critical approach should be taken to the individual products offered. The basic return profile for the index-linked bonds discussed in this article could, in principle, cover a very wide range of investments – from a lotto coupon to a conservative investment comprising a bank deposit combined with a very small equity investment.

The high degree of complexity inherent in the index-linked bonds offered for sale makes it difficult to assess the exact properties underlying each product. This makes it hard to distinguish the lotto coupons from the sound investments.

Calculations presented in this article show that the parties behind the index-linked bonds have historically made a good profit from the sale of these bonds. Investors, on the other hand, have incurred a risk that is higher than on investment in e.g. government bonds, without being rewarded with higher average returns.

PROBABILITY DISTRIBUTIONS AND PAYMENT PROFILES FOR INDEX-LINKED BONDS	Box 3
An index-linked bond may be constructed with countless payment profiles and underlying assets. The fundamental properties of the bond can be investigated by considering one of the simplest structures, an equity-indexed bond linked to a single equity. Assume a 2-year bond with a payment at maturity of kr. 1,000, plus a positive return equivalent to 80 per cent of any percentage increase in the underlying equity. The initial price of the equity is assumed to be kr. 40. If the price increases by 20 per cent, to kr. 48, the bond owner will receive kr. 1,000 + 0.8 * 0.2 * kr. 1,000 = kr. 1,160 at maturity. This is a zero-coupon bond with a plain vanilla option. At maturity of the bond, the structure can be illustrated as shown in Chart 4.
EQUITY-INDEXED BOND	Chart 4
Note: It is assumed that the price of the equity follows a geometrical Brownian motion with a drift rate of 4 per cent p.a. and annual volatility of 14.5 per cent. For simplicity, it is assumed that no dividend is paid.
On comparison with a portfolio comprising a deposit in a non-interest-bearing bank account and a small equity investment, cf. portfolio 2 in Chart 2, it is seen that the index-linked bond gives a lower return at maturity if the price of the underlying stock is below 40 on the maturity date. On the other hand, a relatively higher return is achieved if the equity price increases. If the full amount of kr. 1,040 is invested in equities, an even higher return is achieved if the equity price rises, but the payment is lower than kr. 1,000 if the price falls. Individual preferences govern the choice of investment strategy. The above illustration is based on a very simple index-linked bond. Even so, it can be difficult to understand the structure's exact probability distribution and return profile. For most private investors this type of structure will be impossible to price. The index-linked bonds are often linked to other asset classes than equities, and the built-in option, cf. Box 2, may have special characteristics that influence both the probability distribution and the future return. Chart 5 illustrates the possible consequences. The index-linked bond may be linked to scenarios where the probability of achieving a positive return on the strategy is in fact very small, as illustrated by alternative 1 in Chart 5 (left). If this is compared with the original scenario, indicated by the full blue line, it is seen that for alternative 1 the probability mass is much smaller in the area where the strategy gives a positive return. A bond with this probability distribution is worth less than the original structure. This could be the case if the selected scenario is driven by a speculative strategy rather than by financial rationales.
ALTERNATIVE PROBABILITY DISTRIBUTIONS AND PAYMENT PROFILES	Chart 5
There may also be strategies where the variation in the future value of the underlying index is low. This is illustrated by alternative 2 in Chart 5 (left). This structure may e.g. be seen when the bond is linked to a basket of underlying assets. It is also the case when the return on the bond is determined by the average of a number of observations of the price of the underlying assets during the term of the bond, an Asian option, rather than just the price of the underlying asset when the bond matures. Again, this type of bond is worth less than the original bond, cf. the full line. Chart 5 (right) illustrates how the return profile at maturity depends on the conditions for the index-linked bond. Alternative A shows the consequences of a change in the participation rate, i.e. the percentage of any increases in the underlying asset whereby the index-linked bond return is scaled. If the participation rate is changed from 80 to 30 per cent, the dashed yellow line is seen instead of the full yellow line. It takes greater increases in the value of the underlying asset to achieve a return on the index-linked bond equivalent to that in the original scenario. An upper limit to the final return on the index-linked bond may also be set, as illustrated by alternative B in the right-hand chart. The various structures give an idea of the multitude of combinations of probability distribution and return profile on which an index-linked bond may be based. If an investor cannot identify the properties of the individual product, he may in principle just as well end up with an index-linked bond with a probability distribution corresponding to alternative 1 in Chart 5 (left), and a return profile as in alternative A in Chart 5 (right), as the product illustrated in Chart 4. The value of the index-linked bond illustrated in Chart 4 is clearly greater than the value of the alternative structure.

LITERATURE

Jakobsen, S., 2000, Equity-, currency- and commodity-linked bonds (in Danish only), Finans/Invest Nr. 8.

Stoimenov, P. A., S. Wilkens, 2005, Are structured products " fairly" priced? An analysis of the German market for equity-linked instruments, Journal of Banking and Finance Nr. 29, pp. 2971-2993.

Investors severely affected by Turkish downturn (in Danish only), Børsen, 13 June 2006.

Private investors throw billions into exotic currency lottery (in Danish only), Børsen, 23 March 2007.

Small banks bind clients to investments (in Danish only), Børsen, 7 May 2007.

Large hidden gains in Forstædernes Bank (in Danish only), Børsen, 26 April 2006.