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 discounting times 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 interest 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 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.

§ 18.1. BASIC DEFINITIONS

The two main forms of corporate capital are credit and ordinary shares. In this chapter, we look at valuing bonds, the main type of long-term debt.

A bond is a debt obligation issued by a business company or government under which the issuer (that is, the borrower who issued the bond) guarantees to the lender the payment of a specified amount at a fixed point in time in the future and the periodic payment of specified interest (at a fixed or floating interest rate).

The nominal (nominal) value of a bond is the amount sum of money, indicated on a bond that the issuer borrows and promises to repay at the end of a specified period (maturity).

The maturity date is the day on which the face value of the bond is due to be paid. Many bonds contain a condition under which the issuer has the right to repurchase the bond before maturity. Such bonds are called callable. The issuer of a bond is required to periodically (usually once every year or six months) pay a certain percentage of the face value of the bond.

Coupon interest rate is the ratio of the amount of interest paid to the face value of the bond. It determines the initial market value of the bond: the higher the coupon interest rate, the higher market price bonds. At the time of bond issue, the coupon interest rate is set equal to the market interest rate.

Within a month from the date of issue, the bonds are called new issue bonds. If a bond is traded on the secondary market for more than a month, it is called a marketable bond.

§ 18.2. BASIC METHOD FOR ASSESSING THE VALUE OF A BOND

The bond can be viewed as a simple post-numerando annuity, consisting of coupon interest payments and reimbursement of the bond's face value. Therefore, the present value of the bond is equal to the present value of this annuity.

Let i be the current market interest rate, k be the coupon interest rate, P be the face value of the bond, n be the remaining maturity of the bond, R = kP be the coupon payment, An be the current market value of the bond.

R R R R ... R R R+P

O 1 2 3 4 ... n-2 n-1 n 1 - 1/(1 + i)n

Then An = R - + Р/(1 +ї)п. We took advantage

formula for the modern value of simple annuity post-numerando.

Example 70. The nominal value of the bond is P = 5,000 rubles, the coupon interest rate is k = 15\%, the remaining maturity of the bond is n = 3 years, the current market interest rate is i = 12\%. Let's determine the current market value of the bond.

The amount of coupon payments is equal to R = kP = 0.15x5000 = 750 rubles. Then the current market value of the bond

1-1/(1 + 0* n 1-1/(1 + 0.12)3

An = R - + P/(1 + 0 = 750 --- +

5000 i 5360.27 rubles, that is, in case i< k текущая

the market value of the bond is higher than the par value of the bond R.

Problem 70. Determine the current market value of the bond in example 70, if the current market interest rate i = 18\%.

§ 18.3. BOND RATE OF RETURN

Another the most important characteristic bonds is the rate of return. The rate of return is calculated using the following formula:

rate of return

coupon payment bond price at the end of the period

bond price at the beginning of the period

Example 71. Bond with a nominal value of P = 1000 rubles. with a coupon interest rate k = 10\% was purchased at the beginning of the year for 1200 rubles. (that is, at a price higher than face value). After receiving the coupon payment at the end of the year, the bond was sold for RUB 1,175. Let's determine the rate of profit for the year.

The amount of coupon payments is equal to R = kP = 0.1x1000 =

Then the rate of return = (coupon payment + bond price at the end of the period, bond price at the beginning of the period)/(bond price at the beginning of the period) = (100 + 1175 -

1200)/1200 0,0625 (= 6,25\%).

Problem 71. Bond with a nominal value of P = 1000 rubles. with a coupon interest rate k = 15\% was purchased at the beginning of the year for 700 rubles. (that is, at a price below face value). After receiving the coupon payment at the end of the year, the bond was sold for 750 rubles. Determine the rate of profit for the year.

§ 18.4. BOND YIELD AT MATURITY AT THE END OF THE TERM

Very often, an investor solves the problem of comparing different bonds with each other. How to determine the interest rate (yield) at which a bond generates income? To do this, you need to solve for i the equation Аn = d1-1/(1 + 0" + р/(1 + .)В

We will consider two approximate methods for solving this nonlinear equation.

