# Weapon of a cautious investor: we consider the fair value of investment bonds

Hello, Habr! Today we want to raise a rather non-standard topic for the blog - investment bonds. Why did we decide to write about this? The topic of structural financial products, which include investment bonds, has recently become more popular, and there is very little clear information about what this tool is and how it works. Interested in? Welcome to cat.

What is an investment bond (IO)? In a general sense, it is a “hybrid” of bonds and options.

It is related to the classic bond by the fact that the issuer is obliged to pay the face value (to repay the bond) after the specified period. However, depending on the conditions specified in the prospectus, the paper may be redeemed at a cost lower than the nominal.

As in ordinary bonds, coupon payments are provided for investment bonds. Their parameters are described in the prospectus. However, the rate on them is usually small and significantly lower than the key rate of the Central Bank. Instead, the investment income of the instrument is tied to some underlying asset - this is the relationship between the IO and options.

The underlying asset for an investment bond can be, for example, stocks of some companies (for example, Sberbank, Gazprom, Apple or Agricultural Bank of China) or stock indices (S & P500, Nasdaq Composite, MICEX or FTSE100). Depending on the value of the underlying asset at the maturity date, the investment income that the holder of the paper will receive or not will be determined.

Investment bonds are a good way to test your investment idea. If the rate of the underlying asset does not rise or even falls, the investor will still return his money when paying off the bonds and will not lose anything, except the time value of money (of course, the risk of default of the paper issuer should also be taken into account). If the price of the underlying asset, to which the investment income is tied, rises, the investor will make a profit.

This is critical for unskilled investors who are just starting to invest in risky assets, or for more experienced ones if they want to change the risk profile of their portfolio. Of course, it is worth considering that a significant reduction in risk means a decrease in expected income - this basic rule for evaluating financial instruments also applies here.

Thus, if we compare the IE with ordinary bonds and deposits, it can be noted that due to more flexible payments on structural bonds, the

As a rule, most investors cannot create a portfolio of instruments similar to AI. Nomenclature and liquidity on option boards on exchanges is limited. Moreover, options built into the IO can have an exotic structure. But by adding such an instrument to his portfolio, the investor can create a paper that wins back his investment idea with the appropriate risk-return ratio. As an example, IO with a coupon with a memory property (memory coupon) or IO with built-in barrier knock-in options.

The main buyers of investment bonds are private and professional investors, the main issuers are financial companies. Papers are usually placed on the OTC market in the format of the so-called Principal Protected Notes. But there are bonds traded on exchanges. In Russia, due to the peculiarities of regulation, such an option has become more widespread.

This is very good news. Due to the fact that IOs are mainly traded on the stock exchange, almost anyone can buy or sell them on the secondary market if the current price suits them.

When placing investment bonds, the issuer charges a commission. It should be borne in mind: the smaller the volume of a single transaction with such a “customized” IO, the higher the commission will be. In essence, the commission is the payment for the services of the team of traders and managers who formed this structural product. For investors with a small capital, a profitable solution may be to participate in public offerings of IO. Banks offer such securities in large volumes and spend a lot of resources on optimizing the instrument. If the client generally agrees with the investment idea of the product, participation in the pool of buyers will reduce his commission.

Since the income that an investment bond will bring (or will not bring) to its owner directly depends on the value of the underlying asset, such a link can be considered as an option embedded in the paper. To correctly determine the value of the paper, it is necessary to "divide" it into two components - interest and derivative.

The percentage component is estimated by discounting future payments on a certain yield curve. Typically, this is a risk-free yield curve to which a counterparty credit spread is added.

A derivative of an IO often consists of a variety of built-in options. Their price and risk metrics will be calculated using various valuation methods. For example, if a simple call or put option is used, then most often its price will be determined through the Black-Scholes model.

It should be noted that nevertheless, in most cases, the application of the Black-Scholes approach is a strong simplification. For example, in the case of IO, since they most often have a long maturity, the Black-Scholes model is no longer adequate for their assessment. Also, sometimes for calculation it is necessary to take into account, in addition to the stochastic evolution of the underlying asset, the evolution of interest rates, exchange rates and other market factors. Moreover, the parameters of evolution can, in turn, be completely random processes. Therefore, most often, other methods are used to determine prices and risk metrics. As a rule, numerical: trees (for example, binomial), Monte Carlo approaches.

