11.2. Measuring Bond Yields

Bond yield. Bond yields are characterized by several indicators. Distinguish coupon(coupon rate), tech at shuyu(current, running yield) and full profitability(yield to maturity, redemption yield, yield).

The coupon yield is determined when the bond is issued and therefore does not need to be calculated. The current yield characterizes the ratio of coupon receipts to the purchase price of the bond. This parameter does not take into account the second source of income - receiving the face value or redemption price at the end of the term. Therefore, it is not suitable for comparing the yields of different types of bonds. Suffice it to note that zero coupon bonds have a current yield of zero. At the same time, they can be very profitable if you take into account their entire “life” period.

The most informative is the total return indicator, which takes into account both sources of income. This indicator is suitable for comparing the return on investments in bonds and other securities. So, total return, or to use old commercial terminology, room rate, measures the true investment performance of a bond for an investor in terms of the annual compounding rate. In other words, the accrual of interest at the placement rate on the purchase price of the bond fully ensures the payment of coupon income and the amount to repay the bond at the end of the term.

Let us consider the methodology for determining the yield indicators of various types of bonds in the sequence adopted above when classifying bonds according to the method of payment of income.

Bonds without obligatory repayment with periodic interest payments. Although this type of bond is extremely rare, familiarity with them is necessary to obtain a complete understanding of the methodology for measuring profitability. When analyzing this type of bond, we do not take into account the payment of par value in the foreseeable future.

Let us introduce the following notation:

g - declared rate of annual income (coupon interest rate);

i t - current profitability;

i- total profitability (premises rate).

The current yield is as follows:

i t = 100. (11.2)

If coupons are paid out r once a year (each time at the rate g/ p), then in this case, in practice, “formula (11.2) is applied, although the summation of income paid at different points in time, strictly speaking, is incorrect.

Since the coupon income is constant, the current yield of the bonds being sold changes along with the change in their market price. For a bond owner who has already invested some funds, this value is constant.

Let's move on to total profitability. Since coupon income is the only source of current income, it is obvious that the total yield of the bonds under consideration is equal to the current one in the case when coupon payments are annual: i = i t. If interest is paid r once a year (each time according to the norm g / p), then according to (2.8) we get

(11.3)

Example 11.1. A perpetual annuity yielding 4.5% income was purchased at an exchange rate of 90. What is the financial efficiency of the investment, provided that interest is paid once a year, quarterly ( p = 4)?

i = i t = 100 = 0,05; i = - 1 = 0,0509.

Bonds without interest payments. This type of bond provides its owner with the difference between the par value and the purchase price as income. The rate of such a bond is always less than 100. For

To determine the premises rate, we equate the current value of the face value to the purchase price:

Nvn = P, or vn = ,

Where n - period until the bond is redeemed. After which we get

Example 11.2. Corporation X issued zero coupon bonds maturing in five years. Sales rate - 45. Bond yield at maturity date

those. the bond provides the investor with 17.316% annual income.

Bonds that pay interest and face value at maturity. Interest here is accrued for the entire term and is paid in one lump sum along with the face value. There is no coupon income. Therefore, the current yield can conditionally be considered zero, since the corresponding interest is received at the end of the term.

Let’s find the total return by equating the current value of income to the price of the bond:

(1 + g)nNvn = P, or .

From the last formula it follows that

If the bond rate is less than 100, then i > g.

Example 11.3. A bond yielding 10% per annum relative to par was purchased at an exchange rate of 65, with a maturity period of three years. If par and interest are paid at maturity, the total return to the investor will be

i = - 1 = 0.26956, or 26.956%.

Bonds with periodic interest payments and par value repayment at the end of the term. This type of bonds is most widespread in modern practice. For such a bond, you can get all three yield indicators - coupon, current and total. Current yield is calculated using the above formula (11.2). As for the total yield, to determine it it is necessary to equate the current value of all proceeds to the price of the bond. The discounted value of the nominal value is Nvn. Since receipts from coupons represent a constant post-numerando annuity, the term of such annuity is equal to gN, and its modern cost will be gNa n ; i (if coupons are paid annually) and if these payments are made r once a year (each time at the rate g/ p). As a result, we obtain the following equalities:

for bonds with annual coupons

(11.6)

Divided by N, we find

(11.7)

for a bond with coupon redemptions semiannually and quarterly, we obtain

(11.8)

where is the reduction coefficient p-term annuity ( p = 2, p = 4).

In all the given formulas vn means the discount multiplier for the unknown annual rate of the premises i.

In foreign practice, however, for bonds with semi-annual and quarterly payments of current income, the annual nominal placement rate is used for discounting, and the number of times discounting per year is usually taken equal to the number of coupon income payments. Thus, the initial equality for calculating the premises rate has the form

Where i - nominal annual rate;

rp - total number of coupon payments; g - annual percentage of coupon payments.

When solving the above equalities for an unknown quantity i face the same problems as when calculating i for a given value of the rent reduction coefficient - see paragraph 4.5. The required room rate values ​​are calculated either using interpolation or some iterative method.

Let's evaluate i using linear interpolation:

(11.10)

Where i" And i" - floor and ceiling room rate values ​​that limit the interval within which the unknown rate value is expected to lie;

K" , K" - calculated exchange rate values ​​for bets respectively i" , i" . The rate interval for interpolation is determined taking into account the fact that i > g at K < 100.

You can also apply the approximate estimation method, according to which

. (11.11)

This formula relates the average annual yield of a bond to its average price. The simplicity of the calculation, however, comes at the price of loss of estimation accuracy.

Example 11.4. A bond with a term of five years, on which interest is paid once a year at a rate of 8%, was purchased at an exchange rate of 97.

Current yield on bond 8 / 97 = 0,08247.

To estimate the total profitability, we write the original equality (11.7):

0,97 = (1 + i) -5 + 0,08a 5; i.

For interpolation, we will accept the following bet values: i" = 0,085, i" = 0.095. According to (11.7) we find

1,085 -5 + 0,08A 5;8,5 = 98,03;

= 1,095 -5 + 0,095A 5;9,5 = 94,24.

i = 8,5 + (9,5 - 8,5) = 8,77.

To check, let's calculate the rate for the premises rate of 8.77%. We get

= 1,0877 -5 + 0,08A 5;8,77 = 96,99.

As we can see, the calculated rate is very close to the market rate - 97. An approximate solution according to (11.11) gives

i = = 8,73,

which corresponds to the market rate of 97.2. The error is higher than when using linear interpolation.

Bonds with a redemption price different from their par value. In this case, interest is calculated on the par amount, and capital gains are equal to S - R, Where WITH- redemption price. Accordingly, when assessing the premises rate, it is necessary to make appropriate adjustments

tives into the above formulas. For example, making adjustments to (11.6) and (11.7), we obtain

and instead of (11.11)

(11.14)

Example 11.5. Let's compare the yield of two bonds with annual interest payments (Table 11.1). Bond parameters A taken from the previous example.

