Let's determine the bond's yield using the average method. Methods for determining bond yield

What would you like to achieve? investing in bonds? Save money and get extra income? Saving for an important goal? Or maybe you dream about 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 bond quotes from many brokers show the so-called net price of the bond, 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 par 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. I'm not kidding?

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 shares, 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 term to maturity 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 increased 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 term to maturity 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 particular 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!

✔ 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 repurchase by the issuer under an offer.

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

✔ 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 the coupon set by the issuer.

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 initial investment rate. The effective yield on sale shows the return on investment in bonds for a certain period.

§ 18.1. BASIC DEFINITIONS

The two main forms of corporate capital are debt and common stock. 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 par (face) value of a bond is the amount of money specified on the bond that the issuer borrows and promises to repay upon expiration of a certain 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.

The 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 the market value of the bond. 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 important characteristic of a bond 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 the 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 earns income? To do this, you need to solve for i the equation Аn = d1-1/(1 + 0" + р/(1 + .)В

We will look at 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 of early redemption of the bond).

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.

The financial 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

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 (Pr)

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 par 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 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 of 2 years. Determine the total return to the investor if par 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 the face value, an exchange rate of 80, and a maturity of 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 the information that the market rate of return has increased to 14% affect your decision?

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. Determine the total cost and quantity 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. Securities market: 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: mathematical modeling of 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.

When determining the yield of a bond portfolio, one proceeds from the amount reduced to a certain point in time. t o income streams from each bond in the portfolio. Let's assume that the portfolio includes " M"bonds of various types with the number of bonds of each type equal to Bonds of each type have a nominal value; the period until the bonds are redeemed N m and coupon rates with m. With this formulation of the problem, the total market value of the bond portfolio can be determined by the formula:

(5.27)

where is the market value of the bond m-th type, calculated by the formula:

(5.28)

On the other hand, a bond portfolio creates an income stream that can be characterized by the following parameters: S i– the total income from bonds of all types received at a given time t = t i, and is the yield of the bond portfolio. The present value of a given payment stream can be determined using a formula similar to formula (2.2):

(5.29)

Where N max – maximum term for payment of income on all bonds of the portfolio.

The yield of a bond portfolio can be determined under the condition, i.e., from the solution of the equation:

(5.30)

The yield value of a bond portfolio can be found by solving equation (5.30) using iterative methods or based on the method of linear interpolation between the minimum and maximum values ​​of the portfolio yield, limiting the interval within which the desired value of the bond portfolio yield is located. When using the linear interpolation method, the portfolio return can be determined by the formula:

where is the market value of the bond portfolio, determined by formulas (5.27) and (5.28);

And are the present values ​​of the payment stream, determined by formula (5.29) when used in calculating rates of return and, respectively.

Let's consider the method of calculating profitability using the example of a portfolio consisting of two types of bonds.

Example 5.2. The bond portfolio consists of two types of bonds with the following characteristics:

First bond rub.; WITH 1 = 0.08, years;

Second bond RUR; WITH 2 = 0.05, years.

Determine the yield of a bond portfolio if the number of bonds of the first and second types is the same

Solution: Let’s determine the market value of a bond of the first type using formula (5.28):

rub.

Similarly, we determine the market value of the second type of bond:

rub.

The total market value of the bond portfolio in accordance with formula (5.27) will be:

Let's calculate the total flow of payments S i for bonds of the first and second types. In table 5.2 shows the amounts of payments on bonds of the first and second types and the total flow S i.

Table 5.2

Calculation of the total payment flow

For two bonds, rub.

Since the number of bonds of the first and second types is the same, equation (5.30) can be written as:

(5.32)

Let's calculate the present value of the total flow of payments for various values:

The calculation results are given in table. 5.3.

1.8. Internal yield of a bond.

Term structure of interest rates.

We will study the analysis of financial investments under conditions of certainty using the example of fixed income securities. The most common type of such securities are bonds.

Bond is an obligation to pay predetermined amounts of money at specified times in the future. The main parameters of a bond are the nominal price (face value), maturity date, size and timing of payments on the bond. From the moment of issue until maturity, bonds are bought and sold on the stock market. The market price of a bond is set based on supply and demand and can be equal to, above, or below par.

We will consider bonds under conditions of certainty: the issuer cannot call the bond before the set maturity date, payments on the bond are set at fixed values ​​at certain points in time. In this case, the receipt of future income exactly at the specified time and in full is considered guaranteed. Such bonds are said to have no credit risk. The main risk factor remains interest rate risk – the risk of changes in market interest rates.