When calculating the price, sometimes it is also necessary to take into account the credit properties of the issuer of the IE. The so-called CVA corrections are responsible for this. Numerical methods are also used to calculate them, and due to some features of these corrections, these methods must be very efficient from a computational point of view.

Now there is a fairly wide range of tools that allow, if you do not solve such problems completely, then at least have a certain framework for starting work. These are vendor products with proprietary code (for example, Bloomberg), which are often offered as services, and open-source libraries (such as QuantLib). As a rule, banks that are active in this market develop their own libraries for the evaluation and risk management of exotic options that integrate with all front and back office systems.

Let's look at the IE, which is tied to the behavior of the S & P500 index and pays an additional coupon yield proportional to the index's yield in case of positive dynamics. Simply put, a potential investor buys this instrument, hoping to capitalize on the growth of the S & P500 index.

Suppose, at the time of issuing the IE, the index value is 2500 points. Let the payment by the IE be structured so that at the time of redemption, additional income will be paid only if the index value is above 2500. That is, if the index is below this level, then only the nominal value of the bond will be returned to the owner at maturity. If at the time of repayment S & P500 is equal to 3000, then the additional coupon payment will be equal to$$from the face value of the bond.

Suppose we want to purchase such an IO with a face value of 1000 rubles. Suppose a year has passed from the moment of issue, until the end of repayment there is also a year left, and now the S & P500 is 2800. What is the fair value of the paper?

Payment by AI in a year is$$

Note that the last term is a call option payout. Suppose the time value of money is determined by the risk-free rate r, as a proxy for it we take the key rate of the Central Bank = 7.75%. Then the fair value of the bond, excluding the issuer's credit risk, is now equal to:

$$

$$, here EV is a mathematical expectation for some probability measure.

If the market value of the paper is below this level, obviously, its purchase is generally appropriate. Here the option is calculated using the Black-Scholes formula.

In this post, we presented investment bonds and outlined the essentials about them. Next, it is worth considering the methods for assessing the value of options embedded in the IO. This is a big hot topic. Open access contains only basic information or offers from vendors. If the topic is interesting, we will definitely continue its research, but for now we are ready to answer your questions.

## Bond + Option

What is an investment bond (IO)? In a general sense, it is a “hybrid” of bonds and options.

It is related to the classic bond by the fact that the issuer is obliged to pay the face value (to repay the bond) after the specified period. However, depending on the conditions specified in the prospectus, the paper may be redeemed at a cost lower than the nominal.

As in ordinary bonds, coupon payments are provided for investment bonds. Their parameters are described in the prospectus. However, the rate on them is usually small and significantly lower than the key rate of the Central Bank. Instead, the investment income of the instrument is tied to some underlying asset - this is the relationship between the IO and options.

The underlying asset for an investment bond can be, for example, stocks of some companies (for example, Sberbank, Gazprom, Apple or Agricultural Bank of China) or stock indices (S & P500, Nasdaq Composite, MICEX or FTSE100). Depending on the value of the underlying asset at the maturity date, the investment income that the holder of the paper will receive or not will be determined.

*For players who are not ready to risk their capital, but are interested in betting on some investment idea, this is a very convenient scheme.*Imagine: an investor is confident that Amazon / Netflix / Tesla ... stocks will grow at least 30% over the next 2 years. But he is not ready to purchase the paper directly, because he is afraid of losing his savings, or he does not have time for weekly tracking of quotes and constant analysis of the news flow.Investment bonds are a good way to test your investment idea. If the rate of the underlying asset does not rise or even falls, the investor will still return his money when paying off the bonds and will not lose anything, except the time value of money (of course, the risk of default of the paper issuer should also be taken into account). If the price of the underlying asset, to which the investment income is tied, rises, the investor will make a profit.

This is critical for unskilled investors who are just starting to invest in risky assets, or for more experienced ones if they want to change the risk profile of their portfolio. Of course, it is worth considering that a significant reduction in risk means a decrease in expected income - this basic rule for evaluating financial instruments also applies here.

Thus, if we compare the IE with ordinary bonds and deposits, it can be noted that due to more flexible payments on structural bonds, the

*IEs allow to form a more favorable risk-return ratio*. Therefore, IOs are likely to continue to gain popularity.As a rule, most investors cannot create a portfolio of instruments similar to AI. Nomenclature and liquidity on option boards on exchanges is limited. Moreover, options built into the IO can have an exotic structure. But by adding such an instrument to his portfolio, the investor can create a paper that wins back his investment idea with the appropriate risk-return ratio. As an example, IO with a coupon with a memory property (memory coupon) or IO with built-in barrier knock-in options.