Table 11.1

The yield indicators for these bonds are given in table. 11.2.

Table 11.2

As you can see, in terms of total return, the advantage is on the side of the bond A, although its current yield is lower than that of the second. The approximate method of calculation according to (11.11) - the corresponding indicators are given in parentheses - noticeably overestimated the estimate of the total return on the bond B.

All the formulas discussed above for calculating the total yield assume that the assessment is made at the beginning of the bond's term or at the date of interest payment. For the case where the estimate is made at a time between two interest payment dates, the given formulas will give biased estimates.

The price of a bond selling at a discount, subject to a constant required yield, rises. The reverse process occurs with the price of a bond selling at a premium. The price of both bonds at maturity is equal to the par value. Symmetric differences between the required yield and the coupon rate are converted into asymmetric differences between the bond price and its par value. In particular, the price of a bond rises more when yields fall than it falls when yields rise.

Bond yield. In general, the return on any investment is understood as the interest rate that allows the present value of the cash flows of a competitive investment to be equalized with the price (cost) of the investment.

The yield on a zero-coupon bond is the annual interest rate received by an investor who buys and holds the bond until maturity.

If then.

Determining the yield on a coupon bond. For a coupon bond there are current yield and the internal rate of return or yield to maturity.

Current profitability is determined by the formula:

where rt – current profitability;

C – coupon income on the bond (coupon);

P – current bond price.

Internal yield can be calculated using the formula for estimating the market price of a bond:

Unfortunately, this equation cannot be solved in its final form: profitability can only be determined using a special computer program.

M/use the method of substituting various internal yield values ​​into the bond price formula and calculating the corresponding prices. The operation is repeated until the value of the calculated price coincides with the specified bond price. The block diagram of the algorithm for this calculation is shown in Fig. 4.

Rice. 4. Algorithm for calculating yield coupon bonds

In some cases, to make a financial decision, it is enough to determine only the approximate (approximate) level of bond yield. It can be used as the initial level of profitability in the first block of the algorithm discussed above.

The traditionally used formula for calculating the approximate level of bond yield is:

where r is the internal yield (yield to maturity); N – nominal value of the bond; P – bond price; n – number of years until maturity; C – coupon income;

In some cases, the best approximation is provided by R. Rodriguez's formula

This formula gives a good approximation provided that the coupon rate is low (below 50% per annum) and the bond price and its face value are close. In particular, if the price differs from the face value by more than 2 times, then the use of both formulas for calculating approximate estimates is unacceptable.

The error in calculations using approximate estimation formulas is higher the more years remain until the bond matures.

To speed up the process of calculating the internal yield of a bond, the linear interpolation formula can also be used:

Where r 1 , r 2 – values ​​of respectively underestimated and overestimated levels of estimated bond yields; R 1 , R 2 – estimated market prices of bonds corresponding to yield levels r 1 and r 2 ;

R– actual (actual) price of the bond on the stock market.

Summarizing the above, we note that the yield to maturity allows us to estimate not only the current (coupon) income, but also the amount of profit or loss awaiting the capital of the investor who remains the owner of the bond until its redemption by the issuer. In addition, the yield to maturity takes into account the timing of cash flows.

Correlation of the main parameters of the bond

The bond is for sale	Relationship between bond parameters
At par	Coupon rate = Current yield = Yield to maturity
With discount	Coupon rate< Текущая доходности < Доходность к погашению
With a bonus	Coupon rate > Current yield > Yield to maturity

In accordance with the algorithm for determining the value of a bond, presented in Problem 2.1, the formula for calculating the price of a bond has the form:

where P is the bond price; C - coupon in rubles; N - denomination;

n is the number of years until the bond matures; r is the yield to maturity of the bond. According to formula (2.1), the bond price is equal to:

Problem 2.3.

The face value of the bond is 1000 rubles, the coupon is 10%, paid once a year. The bond has 3 years until maturity. Determine the price of a bond if its yield to maturity should be 9%.

R = 1025.31 rub.

Problem 2.4.

The face value of the bond is 1000 rubles, the coupon is 10%, paid once a year. The bond has 3 years until maturity. Determine the price of a bond if its yield to maturity is 10%.

R = 1000 rub.

Problem 2.5.

The face value of the bond is 1000 rubles, the coupon is 10%. paid once a year. The bond has 3 years until maturity. Determine the price of a bond if its yield to maturity should be 11%.

R = 975.56 rub.

Question 2.6.

The yield to maturity of a bond is less than its coupon. Should the price of a bond be higher or lower than par?

The price of the bond must be higher than its face value. This pattern is illustrated by problems 2.2 and 2.3.

Question 2.7.

The yield to maturity of a bond is greater than its coupon. Should the price of a bond be higher or lower than par?

The price of the bond must be below par. This pattern is illustrated by Problem 2.5.

Question 2.8.

The yield to maturity of a bond is equal to its coupon. How much is the bond worth?

The price of the bond is equal to the face value. This pattern is illustrated by Problem 2.4.

Problem 2.9.

The face value of the bond is 1000 rubles, the coupon is 10%, paid twice a year. The bond has 2 years until maturity. Determine the price of a bond if its yield to maturity is 8%.

When the coupon is paid m times a year, formula (2.1) takes the form:

According to (2.2), the bond price is equal to:

Note.

This problem can be solved using formula (2.1), only in this case the time periods for paying coupons should be taken into account not in coupon periods, but, as before, in years. The first coupon is paid in six months, so its payment time is 0.5 years, the second coupon is paid in a year, its payment time is 1 year, etc. The discount rate is taken into account in this case as an effective interest based on a given yield to maturity , i.e. it is equal to:

(1+0,08/2)^2 – 1 = 0,0816.

According to formula (2.1), the bond price is:

Problem 2.10.

The face value of the bond is 1000 rubles, the coupon is 10%, paid twice a year. The bond has 2 years until maturity. Determine the price of a bond if its yield to maturity is 9%.

According to (2.2), the price of the bond is 1017.94 rubles.

Problem 2.11.

The face value of the bond is 1000 rubles, the coupon is 10%, paid twice a year. The bond has 2 years until maturity. Determine the price of a bond if its yield to maturity is 10%.

R = 1000 rub.

Problem 2.12.

The face value of the bond is 1000 rubles, the coupon is 10%, paid twice a year. The bond has 2 years until maturity. Determine the price of a bond if its yield to maturity should be 11%.

R = 982.47 rub.

Problem 2.13.

The face value of the bond is 1000 rubles, the coupon is 6%, paid twice a year. The bond has 3 years until maturity. Determine the price of a bond if its yield to maturity is 7%.

R = 973.36 rub.

Problem 2.14.