§ 18.4.1. Average method

Find the total amount of payments on the bond (all coupon payments and the face value of the bond):

Then the yield of the bond is calculated using the following formula:

bond yield

average profit for one period average cost of a bond

Example 72. Bond with a nominal value of P = 1000 rubles. with a coupon interest rate k = 10\% and a repayment period n = 10 years was purchased for 1200 rubles. Let's determine the bond's yield using the average method.

The amount of coupon payments is equal to R = kP = 0.їх 1000 = 100 rubles.

Then the total amount of payments is equal to nR + P = 10x100 + + 10U0 = 2000 rubles.

Hence, total profit = total amount of payments, bond purchase price 2000 1200 = 800 rubles.

Therefore, the average profit for one period = (total profit b)/(number of periods) = 800/10 = 80 rubles.

Average cost of a bond = (face value of the bond + purchase price of the bond)/2 = (1000 + + 1200)/2 = 1100 rubles.

Then the yield of the bond * (average profit for one period)/(average cost of the bond) is equal to 80/1100 * 0.073 (= 7.3%).

Problem 72. Bond with a nominal value of P = 1000 rubles. with a coupon interest rate k = 15\% and a repayment period n = 10 years was purchased for 800 rubles. Determine the bond yield using the average method.

§ 18.4.2. Interpolation method

The interpolation method provides a more accurate approximation of a bond's yield than the average method. Using the method of averages, you need to find two different close values ​​of the current market interest rate i$ and ii such that the current market price of the bond An is between An(ii) and An(i0): An(ii)< Ап < An(i0), где значения An(io) и An(ii) вычисляются по следующей формуле: 1 - 1/(1 + i)n

An(i) = R ^ + P/(1 + 0L. Here P is the nominal

bond price, n - remaining term to maturity

bonds, R - coupon payment.

Then the approximate value of the bond's yield is ravAp - AMg)) but: / to + " "l (h io).

Example 73. Let's determine the bond yield using the interpolation method in Example 72.

Using the method of averages, the bond yield value i = 0.073 was obtained. Let's set *o = 0.07 and = 0.08 and determine the current value of the bond at these values ​​of the market interest rate:

An(i0) = Rlzl^f + m + iof . 1001-1/(іу07)У> + i0 0.07

W* 1210.71 rub. (1 + 0.07)10

Anih)=Rizi^±hi+т+ііГ=уо1-^1;^10+

1000 1lo, OL l

+ * 1134.20 rub.

Since Ap = 1200 rubles, then the conditions Ap(i)< Ап< An(io) выполнены (1134,20 < 1200 < 1210,71).

Then the approximate value of the bond yield is:

i. i0 + A" A»™ ih i0) 0.07 + 1200-121°"71 x

An(ig) An(i0) 1 and 1134.20 1210.71

x(0.08 0.07) 0.071 (= 7.1%).

Problem 73. Determine the bond yield using the interpolation method in Problem 72.

§ 18.5. REVOKABLE BONDS YIELD

Callable bonds contain a condition under which the issuer has the right to repurchase the bond before maturity. The investor must take this condition into account when calculating the yield of such a bond.

The yield of a callable bond is found from the following 1 - 1/(1 + i)N

equations: AN = R ~ - + T/(1 + i)N, where AN is the current market value of the bond, P is the par value of the bond, N is the remaining period until the call

bonds, R - coupon payment, T - bond call price (the amount paid by the issuer in case early repayment bonds).

The approximate value of the yield of a callable bond can be determined using the average method or the interpolation method.

Comment. The Excel fx function wizard contains the financial functions PRICE and YIELD, which allow you to calculate the current market value of a bond and the yield of the bond, respectively. For these functions to be available, the Analysis Package add-on must be installed: select Tools -* Add-ons and check the box next to the Analysis Package command. If the Analysis package command is missing, you need to install Excel.

Finance function PRICE returns the current market value of a bond with a nominal value of 100 rubles: fx -+ financial -* PRICE -+ OK. A dialog box appears that you need to fill out. Settlement date is the date on which the current market value of the Ap bond is determined (in date format). Maturity is the maturity date of the bond (in date format). Rate is the coupon interest rate k. Yield (Yld) is the current market interest rate i. Redemption is the face value of the bond (= 100 rubles). Frequency

this is the number of coupon payments per year. Basis is the practice of calculating interest, possible values:

or not specified (American, 1 full month = 30 days,

year = 360 days); 1 (English); 2 (French); 3 (the period is equal to the actual number of days, 1 year = 365 days); 4 (German). OK.