Consider a bond for which, through t 1 , t 2 ,…, t n years from the current moment in time t= 0, where 0< t 1 < t 2 <…< t n, promise to pay sums of money WITH 1 , WITH 2 ,…, WITH n respectively. It's obvious that C i > 0, i = 1, 2,…, n. Let P– market value of the bond at the moment t= 0. Then it is natural to assume that P < WITH 1 + WITH 2 +…+ WITH n. Moment of time t= 0 – this is the moment at which it is supposed to invest in the bond or the moment of purchasing the bond. Moment of time t= t n, when the last payment on the bond is made, is called the moment of maturity of the bond, and the term T = t n(in years) – period until maturity. Two indicators are mainly of interest to the investor - the yield and the price of the bond. Internal Return is the most important and most widely used bond valuation metric. Also known as yield to maturity.

Definition. Annual internal bond yield r is the annual compound interest rate at which the present value of the bond's stream of payments equals the market value of the bond at the time t= 0:

Here, the internal yield of a bond is defined as the annual cash flow yield WITH 1 , WITH 2 ,…,WITH n, the cost of which P(see paragraph 1.4).

In foreign practice, there is a market agreement according to which if payments on a bond are paid at regular intervals m once a year, then the annual nominal internal rate of return is applied to discount cash flow terms j :

.

Properties of a bond's internal yield.

1. A bond's internal rate of return is equal to the prevailing market interest rate for investments in alternative financial instruments with the same degree of risk. Or, in short, the rate of internal return of a bond is equal to the yield of comparable instruments.

2. A bond's annual internal yield is the rate of return an investor receives if two conditions are met:

1) the investor owns the bond until its maturity t= t n ;

2) all payments on the bond are reinvested at a rate equal to the internal yield of the bond r at the time of purchase.

Let us show that if these conditions are met, the average annual return on an investment in a bond is equal to its internal return. We will consider purchasing a bond, then holding it until maturity and reinvesting the proceeds as a financial transaction (see paragraph 1.2). Operation time T = t n years. Monetary valuation of the start of the operation P(0) is the market purchase price of the bond P in the moment t= 0. According to (8.1), P =
. Monetary value of the bond maturity date t= t n for the investor, if conditions 1), 2) are met, this is the amount P(t n) =
. According to the definition of profitability of a financial transaction (2.2):

P(t n) = P
,

Where - average annual return on investment in a bond for a period T = t n years. Let us substitute into this equality the expressions for P And P(t n):

=

.

Where do we get it from? r = .

Thus, the average annual return on an investment in a bond is equal to r, is sold on the bond maturity date if conditions 1), 2) are met. Hence another name for internal yield – yield to maturity. If points 1) or 2) are not met, then the real return received by the investor may be higher or lower than the internal yield of the bond. The risk an investor faces when purchasing a bond is the risk that future reinvestment rates will be lower than the internal rate of return. This risk is called reinvestment risk, or reinvestment rate risk.

A bond's internal yield is used to assess the attractiveness of alternative investment instruments. All other things being equal, the higher the yield to maturity of bonds of a given issue, the more attractive it is.

Let's consider the problem of determining the internal yield of a bond. The internal yield of a bond is the solution to equation (8.1). According to Theorem 4.1, this equation, subject to the condition P < WITH 1 + WITH 2 +…+ WITH n has only one positive solution. This solution is found using approximate methods. One of them is the linear interpolation method (described in paragraph 1.4, examples 4.2, 4.4).

Example 8.1. Determine annual internal return r bonds, the payment flow for which is indicated in the table:

We will find the approximate value of the bond's internal yield using the linear interpolation method. According to the definition of the annual internal yield of a bond

.

It is necessary to find a solution to the equation F(r) = 0, where

F(r) =
.

Since 948< 50 + 1050, то согласно теореме 4.1 существует единственное положительное решение этого уравнения. Так как F(0,07) = – 15,8396, F(0.08) = 1.4979, then the required internal return r (0.07; 0.08). Using formula (4.8) we find the first approximation

r l1 = 0.07 + .

In this case, the value of the function F(r l1) = 0.02567 > 0. Hence, the solution r (0.07; 0.07914). The next step of the method gives

r l2 = 0.07 + .

Therefore we can assume that r 0.07913 or 7.913% accurate to the third decimal place.

Definition. A bond is said to be pure discount if the bond makes only one payment.

Definition. The internal yield of a pure discount bond with no credit risk that has a maturity date of t years is called the annual risk-free interest rate for investments of t years. Another name is annual spot rate.