The face value of the bond is 1000 rubles, the coupon is 10%. paid once a year. The bond has 2 years and 250 days until maturity. Determine the price of a bond if its yield to maturity is 8%. Base 365 days.

The bond price is determined by formula (2.1). If there are not an integer number of years left until the bond matures, then the actual time of payment of each coupon is taken into account. Thus, the payment of the first coupon will occur at time 250/365, the second coupon at time 1*250/365, etc.

The bond price is:

Problem 2.15.

The face value of the bond is 1000 rubles, the coupon is 10%, paid once a year. The bond has 2 years and 120 days until maturity. Determine the price of a bond if its yield to maturity should be 12%. Base 365 days.

The bond price is:

Problem 2.16.

The face value of the bond is 1000 rubles, kunon 10%, paid once a year. The bond has 2 years and 30 days until maturity. Determine the price of a bond if its yield to maturity is 10%. Base 365 days.

R = 1091.47 rub.

Problem 2.17.

The face value of the bond is 1000 rubles, the coupon is 10%, paid once a year. The bond has 15 years until maturity. Determine the price of the bond if its yield to maturity should be 11.5%.

When a bond has many years until maturity, it is quite cumbersome to directly use formula (2.1). It can be converted to a more convenient form. The sum of the discounted values ​​of a bond's coupons is nothing more than the present value of the annuity. Taking into account this remark, formula (2.1) can be written as (Formula (2.1) can also be transformed to the form:):

Problem 2.18.

The face value of the bond is 1000 rubles, the coupon is 8%, paid once a year. The bond has 20 years until maturity. Determine the price of the bond if its yield to maturity should be 9.7%.

According to (2.3), the bond price is equal to:

Problem 2.19.

The face value of the bond is 1000 rubles, the coupon is 4%, paid once a year. The bond has 30 years until maturity. Determine the price of the bond if its yield to maturity should be 4.5%.

R = 918.56 rub.

Problem 2.20.

The face value of the bond is 1000 rubles, the coupon is 3%, paid once a year. The bond has 25 years until maturity. Determine the price of the bond if its yield to maturity should be 4.3%.

R = 803.20 rub.

Problem 2.21.

The face value of the bond is 1000 rubles, the coupon is 5%, paid once a year. The bond has 18 years until maturity. Determine the price of the bond if its yield to maturity should be 4.8%.

P = 1023.75 rub.

Problem 2.22.

The face value of the bond is 1000 rubles, the coupon is 10%, paid twice a year.

The bond has 6 years until maturity. Determine the price of the bond if its yield to maturity should be 8.4% per annum.

If the bond coupon is paid m times a year, formula (2.2) can be converted to the form (Formula (2.4) can also be converted to the form:):

According to formula (2.4), the bond price is equal to:

Problem 2.23.

The face value of the bond is 1000 rubles, the coupon is 7%, paid quarterly. The bond has 5 years until maturity. Determine the price of the bond if its yield to maturity should be 6.5% per annum.

According to (2.4), the bond price is equal to:

Problem 2.24.

The face value of the bond is 1000 rubles, the coupon is 4%, paid quarterly. The bond has 10 years until maturity. Determine the price of the bond if its yield to maturity should be 4.75% per annum.

R = 940.57 rub.

Problem 2.25.

The face value of the bond is 1000 rubles, the coupon is 7%, paid once a year. The bond has 11 years and 45 days until maturity. Determine the price of a bond if its yield to maturity is 8%. Base 365 days.

If there is not an integer number of years left until the bond is redeemed, then formula (2.3) can be transformed to the form:

where t is the number of days until the next coupon is paid;

n is the number of full years until the bond matures, that is, excluding the incomplete coupon period.

According to (2.5), the bond price is equal to:

Problem 2.26.

The face value of the bond is 1000 rubles, the coupon is 5%, paid once a year. The bond has 14 years and 77 days until maturity. Determine the price of the bond if its yield to maturity should be 4.8%. Base 365 days.

R = 1059.52 rub.

Problem 2.27.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 5 years. Determine the price of a bond if its yield to maturity should be 12% per annum.

For a zero-coupon bond, only one payment is made - at the end of its circulation period, the investor is paid the face value. Therefore, its price is determined by the formula:

According to (2.6), the price of the bond is: 1000/1.12^5 = 567.43 rubles.

Problem 2.28.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 3 years. Determine the price of a bond if its yield to maturity should be 8% per annum.

R = 793.83 rub.

Problem 2.29.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 8 years. Determine the price of a bond if its yield to maturity should be 6% per annum.

R = 627.41 rub.

Problem 2.30.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 5 years and 20 days. Determine the price of a bond if its yield to maturity should be 12% per annum. Base 365 days.

According to (2.6), the bond price is equal to:

Problem 2.31.

The face value of a zero-coupon bond is RUB 1,000, and the paper matures in 2 years and 54 days. Determine the price of the bond if its yield to maturity should be 6.4% per annum. Base 365 days.

R = 875.25 rub.

Problem 2.32.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 7 years. Determine the price of a bond if its yield to maturity should be 8% per annum. Coupon bonds pay coupons twice a year.

If coupon bonds pay coupons m times per year, this means that the frequency of compounding on bond investments is m times per year. To obtain a similar frequency of interest accrual on a zero-coupon bond, its price should be determined using the formula:

According to (2.7), the bond price is equal to:

Problem 2.33.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 4 years. Determine the price of a bond if its yield to maturity should be 5% per annum. A coupon bond pays coupons four times a year.

P = 819.75 rub.

Problem 2.34.

The face value of a zero-coupon bond is 1000 rubles, the paper is redeemed after 30 days. Determine the price of a bond if its yield to maturity should be 4% per annum. Base 365 days.

The price of a zero-coupon short-term bond is determined by the formula:

where t is the time until the bond matures.

According to (2.8), the bond price is equal to:

Problem 2.35.

The face value of a zero-coupon bond is 1000 rubles, the paper is redeemed in 65 days. Determine the price of the bond if its yield to maturity should be 3.5% per annum. Base 365 days.

R = 993.81 rub.

Problem 2.36.

The face value of a zero-coupon bond is 1000 rubles, the paper is redeemed in 4 days. Determine the price of a bond if its yield to maturity should be 2% per annum. Base 365 days.

R = 999.78 rub.

Problem 2.37.

The face value of the bond is 1000 rubles, the coupon is 10%. The bond costs 953 rubles. Determine the current yield of the bond.

The current bond yield is determined by the formula:

where rT is the current yield; C - bond coupon; P is the price of the bond.

According to (2.9), the current bond yield is equal to:

Problem 2.38.

The face value of the bond is 1000 rubles, the coupon is 8%. The bond costs 1014 rubles. Determine the current yield of the bond.

Problem 2.39.

The par value of the bond is 1000 rubles, the coupon is 3.5%. The bond costs 1005 rubles. Determine the current yield of the bond.