This is the date on which the market price of the bond is determined, and the maturity date of the bond, respectively. Then Ap 50хЦЯ#А("9.6.2004"; "9.6.2007"; 0.15; 0.12; 100; 1) « * 5360.27 rub.

The financial function INCOME (YIELD) returns the yield of the bond: fx -* financial -* INCOME -+ OK. A dialog box appears that you need to fill out. Price (Pg)

Option	№№ tasks	Option	№№ tasks	Option	№№ tasks
1	1, 30, 31	6	6, 25, 36	11	11, 20, 41
2	2, 29, 32	7	7, 24, 37	12	12, 19, 42
3	3, 28, 33	8	8, 23, 38	13	13, 18, 43
4	4, 27, 34	9	9, 22, 39	14	14, 17, 44
5	5, 26, 35	10	10, 21, 40	15	15, 16, 45

Task 1. The nominal value of an ordinary bond is N = 5,000 rubles. Coupon interest rate c = 15%, remaining bond maturity n = 3 years, current market interest rate i = 18%. Determine the current market value of the bond.

Task 2. Determine the current value of a three-year bond with a face value of 1000 units. and an annual coupon rate of 8%, paid quarterly if the rate of return (market rate) is 12%.

Task 3. Determine the current value of 100 units. par value of a bond with a maturity of 100 years, based on the required rate of return of 8.5%. The coupon rate is 7.72%, paid semi-annually. (The bond is perpetual).

Task 4. What price would an investor pay for a zero-coupon bond with a face value of 1,000 units? and repayment in three years if the required rate of return is 4.4%.

Task 5. The bank's bond has a face value of 100,000 units. and maturity in 3 years. The coupon rate on the bond is 20% per annum, accrued once a year. Determine the cost of the bond if the investor's required return is 25%, and the coupon income is accumulated and paid along with the face value at the end of the circulation period.

Task 6. Perpetual bonds with a coupon of 6% of the face value and a face value of 200 monetary units. should provide the investor with a return of 12% per annum. At what maximum price will an investor buy this financial instrument?

Task 7. You are the holder of a bond with a par value of $5,000 that provides a constant annual income of $100 for 5 years. The current interest rate is 9%. Calculate the current value of the bond.

Task 8. Estimate the market value of a municipal bond proposed for public circulation, the par value of which is 100 rubles. There are 2 years left until the bond matures. The nominal interest rate on the bond (used to calculate the annual coupon income as a percentage of its face value) is 20%, the coupon income is paid quarterly. The yield on government bonds comparable in terms of risks (also risk-free for holding and with the same maturity) is 18%.

Task 9. Estimate the market value of a municipal bond proposed for public circulation, the par value of which is 200 rubles. There are 3 years left until the bond matures. The nominal interest rate on the bond (used to calculate the annual coupon yield as a percentage of its face value) is 15%. The yield on government bonds comparable in terms of risks (also risk-free for holding and with the same maturity) is 17%.

Problem 10. The company announces the issue of bonds with a par value of 1000 thousand rubles. with a coupon rate of 12% and a maturity of 16 years. At what price will these bonds sell in an efficient capital market if investors' required return on bonds with a given level of risk is 10%?

Problem 11. The company issues bonds with a par value of 1000 thousand rubles, with a coupon rate of 11%. The required return for investors is 12%. Calculate the current value of the bond with the bond maturity: a) 30 years; b) 15 years; c) 1 year.

Problem 12. The par value of the bond is 1200 rubles, the maturity period is 3 years, the coupon rate is 15%, the coupon payment is once a year. It is necessary to find the intrinsic value of a bond if the rate of return acceptable to the investor is 20% per annum.

Problem 13. The par value of the bond is 1,500 rubles, the maturity period is 3 years, the coupon rate is 12%, the coupon payment is 2 times a year. It is necessary to find the intrinsic value of a bond if the rate of return acceptable to the investor is 14% per annum.