Let A– the repayable amount of a pure discount bond, t years - term to maturity, R– market price of the bond at the moment t = 0, r(t) – internal yield of the bond. Then, according to the definition of the internal yield of a bond,

.

(8.2)

– annual risk-free interest rate for investments on t years.

An example of a pure discount bond that has no credit risk is the zero-coupon U.S. Treasury bond. Treasury yields serve as the benchmark for valuing all types of bonds.

Let's consider how you can value any bond if there are pure discount bonds on the market. Let there be a bond on the market IN without credit risk, through which t 1 , t 2 ,…, t n years they promise to pay sums of money WITH 1 , WITH 2 ,…, WITH n respectively. Bond IN can be valued by considering it as a portfolio of pure discount bonds IN 1 , IN 2 ,…, IN n with maturities in t 1 , t 2 ,…, t n years respectively. Let's assume the following conditions are met:

1) annual risk-free interest rates are known r(t 1), r(t 2), …, r(t n) for investment on t 1 , t 2 ,…, t n years counted from the moment t = 0;

2) pure discount bonds IN 1 , IN 2 ,…, IN n can be purchased on the market in any quantity without transaction costs. Then for these bonds we have

,

i = 1, 2, …, n, Where P i– current market price of one bond i– th species, A i– the repayable amount of this bond, r(t i) is its internal return. Payment WITH 1 of the portfolio is repaid in bonds IN 1, payment WITH 2 – bonds IN 2, etc., payment WITH n– bonds IN n. Then in the briefcase , i = 1, 2, …, n, bonds of each type. Therefore, the value of the portfolio at the moment t= 0 is equal

.

Then the market value of the bond IN in the moment t= 0 is

. (8.3)

Each bond payment IN individually discounted at the appropriate risk-free interest rate.

Definition. Set of annual risk-free interest rates r(t 1), r(t 2), …, r(t n) for investment on t 1 , t 2 ,…, t n years counted from the moment t= 0, where
, is called the term structure of interest rates.

Thus, if the term structure of interest rates is known, then the value of a bond that does not have credit risk can be calculated using formula (8.3).

Definition. Graph of a function r = r(t), Where r(t) - annual risk-free interest rate for investments on t years is called the yield curve (or spot rate curve).

In a real market, there is always only a finite set of pure discount bonds (for example, there are no zero-coupon U.S. Treasury bonds with maturities greater than one year). Therefore, it is impossible to construct a yield curve based solely on market observations. In this regard, a theoretical yield curve is constructed. To do this, using the yields of actually existing pure discount bonds, theoretical yields are calculated for different investment periods. There are several methods for obtaining theoretical yield values. One of them is called "bootstrap procedure". Let's look at this method with an example.

Example 8.2. There are government bonds A, B, C, D, E on the market, payment flows for which and prices at the time t= 0 are indicated in the table:

A and B are pure discount bonds. Their internal returns r(0.5) = 5.25% and r(1) = 6.3%, determined by formula (8.2), are the risk-free interest rates for investments for 0.5 years and 1 year. Knowing these two rates, you can calculate the theoretical risk-free interest rate for an investment for 1.5 years using bond C. The price of bond C according to formula (8.3) is

118,71 =
,

Where r(0,5) = 0,0525, r(1) = 0.063. Then

118,71 =
.

Where do we get the theoretical annual risk-free interest rate for investments for 1.5 years: r(1.5) = 6.9%. This rate is the rate that the market would offer for 1.5-year pure discount bonds, if such securities actually existed.

Knowing the theoretical 1.5-year risk-free interest rate, we can calculate the theoretical two-year risk-free interest rate using bond D:

Where r(2) = 7.1% - theoretical two-year risk-free interest rate. Applying the described procedure again for bond E, we determine the theoretical 2.5-year risk-free interest rate: r(2,5) = 7,9 %.

Risk-free interest rates r(0,5), r(1), r(1,5), r(2), r(2.5), constructed using such a process, specify the term structure of interest rates over a 2.5-year range relative to the point in time to which bond prices relate.

Knowing the term structure of interest rates r(t 1), r(t 2), …, r(t n), we can construct a yield curve. One of the methods for constructing a curve is linear interpolation. Believe

,
, i = 1, 2, …, n – 1. (8.4)

TO
The yield curve for the term structure obtained in Example 8.2, using linear interpolation, has the form:

Using the yield curve, you can determine the approximate value of the risk-free interest rate for investments for any period from t 1 to t n years. For example, since 1.25
, That

r(1,25) r(1)
= 0,066.