Problem 2.40.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 3 years. The bond costs 850 rubles. Determine the yield to maturity of the bond.

The yield to maturity of a zero-coupon bond is determined by the formula (derived from formula 2.6):

According to (2.10), the bond yield is:

Problem 2.41.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 5 years. The bond costs 734 rubles. Determine the yield to maturity of the bond.

Problem 2.42.

The face value of a zero-coupon bond is 1000 rubles, the paper is repaid in 2 years. The bond costs RUB 857.52. Determine the yield to maturity of the bond.

Problem 2.43.

The face value of a zero-coupon bond is RUB 1,000, and the paper matures in 4 years and 120 days. The bond costs 640 rubles. Determine the yield to maturity of the bond. Base 365 days.

Problem 2.44.

The nominal value of the zero-coupon bond is 1000 rubles. The bond matures after three years. The investor bought the bond for 850 rubles. and sold it after 1 year 64 days for 910 rubles. Determine the profitability of the investor's operation per year. Base 365 days.

Problem 2.45.

The nominal value of the zero-coupon bond is 1000 rubles. The bond matures after three years. The investor bought the bond for 850 rubles. and sold it after 120 days for 873 rubles. Determine the profitability of the investor's operation per year based on: 1) simple interest; 2) effective interest. Base 365 days.

Problem 2.46.

The nominal value of the zero-coupon bond is 1000 rubles. The bond matures in four years. The investor bought the bond at RUB 887.52. and sold it after 41 days for 893.15 rubles. Determine the profitability of the investor's operation per year based on: 1) simple interest; 2) effective interest. Base 365 days.

2) reff = 5.79%.

Problem 2.47.

The face value of the bond is 1000 rubles, the coupon is 7%, paid once a year. The bond has 5 years until maturity. The bond costs 890 rubles. Determine approximately the yield to maturity of the bond.

The yield to maturity of a coupon bond can be approximately determined from the formula:

where r is the yield to maturity; N - bond par value; C - coupon; P - bond price; n is the number of years until maturity.

According to (2.11), the yield is equal to:

Problem 2.48.

The face value of the bond is 1000 rubles, the coupon is 8%, paid once a year. The bond has 6 years until maturity. The bond costs 1053 rubles. Determine its yield to maturity.

Problem 2.49.

The face value of the bond is 1000 rubles, the coupon is 9%, paid twice a year. The bond has 4 years until maturity. The bond costs 1040 rubles. Determine its yield to maturity.

Comment.

For a bond for which the coupon is paid m times a year, the estimated yield formula will take the following form:

However, in this case, r is the yield per coupon period. So, if m = 2, then the yield will be for six months. To convert the resulting return per year, it should be multiplied by the value m. Thus, to calculate the estimated yield on bonds with coupon payments m times a year, you can immediately use formula (2.11).

Problem 2.50.

Determine the exact yield to maturity of the bond in Problem 2.48 by linear interpolation.

The formula for determining the yield of a bond using the linear interpolation method is:

The technique for calculating profitability using formula (2.13) comes down to the following. Having determined the estimated bond yield using formula (2.11), the investor selects the value r1, which is lower than the obtained value of the estimated yield, and calculates the corresponding bond price P1 for it using formula (2.1) or (2.3). Next takes the value of r2, which

higher than the estimated profitability value, and calculates the price P2 for it. The obtained values ​​are substituted into formula (2.13).

In problem 2.48, the estimated return was 6.93% per annum. Let's take r1 = 6% . Then according to formula (2.3):

Let's take r2 = 7%. According to formula (2.3):

Problem 2.51.

Determine the exact yield to maturity of the bond in Problem 2.47 by linear interpolation.

In problem 2.47, the estimated return was 9.74% per annum. Let's take r1 = 9%. According to formula (2.3):

Let's take r2 = 10% . According to formula (2.3):

According to (2.13), the exact yield to maturity of the bond is equal to:

Problem 2.52.

Determine the exact yield to maturity of the bond for Problem 2.49 by linear interpolation.

In problem 2.49, the estimated return was 7.84% per annum. Let's take r1 = 7% . According to formula (2.4):

Let's take r2 = 8%. According to formula (2.4):

The exact yield to maturity of the bond is:

Problem 2.53.

The nominal value of a short-term zero-coupon bond is 1000 rubles, the price is 950 rubles. The bond matures in 200 days. Determine the yield to maturity of the bond. Base 365 days.

The yield to maturity of a short-term zero-coupon bond is determined by the formula:

Problem 2.54.

Bond par value is 1000 rubles, price is 994 rubles. The bond matures in 32 days. Determine the yield to maturity of the bond. Base 365 days.

According to (2.14), the bond yield is equal to:

Problem 2.55.

Bond par value is 1000 rubles, price is 981 rubles. The bond matures in 52 days. Determine the yield to maturity of the bond. Base 365 days.

r = 13.6% per annum.

Problem 2.56.

Bond par value is 1000 rubles, price is 987.24 rubles. The bond matures in 45 days. Determine the yield to maturity of the bond. Base 365 days. Answer. r = 10.48% per annum.

Problem 2.57.

Determine the effective bond yield for Problem 2.54.

Problem 2.58.

Determine the effective bond yield for Problem 2.56.

Answer. reff = 10.97%.

Problem 2.59.

The face value of the bond is 1000 rubles, the coupon is 6%, paid once a year. The bond matures after three years. The investor bought the bond for 850 rubles. and sold it after 57 days for 859 rubles. During the period of ownership of the bond, no coupon was paid on the security. Determine the profitability of the investor's operation: 1) based on 57 days; 2) per year based on simple interest; 3) effective interest on the operation. Base 365 days.

Problem 2.60.

The face value of the bond is 1000 rubles, the coupon is 6%, paid once a year. The bond matures after three years. The investor bought the bond for 850 rubles. and sold it after 57 days for 800 rubles. At the end of the bond holding period, the coupon was paid on the security. Determine the profitability of the investor's operation per year based on simple interest. Base 365 days.

2.3. Realized interest (yield)

Problem 2.61.

The investor buys a bond at par, the par value is 1000 rubles, the coupon is 10%, paid once a year. The bond has 5 years until maturity. The investor believes that during this period he will be able to reinvest the coupons at 12% per annum. Determine the total amount of funds that the investor will receive on this security if he holds it until maturity.

After five years, the investor will be paid the face value of the bond. The sum of the coupon payments and the interest on their reinvestment represents the future value of the annuity. Therefore it will be:

The total amount of funds that the investor will receive over five years is equal to:

1000 + 635.29 = 1635.29 rub.

Problem 2.62.

The investor buys a bond at par, the par value is 1000 rubles, the coupon is 8%, paid once a year. The bond has 4 years until maturity. The investor believes that during this period he will be able to reinvest the coupons at 6% per annum. Determine the total amount of funds that the investor will receive on this security if he holds it until maturity.