Problem 14. Terms of the bond issue: term 5 years, coupon yield - 8%, semi-annual payments. The expected average market return is 10.5% per annum. determine the current bond rate.

Problem 15. There are two options for bond circulation conditions. Coupon rates are 8% and 12%, terms are 5 and 10 years. The expected market rate of return is 10%. Coupon income is accumulated and paid at the end of the circulation period along with the face value. Choose the cheapest option.

Bond yield

Problem 16. There are two 3-year bonds. Bond D with an 11% coupon is selling at 91.00. Bond F with a 13% coupon is sold at par. Which bond is better?

Problem 17. Coupon 3-year bond A with a par value of 3 thousand rubles. sold at 0.925. The coupon payment is provided once a year in the amount of 360 rubles. A 3-year bond B with a 13% coupon is sold at par. Which bond is better?

Problem 18. The nominal value of a zero-coupon bond is 1000 rubles. Current market value is 695 rubles. Repayment period is 4 years. Deposit rate - 12%. Determine the feasibility of purchasing a bond.

Problem 19. Bond with a nominal value of N = 1000 rubles. with a coupon rate of c = 15% was purchased at the beginning of the year for 700 rubles. (at a price below par). After receiving the coupon payment at the end of the year, the bond was sold for 750 rubles. Determine the profitability of the operation for the year.

Problem 20. Bond with a nominal value of 1000 rubles. with a coupon rate of 15% and a maturity of 10 years was purchased for 800 rubles. Determine the bond yield using the interpolation method.

Problem 21. Bond with a nominal value of 1,500 rubles. with a coupon rate of 12% (semi-annual compounding) and a repayment period of 7 years was purchased for 1000 rubles. Determine the bond yield using the interpolation method.

Problem 22. A perpetual bond yielding a 20% coupon income was purchased at an exchange rate of 95. Determine the financial efficiency of the investment, provided that interest is paid: a) once a year, and b) quarterly.

Problem 23. The corporation issued zero coupon bonds maturing in 5 years. The selling rate is 45. Determine the yield of the bond on the maturity date.

Problem 24. A bond yielding 10% per annum relative to par was purchased at an exchange rate of 60, with a maturity period of 2 years. Determine the total return to the investor if par value and interest are paid at the end of the maturity date.

Problem 25. A zero coupon bond has been issued with a maturity of 10 years. The bond rate is 60. Find the total yield on the maturity date.

Problem 26. A bond with an income of 15% per annum of par value, exchange rate 80, maturity 5 years. Find the total yield if par and interest are paid at maturity.

Problem 27. A bond with a maturity of 6 years with an interest rate of 10% was purchased at an exchange rate of 95. Find the total yield using the interpolation method.

Problem 28. The current market rate of the bond is 1200 rubles, the par value of the bond is 1200 rubles, the maturity period is 3 years, the coupon rate is 15%, coupon payments are annual. Determine the total yield of the bond using the average method and the interpolation method.

Problem 29. A five-year bond paying interest once a year at a rate of 8% is purchased at an exchange rate of 65. Determine the current and total yield.

Problem 30. Coupon 5-year bond W with a par value of 10 thousand rubles. sold at the rate of 89.5. The coupon payment is provided once a year in the amount of 900 rubles. A 6-year V bond with an 11% coupon is sold at par. Which bond is better?

Bond Risk Assessment

Problem 31. The possibility of purchasing OJSC bonds, the current quote of which is 84.1, is being considered. The bond has a maturity of 6 years and a coupon rate of 10% per annum, payable semi-annually. The market rate of return is 12%.

c) How will your decision be affected by the information that the market rate of return has increased to 14%?

Problem 32. The OJSC issued 5-year bonds with a coupon rate of 9% per annum, payable semi-annually. At the same time, 10-year OJSC bonds with exactly the same characteristics were issued. The market rate at the time of issue of both bonds was 12%.

Problem 33. The OJSC issued 6-year bonds with a coupon rate of 10% per annum, payable semi-annually. At the same time, 10-year OJSC bonds were issued with a coupon rate of 8% per annum, paid once a year. The market rate at the time of issue of both bonds was 14%.

a) At what price were the enterprise bonds placed?

b) Determine the durations of both bonds.

