## Nominal value of the bond and income

* The nominal value of a bond * indicates the amount that is borrowed and is refundable after the bonded loan expires, i.e., when the bond is redeemed. The percentage of income is set to face value. The ratio of the amount of interest payments received for the year to the nominal value of a bond is called its

**"coupon yield";**As a rule, bonds are issued with a high nominal value. This, in particular, they differ from the shares, the selling price of which (or denomination, if it is indicated), the issuer establishes in order to purchase them by representatives of the general population. Bonds are oriented either to rich individual investors or institutional investors. In the US, the most common bonds with a face value of $ 1,000, ie, about 30 times higher than the average value of shares on the New York Stock Exchange. This does not mean that bonds are generally inaccessible to persons with relatively low incomes. The amount sufficient for the acquisition of a share can be invested in a bond, not directly but indirectly through investment funds that combine the funds of small investors.

## Market value and real yield

Bonds, as a rule, are highly liquid investment objects due to the fact that there is their active secondary market. In order to determine the market value of a bond, we will give definitions to the concepts of future (final) and current value.

* Future value * (7Y )

*value in the future (through*

**-***years) of the amount of money currently invested (ho) - is determined by the formula of compound interest:*

where/ is the loan (coupon) interest.

For example, with * Ho = * 100 rubles. and

*8% for the three-year period*

**d =***3)*

**(n =***will be:*

**/strong>**

If interest is paid more than once a year, and * t * times,

then

For example, if interest is paid quarterly, then in our example

* Current value - * discounted (recalculated for the current time) value of the future cash flow. It is determined by the formula:

where * RU - * is the current value of the cash flow

*which should be received through*

**x,***< strong> n*years;

*discount rate.*

**To -** If * x = TU & quot ;, * then

and at a discount rate equal to the loan interest * K * = r, we get

*This means that the present value of the future value of the cash flow is equal to the amount currently invested. For example, the future value of the amount of 100 rubles, now invested at 8% per annum, in three years will be 125.97 rubles. The current value of this future value, that is, the amount of 125.97 rubles, which we expect to receive in three years, will be 100 rubles.*

**RU = x $.**Pa Fig. 9.1 shows the current value of 100 rubles., Received after 1-10 years at a discount rate of 5, 10, 15%.

It is seen that * RU * 100 rubles. decreases in descending order more than the later they will be received. Of course, the higher the interest rate, the less

*the more steep the curve. At a discount rate of 15%*

**RUN***100 rubles, which will be received after 10 years, is equal to 24.72 rubles.*

**RU**

* Fig. 9.1. *

**The current price is 100 rubles. at a discount rate of 5.10 and 15%**

* RU * cash flow, representing an annuity from a number of equal payments for a certain number of time intervals, is:

where * t - * is the number of time slots for which

*are paid after each of them; & pound; - the amount of discounted payments at the end of the time intervals from 1 to*

**x**

**t.**If 1000 rubles. were expected at the end of each of the next two years at a discount rate of 8%, we would have:

The market value of a bond is the current value of the cash flow of a bond. This flow is formed by the payment of interest income during the specified period and payment of the nominal value of the bond at the end of the period.

The equation for estimating the current value of a bond with a par value of a Block is as follows:

where * P * - the discounted (current) value of the payment flow;

*annual interest payments determined by*

**C -** the nominal interest income; * to - * the necessary rate of return, determined by the risk of the bond. The latter is known as income received before maturity.

** Example: **

1. It is required to determine the market foam of a bond with a nominal value of 1,000 rubles. and a nominal income of 12%, which will ensure a 14% return in the remaining term (10 years).

The memorial interest income corresponds to the annual interest payments of 120 rubles. Therefore,

= 625.93 * (the value of the flow of future interest payments) + * + 269.74

*895.67 rubles.*

**(discounted face value) =**In this case, the market value of the bond is lower than the nominal value.

2. Instead of a 12% nominal income, the bond provides for the remaining period only 10% revenue.

In this case, the market value of the bond is higher than the nominal value.

Thus, the market value of a bond varies depending on the required level of income:

1. When the required level of income exceeds the interest income established on the bond, the value of the bond * is inferior * of its nominal value. Such a bond is said to be sold with a

**discount.** * 2. * When the required level of income is inferior to the established interest income, the value of the bond

*its nominal value. Such a bond is said to be sold with a*

**exceeds**

**premium.**3. When the required level of income is equal to the established interest income, the value of the bond is equal to its face value.

4. Given the change in the level of income, the value of the bond will change the stronger, the longer the period of its redemption.

From the table below, in which examples are considered for 5-year and 10-year bonds, it is clear that with a change in the required rate of return from 14% to 10%, the market value of a 10-year bond with a 12% nominal income amounted to 227.22 rubles. And 5-year - 144.48 rubles.

** Change in the market value of a bond with a given income level change **

(The nominal value of the bond is 1000 rubles.)

5. Given the change in income level, the value of the bond will change the more, the lower its nominal interest income. It can be seen from the table that with a change in the necessary rate of profit from 14% to 10%, the relative increase in the market value of a 10-year bond with a 4% nominal income was 32.0%, and bonds with a 12% nominal income of -25.4 %.

As the repayment period increases, the rate of change in market value is reduced: in our example, with a nominal income of 12%, the relative change in the value of the bond over the five-year redemption period was 15.5%, and in the last five years 9.9%.

Thus, the lower the nominal interest income and the longer the maturity of a bond, the greater the degree of interest rate volatility in bond markets.

## The current value of a bond with semi-annual income payment

Interest for the most part of bonds is paid every six months. The value of such bonds at a nominal of 1000 rubles. can be estimated by the formula:

where * P - * the current market price of the bond;

*annual payments of nominal interest income;*

**C -***number of years remaining until maturity of the issue; & amp; - the necessary income received before maturity.*

**n -** Note that the cost equation contains four parameters. If you specify three of them, you can always define the fourth parameter. So, if * P = * 960 rubles;

*$ 100;*

**C =***12 years, you can calculate the required income*

**n =***The equation of value at a nominal of 1000 rubles. will be written in the form:*

**to.**

Solving this equation, we get * to = * 10,60%. To solve such problems, special calculators and tables are used.

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