The amount of coupon payments and interest on their reinvestment for four years is equal to:

Taking into account the payment of the par value, the total amount of funds on the bond after four years will be:

1000 + 349.97 = 1349.97 rub.

Problem 2.63.

The investor buys a bond at par, the par value is 1000 rubles, the coupon is 8%. paid once a year. The bond has six years until maturity. The investor believes that over the next two years he will be able to reinvest the coupons at 10%, and in the remaining four years at 12%. Determine the total amount of funds that the investor will receive on this security if he holds it until maturity.

The amount of coupons and interest on their reinvestment for the first two years (for the first two coupons) will be:

(That is, after a year the investor will receive the first coupon and reinvest it for a year at 10%, and a year later he will receive the next coupon. In total, this will give 168 rubles.) The amount received is invested at 12% for the remaining four years:

168*1.12^4 = 264.35 rubles.

The amount of coupon payments and interest from their reinvestment at 12% over the last four years will be:

1000 + 264.35 + 382.35 = 1646.7 rubles.

Problem 2.64.

The investor buys a bond at par, the par value is 1000 rubles, the coupon is 6%, paid once a year. The bond has three years until maturity. The investor believes that over the next two years he will be able to reinvest the coupons at 7%. Determine the total amount of funds that the investor will receive on this security if he holds it until maturity.

The investor has the opportunity to reinvest the first and second coupons at 7%. The third coupon will be paid upon maturity of the bond. Therefore, the sum of coupons and interest on their reinvestment is nothing more than a three-year annuity. Fro future value is:

The total amount that the investor will receive on the bond is:

1000 + 192.89 = 1192.89 rub.

Problem 2.65.

Determine the realized percentage for the conditions of problem 2.64.

Realized interest is the interest that allows the sum of all future earnings that an investor expects to receive on a bond to be equal to its today's price. It is determined by the formula:

Problem 2.66.

The face value of the bond is 1000 rubles, the coupon is 6%, paid once a year. An investor buys a bond for 950 rubles. The bond has three years until maturity. The investor believes that he will be able to reinvest the coupons at 8%. Determine the realized interest on the bond if the investor holds it to maturity.

The total amount of funds at the time of maturity of the bond will be:

According to (2.15), the realized interest on the bond is equal to:

Problem 2.67.

Prove that with a horizontal structure of the yield curve, the total amount of funds, taking into account the reinvestment of coupons, that the investor will receive from owning the bond at its maturity is equal to P(1+r)n, where n is the time remaining until the paper matures.

The bond price is:

Let us multiply the left and right sides of equality (2.16) by (1+r)n:

Equality (2.17) shows that the total amount of funds, taking into account the reinvestment of coupons, that an investor will receive from owning a bond with a horizontal structure of the yield curve is equal to P(1+r)n. This follows from the right side of equality (2.17). On the right side, the first coupon, which the investor receives in a year, is reinvested for a period of (n – 1), the second coupon

for a period (n – 2), etc. When the bond is redeemed, the last coupon and face value are paid. Formula (2.17) shows that the total amount of funds on the bond, taking into account the reinvestment of coupons, is equal to investing an amount equal to the price of the bond at the existing interest until the paper matures.

Problem 2.68.

The investor bought the bond and will sell it t years before maturity immediately after the next coupon is paid. Prove that with a horizontal structure of the yield curve, the total amount of funds, taking into account the reinvestment of coupons, that the investor will receive from owning the bond is equal to P(1+r)^(n – t), where n – t is the time that the investor will own the bond.

The bond price is:

The investor plans to sell the security t years before its maturity immediately after paying the next coupon, i.e. he will hold it for n – t years. Let's multiply the left and right sides of equality (2.18) by (1+r)^(n – t):

In equality (2.19), the last terms represent nothing more than the price of the bond when t years remain until its maturity, let’s denote it by Рt:

Therefore, we write (2.19) as:

Equality (2.20) shows that the total amount of funds, taking into account the reinvestment of coupons, that the investor will receive from owning the bond is equal to P(1+r)^(n – t).

Problem 2.69.

An investor bought a coupon bond with ten years left until maturity for RUB 887. The coupon on the bond is paid once a year. The next day, the yield to maturity of the bond fell to 11%, and its price rose to 941.11 rubles. Determine the annualized return that an investor would receive on the bond, taking into account the reinvestment of coupons (realized yield), if the interest rate remains at 11% and he sells the paper in three years.

According to formula (2.20), the total amount of funds on the bond, taking into account the reinvestment of coupons, that the investor will receive from owning the bond and selling it at time t is equal to P(1+r)^(n – t). The total amount of income received by the investor on the bond after three years is:

The investor bought the paper for 887 rubles. The realized return is:

Note.

In Problem 2.69, the formula for determining realized profitability can be presented in one action:

where rr is realized profitability;

Pn - new bond price after a change in the interest rate on the market;

P is the price at which the bond was purchased;

r is the interest rate corresponding to the new bond price.

Problem 2.70.

For the conditions of problem 2.69, determine the annual yield that the investor will receive on the bond, taking into account the reinvestment of coupons, if he sells the paper in nine years.

According to formula (2.21), the realized yield on the bond for nine years is equal to:

Problem 2.71.

An investor bought a coupon bond with ten years left until maturity for RUB 1,064.18. The coupon on the bond is paid once a year. The next day, the yield to maturity of the bond fell to 8%, and its price rose to RUB 1,134.20. Determine the annual return that an investor will receive on the bond, taking into account the reinvestment of coupons, if the interest rate remains at 8% and he sells the paper in three years.

According to (2.21), the realized yield on the bond for three years is equal to:

Problem 2.72.

For the conditions of problem 2.71, determine the annual yield that the investor will receive on the bond, taking into account the reinvestment of coupons, if he sells the paper in nine years.

Problem 2.73.

In Problem 2.71, the investor, after holding the bond for three years, received a realized return of 10.32%. In Problem 2.72, the investor, after holding a similar bond for 9 years, received a realized return of 8.77%. Explain why in the second case the yield from owning the bond decreased.

In problems 2.71 and 2.72, after purchasing a bond, its yield to maturity fell, therefore, the price increased. The short-term investor benefited from the rate fall. For a long-term investor, this effect is less pronounced or absent, since as the bond's maturity approaches, its price approaches its face value. At the same time, the short-term investor reinvests the coupons at a lower interest rate (8%) for a shorter time than the long-term investor. Therefore, the realized return of a long-term investor will be lower than that of a short-term investor.

Problem 2.74.

An investor bought a coupon bond with fifteen years left until maturity for RUB 928.09. The coupon on the bond is paid once a year. The next day, the yield to maturity of the bond rose to 12%, and its price fell to 863.78 rubles. Determine the annual return that an investor will receive on the bond, taking into account the reinvestment of coupons, if the interest rate remains at 12% and he sells the paper in four years.