Problem 34. The possibility of purchasing Eurobonds of the OJSC is being considered. Release date: 06/16/2008. Repayment date – 06/16/2018. Coupon rate – 10%. Number of payments – 2 times a year. The required rate of return (market rate) is 12% per annum. Today is December 16, 2012. The average exchange price of the bond is 102.70.

b) How will the price of a bond change if the market rate: a) increases by 1.75%; b) will fall by 0.5%.

Problem 35. The initial price of a 5-year bond is 100 thousand rubles, the coupon rate is 8% per annum (paid quarterly), the yield is 12%. How will the bond price change if the yield increases to 13%.

Problem 36. You need to pay off $200,000 in three years from your bond portfolio. The duration of this payment is 3 years. Let’s say you can invest in two types of bonds:

1) zero-coupon bonds with a maturity of 2 years (current rate - $857.3, par value - $1000, placement rate - 8%);

2) bonds with a maturity of 4 years (coupon rate - 10%, par value - $1000, current rate - $1066.2, placement rate - 8%).

Problem 37. The possibility of purchasing OJSC bonds, the current quote of which is 75.9, is being considered. The bond has a circulation period of 5 years and a coupon rate of 11% per annum, payable semi-annually. The market rate of return is 14.5%.

a) Is buying a bond a profitable transaction for an investor?

b) Determine the duration of the bond.

c) How will your decision be affected by the information that the market rate of return has decreased to 14%?

Problem 38. The OJSC issued 4-year bonds with a coupon rate of 8% per annum, payable quarterly. At the same time, 8-year OJSC bonds were issued with a coupon rate of 9% per annum, paid semi-annually. The market rate at the time of issue of both bonds was 10%.

a) At what price were the enterprise bonds placed?

b) Determine the durations of both bonds.

c) Shortly after release, the market rate increased to 14%. Which bond's price will change more?

Problem 39. The OJSC issued 5-year bonds with a coupon rate of 7.5% per annum, payable quarterly. At the same time, 7-year OJSC bonds were issued with a coupon rate of 8% per annum, paid semi-annually. The market rate at the time of issue of both bonds was 12.5%.

a) At what price were the enterprise bonds placed?

b) Determine the durations of both bonds.

c) Shortly after issuance, the market rate dropped to 12%. Which bond's price will change more?

Problem 40. The possibility of purchasing OJSC bonds is being considered. Release date: 01/20/2007. Repayment date – 01/20/2020. Coupon rate – 5.5%. Number of payments – 2 times a year. The required rate of return (market rate) is 9.5% per annum. Today is 01/20/2013. The average exchange rate price of the bond is 65.5.

a) Determine the duration of this bond on the date of the transaction.

b) How will the price of a bond change if the market rate: a) increases by 2.5%; b) will fall by 1.75%.

Problem 41. The face value of a 16-year bond is 100 rubles, the coupon rate is 6.2% per annum (paid once a year), the yield is 9.75%. How will the bond price change if the yield increases to 12.5%. Perform analysis using duration and convexity.

Problem 42. You need to pay off $50,000 in three years from your bond portfolio. The duration of this payment is 5 years. There are two types of bonds available on the market:

1) zero-coupon bonds with a maturity of 3 years (current rate - $40, par value - $50, placement rate - 12%);

2) bonds with a maturity of 7 years (coupon rate - 4.5%, coupon income is paid semi-annually, par value - $50, current rate - $45, placement rate - 12%).

Build an immunized bond portfolio. Define total cost and the number of bonds purchased.

Problem 43. The face value of a 10-year bond is 5,000 rubles, the coupon rate is 5.3% per annum (paid once a year), the yield is 10.33%. How will the bond price change if the yield increases to 11.83%. Perform analysis using duration and convexity.

Problem 44. The possibility of purchasing OJSC bonds, the current quote of which is 65.15, is being considered. The bond has a maturity of 5 years and a coupon rate of 4.5% per annum, payable quarterly. The market rate of return is 9.75%.

a) Is buying a bond a profitable transaction for an investor?

b) Determine the duration of the bond.

c) How will your decision be affected by the information that the market rate of return has increased to 12.25%?