The main buyers of investment bonds are private and professional investors, the main issuers are financial companies. Papers are usually placed on the OTC market in the format of the so-called Principal Protected Notes. But there are bonds traded on exchanges. In Russia, due to the peculiarities of regulation, such an option has become more widespread.

*Quotes on IO Mosbirzhe can see anyone*This is very good news. Due to the fact that IOs are mainly traded on the stock exchange, almost anyone can buy or sell them on the secondary market if the current price suits them.

When placing investment bonds, the issuer charges a commission. It should be borne in mind: the smaller the volume of a single transaction with such a “customized” IO, the higher the commission will be. In essence, the commission is the payment for the services of the team of traders and managers who formed this structural product. For investors with a small capital, a profitable solution may be to participate in public offerings of IO. Banks offer such securities in large volumes and spend a lot of resources on optimizing the instrument. If the client generally agrees with the investment idea of the product, participation in the pool of buyers will reduce his commission.

## Theory: how to evaluate the fair price of AI

Since the income that an investment bond will bring (or will not bring) to its owner directly depends on the value of the underlying asset, such a link can be considered as an option embedded in the paper. To correctly determine the value of the paper, it is necessary to "divide" it into two components - interest and derivative.

The percentage component is estimated by discounting future payments on a certain yield curve. Typically, this is a risk-free yield curve to which a counterparty credit spread is added.

A derivative of an IO often consists of a variety of built-in options. Their price and risk metrics will be calculated using various valuation methods. For example, if a simple call or put option is used, then most often its price will be determined through the Black-Scholes model.

It should be noted that nevertheless, in most cases, the application of the Black-Scholes approach is a strong simplification. For example, in the case of IO, since they most often have a long maturity, the Black-Scholes model is no longer adequate for their assessment. Also, sometimes for calculation it is necessary to take into account, in addition to the stochastic evolution of the underlying asset, the evolution of interest rates, exchange rates and other market factors. Moreover, the parameters of evolution can, in turn, be completely random processes. Therefore, most often, other methods are used to determine prices and risk metrics. As a rule, numerical: trees (for example, binomial), Monte Carlo approaches.

When calculating the price, sometimes it is also necessary to take into account the credit properties of the issuer of the IE. The so-called CVA corrections are responsible for this. Numerical methods are also used to calculate them, and due to some features of these corrections, these methods must be very efficient from a computational point of view.

Now there is a fairly wide range of tools that allow, if you do not solve such problems completely, then at least have a certain framework for starting work. These are vendor products with proprietary code (for example, Bloomberg), which are often offered as services, and open-source libraries (such as QuantLib). As a rule, banks that are active in this market develop their own libraries for the evaluation and risk management of exotic options that integrate with all front and back office systems.

## Practice: how to evaluate the fair price of AI

Let's look at the IE, which is tied to the behavior of the S & P500 index and pays an additional coupon yield proportional to the index's yield in case of positive dynamics. Simply put, a potential investor buys this instrument, hoping to capitalize on the growth of the S & P500 index.

Suppose, at the time of issuing the IE, the index value is 2500 points. Let the payment by the IE be structured so that at the time of redemption, additional income will be paid only if the index value is above 2500. That is, if the index is below this level, then only the nominal value of the bond will be returned to the owner at maturity. If at the time of repayment S & P500 is equal to 3000, then the additional coupon payment will be equal to$$from the face value of the bond.

Suppose we want to purchase such an IO with a face value of 1000 rubles. Suppose a year has passed from the moment of issue, until the end of repayment there is also a year left, and now the S & P500 is 2800. What is the fair value of the paper?

Payment by AI in a year is$$

Note that the last term is a call option payout. Suppose the time value of money is determined by the risk-free rate r, as a proxy for it we take the key rate of the Central Bank = 7.75%. Then the fair value of the bond, excluding the issuer's credit risk, is now equal to:

$$

$$, here EV is a mathematical expectation for some probability measure.

If the market value of the paper is below this level, obviously, its purchase is generally appropriate. Here the option is calculated using the Black-Scholes formula.

## Conclusion

In this post, we presented investment bonds and outlined the essentials about them. Next, it is worth considering the methods for assessing the value of options embedded in the IO. This is a big hot topic. Open access contains only basic information or offers from vendors. If the topic is interesting, we will definitely continue its research, but for now we are ready to answer your questions.