According to (2.21), the realized yield on the bond for four years is equal to:

Problem 2.75.

For the conditions of problem 2.74, determine the annual yield that the investor will receive on the bond, taking into account the reinvestment of coupons, if he sells the paper in ten years.

Problem 2.76.

In Problem 2.74, the investor, after holding the bond for four years, received a realized return of 10%. In Problem 2.75, the investor, after holding a similar bond for 10 years, received a realized return of 11.2%. Explain why in the second case the yield from owning the bond increased.

In problems 2.74 and 2.75, after purchasing a bond, its yield to maturity increased, therefore, the price decreased. The short-term investor loses when the rate rises. For a long-term investor, this effect is less pronounced or absent, since as the bond's maturity approaches, its price approaches its face value. In addition to this, the short-term investor reinvests the coupons at a higher interest rate (12%) for a shorter time than the long-term investor. Therefore, the realized return for a long-term investor will be higher than that of a short-term investor.

Problem 2.77.

An investor bought a coupon bond with ten years until maturity for RUB 887. Yield to maturity of the bond is 12%. The coupon on the bond is paid once a year. The next day, the yield to maturity of the bond fell to 11%, and its price rose to 941.11 rubles. Determine how long an investor must hold the bond for the realized return to be equal to 12% if the market interest rate remains at 11%.

The realized return is:

where T is the time the investor holds the bond.

Let us find the value of T from (2.22). To do this, we transform (2.22) as follows:

Let’s take the natural logarithm from both sides of (2.23) and take the exponent out of the sign of the logarithm:

In order for the investor's realized return to be 12% per annum, he must sell the bond through:

Problem 2.78.

An investor bought a coupon bond with ten years until maturity for RUB 887. The face value of the bond is 1000 rubles, the coupon is 10%, paid once a year. Yield to maturity of the bond is 12%. The next day, the bond's yield to maturity rose to 13%. Determine how long an investor must hold the bond for the realized return to be equal to 12% if the market interest rate remains at 13%.

With the yield to maturity rising to 13%, the bond price fell to RUB 837.21. In order for the investor's realized return to be 12% per annum, he must sell the bond through:

Problem 2.79.

For the conditions of problem 2.78, determine how long the investor must hold the bond so that the realized yield is equal to 12.3% if the interest rate on the market remains at 13%.

Problem 2.80.

An investor bought a coupon bond with a yield to maturity of 8%. The face value of the bond is 1000 rubles, the coupon is 8.5%, paid once a year. The next day, the bond's yield to maturity rose to 8.2%. Determine how long an investor must hold the bond for the realized return to be equal to 8% if the market interest rate remains at 8.2%. The bond has 5 years until maturity.

The investor bought the bond at a price of RUB 1,019.96. After the yield to maturity increased, the bond price fell to RUB 1,011.92. The investor must sell the bond through:

2.4. Duration

Problem 2.81.

Derive the Macaulay duration formula based on the definition of duration as the elasticity of the bond price with respect to the interest rate.

According to the definition of duration as the elasticity of the bond price with respect to the interest rate, we can write:

where D is the Macaulay duration; P - bond price; dP - small change in bond price; r is the yield to maturity of the bond; dr is a small change in yield to maturity.

In formula (2.25), there is a minus sign to make the duration indicator a positive value, since the bond price and interest rate change in opposite directions.

In equation (2.25), the ratio dP/dr is the derivative of the bond price with respect to the interest rate. Based on the formula for the price of a bond with coupons paid once a year (2.1), it is equal to:

Let us substitute the value dP/dr from equality (2.26) into equality (2.25):

Problem 2.82.

He commemorated bonds of 1000 rubles. coupon 10%, paid once a year, until maturity of the paper 4 years, yield to maturity 8%. Determine the Macaulay duration of the bond.

The bond price is:

Duration is:

Problem 2.83.

The face value of the bond is 1000 rubles. coupon 10%, paid once a year, until maturity of the paper 4 years, yield to maturity 10%. Determine the Macaulay duration of the bond.

According to (2.27), the duration is equal to:

Problem 2.84.

The face value of the bond is 1000 rubles. coupon 10%, paid once a year, until maturity of the paper 4 years, yield to maturity 12%. Determine the Macaulay bond duration.

The bond price is:

Duration is:

Problem 2.85.

The face value of the bond is 1000 rubles. coupon 10%, paid once a year, until maturity of the paper 4 years, yield to maturity 13%. Determine the Macaulay bond duration.

D = 3.46 years.

Question 2.86.

How does the Macaulay duration depend on the yield to maturity of the bond?

The higher the yield to maturity, the lower the duration. This pattern is illustrated by problems 2.82 – 2.85.

Problem 2.87.

The face value of the bond is 1000 rubles. coupon 6%, paid once a year, until maturity of the paper 8 years, yield to maturity 5%. Determine the Macaulay bond duration.

D = 6.632 years.

Problem 2.88.

The face value of the bond is 1000 rubles. coupon 6.5%, paid once a year, until maturity of the paper 8 years, yield to maturity 5%. Determine the Macaulay bond duration.

D = 6.562 years.

Problem 2.89.

The face value of the bond is 1000 rubles. coupon 7%, paid once a year, until maturity of the paper 8 years, yield to maturity 5%. Determine the Macaulay bond duration.

D = 6.495 years.

Question 2.90.

How does the Macaulay duration depend on the bond coupon?

The larger the coupon, the lower the duration. This pattern is illustrated by tasks 2

Problem 2.91.

The face value of the bond is 1000 rubles. coupon 10%, paid twice a year, until maturity of the paper 4 years, yield to maturity 10%. Determine the Macaulay bond duration.

What would you like to achieve? investing in bonds? Save money and get extra income? Saving for an important goal? Or maybe you dream of how to gain financial freedom with the help of these investments? Whatever your goal, it pays to understand the return your bonds provide and be able to tell a good investment from a bad one. There are several principles for assessing income, knowledge of which will help with this.

What types of income do bonds have?

Bond yield- this is the amount of income as a percentage received by an investor from investing in a debt security. Interest income according to them, it is formed from two sources. On the one hand, fixed coupon bonds, like deposits, have interest rate, which is charged on the face value. On the other hand, have bonds, like stocks, have a price, which may change depending on market factors and the situation in the company. True, changes in the price of bonds are less significant than those of stocks.

Total bond yield includes coupon yield and takes into account its acquisition price. In practice, different profitability estimates are used for different purposes. Some of them only show coupon yield, others additionally take into account purchase price, still others show return on investment depending on tenure- before sale on the market or before redemption by the issuer who issued the bond.

To make the right investment decisions, you need to understand what types of bond returns there are and what they show. There are three types of returns, the management of which turns an ordinary investor into a successful rentier. These are the current yield from interest on coupons, the yield on sale and the yield on securities to maturity.

What does the coupon rate indicate?