Problem 45. You need to pay off $100,000 in three years from your bond portfolio. The duration of this payment is 4 years. There are two types of bonds available on the market:

1) zero-coupon bonds with a maturity of 2.5 years (current rate - $75, par value - $100, placement rate - 10%);

2) bonds with a maturity of 6 years (coupon rate - 6.5%, coupon income is paid quarterly, par value - $100, current rate - $85, placement rate - 10%).

Build an immunized bond portfolio. Determine the total cost and quantity of bonds purchased.

1. Anshin V.M. Investment analysis. - M.: Delo, 2002.

2. Galanov V.A. Market valuable papers: textbook. - M.: INFRA-M, 2007.

3. Kovalev V.V. Introduction to financial management. - M.: Finance and Statistics, 2007

4. Handbook of financiers in formulas and examples / A.L. Zorin, E.A. Zorina; Ed. E.N. Ivanova, O.S. Ilyushina. - M.: Professional publishing house, 2007.

5. Financial mathematics: math modeling financial transactions: textbook allowance / Ed. V.A. Polovnikov and A.I. Pilipenko. - M.: University textbook, 2004.

6. Chetyrkin E.M. Bonds: theory and yield tables. - M.: Delo, 2005.

7. Chetyrkin E.M. Financial mathematics. – M.: Delo, 2011.

The practice of forming investment portfolios of international companies shows that investors often do not have enough information about market prices for bonds to optimize their portfolio. Thus, when selecting specific bonds for an optimal investment portfolio, they need to evaluate the financial effectiveness of their decisions, which is almost impossible to do without calculating the profitability of the securities selected for the investment portfolio. Calculation of bond yield, or so-called investment rate, what a bond will provide when purchased for a given price remains perhaps the most important concern regarding bonds. Only by deciding it can an investor determine which of several bonds will provide him with the best investment.

In the most general case, under profitability any investment is subject to an interest rate that allows the present value to be equalized cash flows a specific investment with the price (cost) of the investment.

In the case of investments in bonds, the bond yield is the interest rate r satisfying the following equations:

1) zero-coupon bonds:

Determining the yield of a zero-coupon bond

Zero coupon bond yield - this, in accordance with the above, is the annual interest rate received by an investor who purchases and holds a given bond until its maturity.

To determine the yield on zero-coupon bonds that mature in more than one year, use the bond's present value formula

Example. Consider a zero-coupon bond with a maturity of 2 years. (n = 2), the nominal value of which is $1000, and the purchase price is $880. The required return is 8% per annum.

Its profitability will be

2) bonds with coupon payments:

The calculation indicates that it is inappropriate for the investor to purchase the bond in question.

Determining the yield on a coupon bond

For a coupon bond, in contrast to a zero-coupon bond, a distinction is made between current yield And internal rate of return or yield to maturity.

Current yield is calculated using the formula

where is the current yield; WITH – coupon income on the bond (coupon); R - current price of the bond.

Note. It is the current price that is used here, not the price that the investor paid for the bond.

When calculating the current yield, only coupon payments are taken into account. Other other sources of income flowing to the bond owner are not considered. It does not take into account, for example, capital gains realized by an investor who purchases a bond at a discount and holds it to maturity; at the same time, the loss that an investor suffers if he holds a bond purchased at a premium to maturity is not considered. Time value Money is also not taken into account here.

Therefore, the current yield is, figuratively speaking, a photograph of the yield on this moment time, which is in next moment may change in accordance with changes in the market price of the bond. It is advisable to use the current yield indicator when there is little time left before the bond matures, since in this case its price is unlikely to experience significant fluctuations.

A more objective indicator of profitability is the yield to maturity, or internal yield, since its calculation takes into account not only the coupon yield and the price of the bond, but also the period of time that remains until maturity. Internal yield can be calculated using the formula for estimating the market price of a bond

Bonds are the subject of lively trading, so participants stock market not only the nominal value and coupon interest rate are known, but also the market price of each security. If we assume that the market is characterized by a state of perfect competition, we can assume that the price of a bond is equal to its present value.

Thus, the buyer of the bond knows the weight of the parameters of the bond price equation, except for the discount rate r. Therefore, the present value formula can be used to market information calculate the value of the discount rate, or internal rate of return r .