Another way to construct a yield curve is interpolation ( n– 1) – th order:

r(t)

+
(8.5)

…………………..

+
,

Where t [t 1 , t n]. r(t Then n) – polynomial of degree ( t– 1) relative to the variable t = t 1 , t 2 , …, t n. At r(t 1), r(t 2), …, r(t n the values ​​of the polynomial coincide with

r(t) 0,00633 t 4 - 0,031 t 3 + 0,04442 t 2 - 0,00325 t) respectively. The yield curve equation for the term structure obtained in Example 8.2 is: t .

+ 0.0465, where t Using the resulting curve, we calculate the cost of a bond without credit risk, payments on which relative to the moment

= 0 are indicated in the table: t The market value of this bond at the time

P =
.

= 0 is, according to (8.3):

r(0,7) 0,00633(0,7) 4 - 0,031(0,7) 3 + 0,04442(0,7) 2 - 0,003250,7 + 0,0465 = 0,0569,

r(1,7) 0,00633(1,7) 4 - 0,031(1,7) 3 + 0,04442(1,7) 2 - 0,003251,7 + 0,0465 = 0,0699.

Then the market value of this bond

P =
= 112,14.

The considered “bootstrapping procedure” for obtaining theoretical values ​​of risk-free interest rates can be used if there are bonds suitable for this procedure on the market. Let's consider another method for obtaining theoretical interest rates.

Suppose the term structure of interest rates is known r(t 1), r(t 2), …, r(t k) for investment on t 1 , t 2 ,…, t k years, and there is a bond on the market without credit risk worth P, along which through t 1 , t 2 ,…,t k , t k + 1 , …, t n years of promised payments WITH 1 , WITH 2 ,…,WITH k , WITH k +1 ,…,WITH n respectively. Approximate values ​​of risk-free interest rates r(t k +1), r(t k +2), …, r(t n) can be found using linear interpolation on the segment [ t k , t n]. For this it is believed r(t n) = r. Risk-free interest rate r(t k) is known. Then

,

,

……………….. (8.6)

,

r(t n) = r,

Where t k + 1 , t k + 2 , …, t n – 1 [t k , t n ].

Since the bond price P in the moment t= 0 is known, then

Substituting into this expression instead r(t k + 1), r(t k + 2), …, r(t n) equality (8.6), we obtain an equation with one unknown r. The solution to this equation is found by linear interpolation. Knowing r, using formulas (8.6) we find risk-free interest rates r(t k +1), r(t k + 2), …, r(t n). Thus, we have the term structure of interest rates r(t 1), r(t 2), …,r(t k), r(t k +1),…, r(t n) By t n– summer range relative to the moment t= 0.

Example 8.3. Using linear interpolation, construct a yield curve if annual risk-free interest rates are known:

r(0,5) = 0,06; r(1) = 0,07; r(1,5) = 0,08

and given a bond (without credit risk) with the following payment stream:

Equation (8.7) for this bond is:

We use linear interpolation on the segment. Because r(1,5) = 0,08, r(2,5) = r, That r(2)0,08
+ r
= 0,04 + 0,5r. Then it is enough to solve the equation

86,01581 =
.

Solving this equation by linear interpolation, we find r 0,10489.

Hence, r(2) 0,04 + 0,5r = 0,09245, r(2,5)0.10489. Thus, according to the given r(0,5) = 0,06; r(1) = 0,07; r(1.5) = 0.08 and calculated

r (2) 0,092; r(2,5)Using 0.105 values ​​of risk-free interest rates, we can construct a yield curve:

The yield curve obtained for bonds that do not have credit risk is also used to evaluate risky instruments in the market. Theoretical risk-free interest rates plus a risk premium are used to value all types of bonds. In addition, the shape of the yield curve is seen as reflecting the likely direction of future changes in money market interest rates. In Fig. Figure 1.8.3 shows four main forms of the yield curve: 1 – normal (increasing) curve; 2 – reverse (decreasing) curve; 3 – “humpbacked” curve; 4 – flat (horizontal) curve.

There are two main theories that explain the shape of yield curves - the theory of expectations and the theory of market segmentation. A rising curve most often means an expected increase in the inflation rate. A decreasing curve most often indicates an expected decrease in the inflation rate. A horizontal yield curve means that the annual risk-free interest rates for investments are the same for all maturities. The horizontal curve is used to study a number of the most important concepts in the theory of fixed income financial investments. For example, such as the duration and convexity indicator of a bond, the cost of investing in a bond, immunization of a bond portfolio.