Coupon rate is the base percentage of the bond's face value, also called coupon yield . The issuer announces this rate in advance and periodically pays it on time. Coupon period for most Russian bonds - six months or a quarter. An important nuance is that the coupon yield on the bond is accrued daily, and the investor will not lose it even if he sells the paper ahead of schedule.

If a bond purchase and sale transaction occurs within the coupon period, then the buyer pays the seller the amount of interest accumulated from the date of the last coupon payments. The amount of this interest is called accumulated coupon income(NKD) and added to current market price of the bond. At the end of the coupon period, the buyer will receive the coupon in its entirety and thus compensate for its expenses associated with the compensation of the accrued income to the previous owner of the bond.

Exchange quotes of bonds from many brokers show the so-called net bond price, excluding NKD. However, when an investor orders a purchase, the NCD will be added to the net price, and the bond may suddenly be worth more than expected.

When comparing bond quotes in trading systems, online stores and applications of different brokers, find out what price they indicate: net or with accrued income. After this, estimate the final costs of purchasing from a particular brokerage company, taking into account all costs, and find out how much money will be written off from your account if you purchase securities.

Coupon yield

As the accumulated coupon yield (ACY) increases, the value of the bond increases. After the coupon is paid, the cost is reduced by the amount of the NKD.

NKD- accumulated coupon income
WITH(coupon) - the amount of coupon payments for the year, in rubles
t(time) - number of days from the beginning of the coupon period

Example: the investor bought a bond with a face value of 1000 rubles with a semi-annual coupon rate of 8% per year, which means a payment of 80 rubles per year, the transaction took place on the 90th day of the coupon period. His additional payment to the previous owner: NKD = 80 * 90 / 365 = 19.7 ₽

Is the coupon yield the investor's interest?

Not really. Every coupon period the investor receives a certain amount of interest in relation to face value bonds to the account that he indicated when concluding an agreement with the broker. However, the real interest that an investor receives on invested funds depends on bond purchase prices.

If the purchase price was higher or lower than face value, then profitability will differ from the base coupon rate set by the issuer in relation to the face value of the bond. The easiest way to evaluate real investment income- correlate the coupon rate with the purchase price of the bond using the current yield formula.

From the presented calculations using this formula, it can be seen that profitability and price are related to each other by inverse proportionality. An investor receives a lower yield to maturity than the coupon when he purchases a bond at a price higher than its face value.

C.Y.
C g (coupon) - coupon payments for the year, in rubles
P(price) - purchase price of the bond

Example: the investor bought a bond with a par value of 1000 rubles at a net price of 1050 rubles or 105% of the par value and a coupon rate of 8%, that is, 80 rubles per year. Current yield: CY = (80 / 1050) * 100% = 7.6% per annum.

Yields fell - prices rose. Is this not a joke?

This is true. However, for novice investors who do not clearly understand the difference between return to sale And yield to maturity, this is often a difficult moment. If we consider bonds as a portfolio of investment assets, then its profitability for sale in the event of a rise in price, like that of stocks, will, of course, increase. But the bond yield to maturity will change differently.

The whole point is that a bond is a debt obligation, which can be compared with a deposit. In both cases, when purchasing a bond or placing money on deposit, the investor actually acquires the right to a stream of payments with a certain yield to maturity.

As you know, interest rates on deposits rise for new depositors when money depreciates due to inflation. Also, the yield to maturity of a bond always rises when its price falls. The opposite is also true: the yield to maturity falls when the price rises.

Beginners who evaluate the benefits of bonds based on comparisons with stocks may come to another erroneous conclusion. For example: when the price of a bond has increased, say, to 105% and has become more than the face value, then it is not profitable to buy it, because when the principal is repaid, only 100% will be returned.

In fact, it is not the price that is important, but bond yield- a key parameter for assessing its attractiveness. Market participants, when bidding for a bond, agree only on its yield. Bond price is a derived parameter from profitability. In effect, it adjusts the fixed coupon rate to the rate of return that the buyer and seller have agreed upon.

See how the yield and price of a bond are related in the video of the Khan Academy, an educational project created with money from Google and the Bill and Melinda Gates Foundation.

What will be the yield when selling the bond?

The current yield shows the ratio of coupon payments to the market price of the bond. This indicator does not take into account the investor's income from changes in its price upon redemption or sale. To evaluate the financial result, you need to calculate a simple return, which includes a discount or premium to the nominal value when purchasing:

Y(yield) - simple yield to maturity/put
C.Y.(current yield) - current yield, from the coupon
N
P(price) - purchase price
t(time) - time from purchase to redemption/sale
365/t- multiplier for converting price changes into percentage per annum.

Example 1: an investor purchased a two-year bond with a par value of RUB 1,000 at a price of RUB 1,050 with a coupon rate of 8% per annum and a current coupon yield of 7.6%. Simple yield to maturity: Y 1 = 7.6% + ((1000-1050)/1050) * 365/730 * 100% = 5.2% per annum

Example 2: The issuer's rating was increased 90 days after purchasing the bond, after which the price of the security rose to 1,070 rubles, so the investor decided to sell it. In the formula, let's replace the par value of the bond with its sale price, and the maturity date with the holding period. We get simple return on sale: Y 2 7.6% + ((1070-1050)/1050) * 365/90 *100% = 15.3% per annum

Example 3: The buyer of a bond sold by a previous investor paid 1,070 rubles for it - more than it cost 90 days ago. Since the price of the bond has increased, the simple yield to maturity for the new investor will no longer be 5.2%, but less: Y 3 = 7.5% + ((1000-1070)/1070) * 365/640 * 100% = 3 .7% per annum

In our example, the bond price increased by 1.9% over 90 days. In terms of annual yield, this already amounted to a serious increase in interest payments on the coupon - 7.72% per annum. With a relatively small change in price, bonds over a short period of time can show a sharp jump in profit for the investor.

After selling the bond, the investor may not receive the same 1.9% return for every three months within a year. Nevertheless, profitability converted into annual percentages, is an important indicator characterizing current cash flow investor. With its help, you can make a decision on early sale of a bond.

Let's consider the opposite situation: as yields rise, the price of the bond decreases slightly. In this case, the investor may receive a loss upon early sale. However, the current yield from coupon payments, as can be seen in the above formula, will most likely cover this loss, and then the investor will still be in the black.

The lowest risk of losing invested funds during early sale is bonds of reliable companies with a short period until maturity or redemption under an offer. Strong fluctuations in them can be observed, as a rule, only during periods of economic crisis. However, their exchange rate recovers fairly quickly as the economic situation improves or the maturity date approaches.

Transactions with safer bonds mean lower risks for the investor, but also yield to maturity or offer it will be lower on them. This is a general rule for the relationship between risk and return, which also applies when buying and selling bonds.

How to get the maximum benefit from a sale?