Unfortunately, this equation cannot be solved in its final form: profitability can only be calculated using a special computer program. You can also 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 (Fig. 3.8).

Rice. 3.8.

Sometimes, to make a financial decision, it is enough to determine only the approximate (approximate) level of bond yield. By the way, 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 has the form

Where r – internal yield (yield to maturity); N – face value of the bond; R – bond price; P – number of years until maturity; WITH – coupon income; – average annual income; – average bond price.

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

For example, when estimating the internal yield of a bond with a five-year maturity and a 10% coupon rate with a par value of $1,000 and current price$1059.12 the exact solution is 8.5%; the traditional formula gives a value of 8.56%, and the R. Rodriguez formula gives a value of 8.48%. This formula provides 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. It should also be noted that the error in calculations using approximate estimation formulas is higher the more years remain until the bond matures. If a bond is sold at a discount, the formulas under consideration give an underestimated value of the bond's yield; if at a premium, then an overestimated value.

The ability to calculate the internal yield of bonds is so important that special computer programs have been developed to determine the values G for any combination of bond price, maturity, coupon rate, and face value. Nowadays, even pocket calculators are produced that can perform calculations of this kind.

Example. An 8% coupon bond with a par value of $1,000 was purchased for $1,050 with four years to maturity. Assuming that coupons are redeemed once a year, determine the internal rate of return.

Solution.

Let's use the formula to calculate the approximate value of the bond's internal yield:

Applying the substitution method, we get:

Since (1047.20 from 1050), we repeat the calculation for the value of r adjusted downward, taking for this, for example, r = 0.0655. In this case, it practically coincides with the market (actual) price of the bond, which allows us to complete the calculation of the internal yield indicator at the level G = 0.0655, or 6.55%.

The procedure for repeated calculations using the substitution method can be significantly accelerated if there is a graph of the dependence of the present value of the bond on the level of its internal yield. It can be built from several points, the coordinates of which (pairs of values G and present value) can be easily determined from the special tables given in each textbook on financial calculations. For the example we are considering, a graphical interpretation of the calculation of the level of internal return is shown in Fig. 3.9.

Rice. 3.9.

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

where Г[, G 2 – values ​​of respectively underestimated and overestimated levels of estimated bond yields; R, R 2 – estimated market prices of bonds corresponding to yield levels Г] 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. The relationship between the levels of the coupon rate, current yield, and yield to maturity is presented in table. 3.3.

Table 3.3

Correlation of the main parameters of the bond

- certifying the loan relationship between the creditor - the owner of the bond and the debtor - the issuer of the bond.

The bond certifies the contribution of funds by its owner and confirms the obligation to reimburse him the face value of the bond in advance fixed time with payment of a fixed percentage.

TO main parameters of the bond include: nominal price, redemption price if it differs from nominal, rate of return and interest payment terms. The moment of interest payment is stipulated in the terms of the issue and can be made once a year, semi-annually or quarterly.

Methods of paying income on a bond

In world practice, several methods are used to pay income on bonds, including:

• establishing a fixed interest payment;
• application of stepped interest rates;
• use of a floating interest rate;
• indexing the nominal value of the bond;
• sale of bonds at a discount (discount) against their nominal price;
• carrying out winning loans.

Establishing a fixed interest payment is the most common and simplest form of payment of income on bonds.

Using stepped interest rate several dates are set after which bondholders can either redeem them or leave them until the next date. In each subsequent period the interest rate increases.

The interest rate on bonds can be floating, i.e. changing regularly (every six months, etc.) in accordance with the dynamics discount rate central bank or the level of profitability placed through auction sales.

In some countries, as an anti-inflationary measure, issue of bonds with a face value indexed to growth .

Some bonds do not pay interest. Their owners earn income by buying these discounted bonds(discount against face value), and repay at face value.

Bond income may be paid in the form of winnings, which go to their individual owners based on the results of regularly held draws.

Bond rate

Bonds, being the object of purchase and sale on , have a market price, which at the time of issue may be equal to the par value, as well as lower or higher than it. Market prices differ significantly from each other, therefore, to achieve their comparability, it is calculated bond rate. The bond rate is the purchase price of one bond per 100 monetary units denomination The bond price depends on average size loan market interest rate existing at the moment, repayment period, degree of reliability of the issuer and a number of other factors.