So, as the price rises, the bond's yield falls. Therefore, to get the maximum benefit from rising prices When selling early, you need to choose bonds whose yield may decrease the most. Such dynamics, as a rule, are shown by securities of issuers that have the potential to improve their financial position and increase credit ratings.

Large changes in yield and price can also be seen in bonds with long term to maturity. In other words, long bonds are more volatile. The thing is that long bonds generate a larger cash flow for investors, which has a greater impact on price changes. It is easiest to illustrate how this happens using the same deposits as an example.

Suppose an investor a year ago deposited money at a rate of 10% per annum for three years. And now the bank accepts money for new deposits at 8%. If our depositor could assign the deposit, like a bond, to another investor, then the buyer would have to pay the difference of 2% for each remaining year of the deposit agreement. The additional payment in this case would be 2 g * 2% = 4% on top of the amount of money in the deposit. For a bond purchased under the same conditions, the price would increase to approximately 104% of the par value. The longer the term, the higher the additional payment for the bond.

Thus, the investor will receive more profit from the sale of bonds if he chooses long papers with fixed coupon when rates in the economy decrease. If interest rates, on the contrary, rise, then holding long bonds becomes unprofitable. In this case, it is better to pay attention to securities with a fixed coupon that have short maturity, or bonds with floating rate .

What is the effective yield to maturity?

Effective yield to maturity- this is the investor’s total income from investments in bonds, taking into account the reinvestment of coupons at the rate of the initial investment. To estimate the full yield to maturity of a bond or its redemption under an offer, use the standard investment indicator - cash flow internal rate of return. She shows average annual return on investment taking into account payments to the investor over different periods of time. In other words, this return on investment in bonds.

You can independently calculate the estimated effective profitability using a simplified formula. The calculation error will be tenths of a percent. The exact yield will be slightly higher if the purchase price exceeded the par value, and slightly less if it was below the par value.

YTM OR (Yield to maturity) - yield to maturity, approximate
C g (coupon) - the amount of coupon payments for the year, in rubles
P(price) - current market price of the bond
N(nominal) - bond face value
t(time) - years to maturity

Example 1: the investor purchased a two-year bond with a par value of 1000 at a price of 1050 rubles with a coupon rate of 8% per annum. Estimated effective yield to maturity: YTM 1 = ((1000 – 1050)/(730/365) + 80) / (1000 + 1050) / 2 * 100% = 5.4% per annum

Example 2: The issuer's rating was increased 90 days after purchasing the bond, and its price rose to 1,070 rubles, after which the investor decided to sell the bond. In the formula, let's replace the par value of the bond with its sale price, and the maturity date with the holding period. Let's get the approximate effective yield for sale (horizon yield): HY 2 = ((1070 – 1050)/(90/365) + 80) / (1000 + 1050) / 2 * 100% = 15.7% per annum

Example 3: The buyer of a bond sold by a previous investor paid 1,070 rubles for it - more than it cost 90 days ago. Since the price of the bond has increased, the effective yield to maturity for the new investor will no longer be 5.4%, but less: YTM 3 = ((1000 – 1070)/(640/365) + 80) / (1000 + 1050) / 2 * 100% = 3.9% per annum

The easiest way to find out the effective yield to maturity for a specific bond is to use bond calculator on the website Rusbonds.ru. An accurate calculation of effective profitability can also be obtained using financial calculator or Excel programs through the special function “ internal rate of return"and its varieties (XIRR). These calculators will calculate the rate effective yield according to the formula below. It is calculated approximately using the method of automatic selection of numbers.

How to find out the yield of a bond, watch the video from the Higher School of Economics with Professor Nikolai Berzon.

The most important thing!

✔ The key parameter of a bond is its yield, the price is a derived parameter from the yield.

✔ When a bond's yield falls, its price rises. And vice versa: when yields rise, the price of the bond falls.

✔ You can compare comparable things. For example, the net price without taking into account the accrued income is with the net price of the bond, and the full price with the accrual income is with the full price. This comparison will help you make a decision when choosing a broker.

✔ Short one- and two-year bonds are more stable and less dependent on market fluctuations: investors can wait for the maturity date or redemption by the issuer under an offer.

✔ Long bonds with a fixed coupon, when rates in the economy decrease, allow you to earn more by selling them.

✔ A successful rentier can receive three types of income from bonds: from coupon payments, from changes in the market price upon sale, or from reimbursement of the face value upon redemption.

An intelligible dictionary of terms and definitions of the bond market. A reference base for Russian investors, depositors and rentiers.

Discount Bond- discount to the face value of the bond. A bond whose price is below par is said to be selling at a discount. This occurs if the seller and buyer of the bond have agreed on a higher rate of return than the coupon set by the issuer.

Coupon yield of bonds- this is the annual interest rate that the issuer pays for the use of borrowed funds raised from investors through the issue of securities. Coupon income is accrued daily and calculated at a rate based on the face value of the bond. The coupon rate can be constant, fixed or floating.

Bond coupon period- the period of time after which investors receive interest accrued on the face value of the security. The coupon period of most Russian bonds is a quarter or six months, less often - a month or a year.

Bond Premium- an increase to the face value of the bond. A bond whose price is higher than its face value is said to sell at a premium. This occurs if the seller and buyer of the bond have agreed on a lower rate of return than that set by the issuer for the coupon.

Simple yield to maturity/offer- calculated as the sum of the current yield from the coupon and the yield from the discount or premium to the face value of the bond, as a percentage per annum. Simple yield shows an investor the return on an investment without reinvesting coupons.

Simple return to sale- calculated as the sum of the current yield from the coupon and the yield from the discount or premium to the sale price of the bond, as a percentage per annum. Since this yield depends on the price of the bond at sale, it can differ greatly from the yield to maturity.

Current yield, from coupon- is calculated by dividing the annual cash flow from coupons by the market price of the bond. If you use the purchase price of the bond, the resulting figure will show the investor the annual return on his cash flow from coupons on the investment.

Full bond price- the sum of the market price of the bond as a percentage of the nominal value and the accumulated coupon income (ACI). This is the price an investor will pay when purchasing the paper. The investor compensates for the costs of paying the NKD at the end of the coupon period, when he receives the coupon in full.

Bond price net- the market price of the bond as a percentage of the nominal value without taking into account the accumulated coupon income. It is this price that the investor sees in the trading terminal; it is used to calculate the return received by the investor on the invested funds.

Effective yield to maturity/put- average annual return on initial investments in bonds, taking into account all payments to the investor over different periods of time, redemption of par value and income from reinvestment of coupons at the rate of initial investments. To calculate profitability, the investment formula for the rate of internal return on cash flow is used.

Effective return on sale- average annual return on initial investments in bonds, taking into account all payments to the investor over different periods of time, proceeds from sales and income from reinvestment of coupons at the rate of initial investment. The effective yield on sale shows the return on investment in bonds for a certain period.