The exchange rate is calculated using the formula:

• R k— bond rate;
• R- market price;
• N— nominal price of the bond.

Bond yield

The yield of a bond is characterized by a number of parameters that depend on the conditions proposed by the issuer. So, for example, for bonds that mature at the end of the period for which they are issued, the yield is measured:

• coupon yield;
• current profitability;
• full profitability.

Coupon yield

Coupon yield- the rate of interest that is indicated on the security and which the issuer undertakes to pay for each coupon. Coupon payments can be made quarterly, semiannually or annually.

For example, bonds have a coupon yield of 11.75% per annum. The par value of the bond is 1.0 thousand rubles. There are two coupons for each year. This means that the bond will bring a semi-annual profit of 58.75 rubles. (1.0 . 0.1175 . 0.50), and for the year - 117.5 rubles.

Current yield

Current yield (C.Y.) bonds with fixed rate coupon - defined as the ratio of the periodic payment to the purchase price.

Current yield characterizes the annual interest paid on invested capital, i.e. for the amount paid at the time of purchasing the bond. Current profitability is determined by the formula:

For example, if the coupon yield is 11.75% and the bond rate is 95.0, then its current yield will be:

At the same time, the current yield does not take into account changes in the price of the bond during its storage, i.e. another source of income.

The current yield of the bonds being sold changes in accordance with changes in their prices on the market. However, from the moment of purchase it becomes a constant (fixed) value, since the coupon rate remains unchanged. It is easy to see that the current yield of a bond purchased at a discount will be higher than the coupon, and that of a bond purchased at a premium will be lower.

The current yield indicator does not take into account the exchange rate difference between the purchase and redemption prices. Therefore, it is not suitable for comparing the performance of operations of operations with different initial conditions. Yield to maturity is used as a measure of the overall performance of a bond investment.

Yield to maturity

Yield to maturity (YTM) is the interest rate in the discount factor that equates current value the flow of payments on the bond and its market price.

Let's look at some of the most important properties of this indicator. It essentially represents the internal return on an investment (IRR). However, the real bond yield to maturity will be equal to YTM only if the following conditions are met:

It is obvious that, regardless of the investor’s wishes, the second condition is quite difficult to fulfill in practice.

The table shows the results of calculating the yield to maturity of a bond purchased at the time of issue at a par value of 1000 with maturity in 20 years and a coupon rate of 8%, paid once a year, at different reinvestment rates.

Dependence of yield to maturity on the reinvestment rate

From the above calculations it follows that there is a direct relationship between the yield to maturity and the coupon income reinvestment rate. With a decrease, the value will also decrease, and with an increase, the value will also increase.

Total return

Total return takes into account all sources of income. In a number of economic publications, the indicator of total profitability is called room rate. By determining the placement rate in the form of an annual compound or simple interest rate, one can judge the effectiveness of the purchased security.

The accrual of interest at the placement rate on the purchase price gives income equivalent to the income actually received on it for the entire period of circulation of this bond until its redemption. The placement rate is a calculated value and does not appear explicitly on the securities market.

When determining the yield of a bond, the purchase price (market price) is taken into account, which itself depends on a number of factors. The buyer of a bond, at the time of its acquisition, expects to receive income in the form of a series of fixed payments in the form of fixed interest, which are made throughout the entire period of its circulation, as well as reimbursement of its face value at the end of this period.

Therefore, if the payments received annually on bonds are placed on bank deposit or invested in any other way and will begin to generate annual interest income, then the value of the bond will be equal to the sum of two terms - the modern value of its annuities (a series of annual interest payments) and the modern value of its face value:

(9.3)

In the case where a bond provides for interest payments semi-annually or quarterly, the market value of the bond is calculated using the formulas:

Example. For a bond with a nominal value of 10.0 thousand rubles. for 10 years (the period until its repayment) interest payments in the amount of 1.0 thousand rubles will be paid annually at the end of the year. (g= 10%), which can be placed in a bank at 11% per annum. Let's determine the bond price at different interest rates.

The market price of the bond according to formula (9.3) will be: