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Current yield - the ratio of coupon income to the purchase price.
Yield to maturity takes into account coupon yield and redemption yield (sometimes called the premise rate).
Yield by types of bonds. /. Bonds without mandatory redemption with periodic interest payments. If c is the coupon rate, rt is the current yield, then
r, \u003d Ms / P \u003d c 100 / K. (9.1)
2. Bonds without paying interest. Yield is formed as the difference between the face value and the purchase price. The price of this bond is less than 100.
The balance of the operation will be written as follows: P = M(I + r)~", where n is the maturity of the bond, r is the total yield of the bond, (1 + r)~n = A/100;
g "1 / 4JK / 100 - 1. (9 2)
EXAMPLE. A zero-coupon bond with a maturity of 10 years was issued. Bond rate - 60. Find the total yield on the maturity date.
Solution, r = 1 / (^60/100) -1 - 0.052, or 5.2%.
3. Bonds with payment of interest and face value at the end of the term (reinvestment of coupon income). Operation balance: M (1 + c)n (1 + r)~n = P or [(1 + c)/(1 + r)]" = /r/100;
r'(1+s)/^AG/100-1. (9 3)
EXAMPLE. Bonds with an income of 15% per annum of face value, a rate of 80, a maturity of 5 years. Find the total yield if the face value and interest are paid at the end of the term.
Solution, r = (1 + 0.15) / ^ / 80/100 -1 = 0.202, or 20.2%.
4. Bonds with periodic interest payments and par value redemption at the end of the term. Operation balance:
CM CM CM M
1 + r (1 + r)2 (1 + r)" (1 + r)"
P \u003d M (I + r) "n + cM ^j (I + r)"", where / is the period from the purchase of bonds to the payment of coupon income.
The determination of the unknown value of the total return can be made by three methods: the so-called approximate method, the method of linear extrapolation and the trial and error method.
For the approximate method, the formula is used
CM + (M - P)In
(M+P)? KU"
c + (1 -Yu / p G-- (1-L) / 2 (96)
To use the linear interpolation method (description of the method is given in Section 3.6), we divide both parts of formula (9.4) by M:
A / 100 \u003d (1 + r) - "+ cV, (9.7)
where ar is the coefficient of reduction of rent at the rate r for the period n.
The total return r can be found by linear interpolation:
where н and гв - the lower and upper limits of the total profitability; Kn and K3 - the lower and upper limits of the course calculated for h and g according to the formula (9.7); Kv< К < Кн.
It should be noted that as the yield increases, the price of the bond decreases.
EXAMPLE. Bond with a maturity of 6 years interest rate 10% bought at the rate of 95. Find the total yield.
Solution. To determine the coefficients of reduction of rent аг we use the already known formula (3.20).
Let's put rI = 10%, /"v = 15%. Then:
KJlOO \u003d 1.10 "6 + 0.1<76;IO = 0,564 + 0,1 4,355 = 0, 99;
Kjm \u003d 1.15 "6 + 0.1 i6: 15 \u003d 0.432 + 0.1 3.784 \u003d 0.81;
/*= 0,10 + [(0,99 - 0,95)/(0,99 - 0,81)] (0,15 - 0,10) = 0,11.
Check: 1.11 "6 + 0.1 a.i = 0.535 + 0.1 4.23 = 0.958.
The trial and error method consists in choosing the value of r in such a way that the equality (9.4) (or (9.7)) turns out to be true.
Duration is one of the indicators of bond volatility. This term is a tracing paper from the English duration, which translates as "duration". This indicator was first studied by Frederick Macaulay in 1938. He defined this indicator as the weighted average maturity of the cash flow of a security1. Macaulay duration is calculated using the formula:
where t is the maturity or cash flow element of the bond; CF1 is the value of the bond cash flow element in the year /; r - yield to maturity (full yield).
The Macaulay duration index, calculated by formula (9.9), is measured in years.
Special attention should be paid to the fact that discounting is carried out at the rate of return to maturity, which must first be determined, for which the methods discussed above can be used. In addition, we note that the denominator of the formula for calculating duration is the price of a bond, since
For bonds for which coupon income is paid m times a year, the calculation formula takes the form:
9.4. Duration
(average duration of payments)
2 CF1(I + gG<
¦2 CZ)(I + g/tG
The Handbook of Fixed Income Securities. P. 85.
EXAMPLE. A bond with a maturity of 6 years, coupon rate - 10%, face value - $100. Yield to maturity - 11%.
Table 9.2
1
(1+r)""
CF1
CF1(X + r)""
tCFt(\+r)-"
I
0,9009
10
9,009
9,009
2
0,8116
10
8,P6
16,232
3
0,7312
10
7,312
21,936
4
0,6587
10
6,587
26,348
5
0,5935
10
5,935
29,675
6
0,5346
on
58,806
352,836
95,765
451,4272
We get:
D = 451.4272/95.765 = 4.7 years.
Duration can also be viewed as the elasticity of the price of a bond to a change in the interest rate (more precisely, the value of 1 + r). In general terms, the elasticity coefficient is the ratio of the relative increase in one indicator to the relative increase in another indicator. In this case, these indicators are the price of the bond and the interest rate.
coupon yield (dk), established when issuing a bond, is calculated by the formula:
dk = C 100% / N, (12.1)
where FROM– annual coupon yield in monetary units;
N is the nominal price of the bond.
Coupon yield on bonds is paid periodically. When selling bonds on days that do not coincide with the days of payment of current income, the buyer and seller must share the amount of interest. To this end, the buyer pays the seller, in addition to the market price of the bond, the interest due for the period that has elapsed since their last payment, the so-called accumulated coupon income. The buyer himself, upon the next date of coupon income payment, will receive it in full for the entire coupon period. Thus, the amount of interest is distributed among the various owners of the bond.
Accumulated coupon income(A) can be calculated using the formula:
A \u003d C t / 365,(12.2)
where t- the number of days from the date of payment of the last coupon yield to the date of sale.
Current yield (dT), which evaluates only the current income in relation to the current market rate:
d T = C 100% / PV, (12.3)
where PV is the current market price of the bond.
The second form of income comes from the change in the market price of the bond over time. In accounting, taxation, and finance terminology, these rate changes are known as capital gain or capital loss.
The most commonly used measure of return is reported yield or yield to maturity (d n), which takes into account both interest income and appreciation. To determine it, the method of calculating the approximate yield is used, which is quite accurate:
where N– nominal bonds;
n is the number of years to maturity of the bond.
Index realized yield (db) assumes that the investor will not hold the bond to maturity. To calculate this indicator, it is necessary to estimate the expected selling rate:
where PVs- the expected selling rate of the bond;
PVb- bond purchase rate;
When determining the yield of a portfolio of bonds, they proceed from the amount reduced to a certain point in time t o income streams from each bond in the portfolio. Suppose the portfolio contains " M"bonds of various types, with the number of bonds of each type equal to Bonds of each type have a nominal value maturity of bonds 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: Si- the total income from bonds of all types, incoming at the time t = ti, and - the yield of the bond portfolio. The present value of this stream of payments can be determined by a formula similar to formula (2.2):
(5.29)
where N max is the maximum yield payment term for all bonds in 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 value of the bond portfolio yield can be found by solving equation (5.30) by 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 - the market value of the bond portfolio, determined by formulas (5.27) and (5.28);
And - the present value of the flow of payments, determined by the formula (5.29) when used in the calculation of the rates of return and, respectively.
Consider the methodology for calculating the yield on 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.; FROM 1 = 0.08, years;
Second bond rub.; FROM 2 = 0.05, years.
Determine the yield of the bond portfolio if the number of bonds of the first and second types is the same
Solution: Let's determine the market value of the first type of bond using the 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:
Calculate the total flow of payments Si on 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 Si.
Table 5.2
Calculation of the total flow of payments
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 us calculate the present value of the total flow of payments for various values of :
The calculation results are given in Table. 5.3.
What would you like to achieve investing in bonds? Save money and earn extra income? Make savings for an important goal? Or maybe you dream about how to get financial freedom with the help of these investments? Whatever the goal, it pays to understand how much your bonds yield and to be able to tell a good investment from a bad one. There are several principles for assessing income, the knowledge of which will help in this.
Bond yield- this is the amount of income in percent received by the investor from investments in debt securities. Interest income they are formed from two sources. On the one hand, have fixed coupon bonds, like deposits, have interest rate, which is charged at face value. On the other hand, have bonds, like stocks, have a price, which may vary 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 yield of a bond includes coupon yield and takes into account purchase price. In practice, different estimates of profitability are used for different purposes. Some of them only show coupon yield, others additionally take into account purchase price, the third show return on investment depending on the tenure- prior to sale on the market or prior to redemption by the issuer that issued the bond.
To make the right investment decisions, you need to figure out what types of bond yields are and what they show. In total there are three types of profitability, the management of which turns an ordinary investor into a successful rentier. These are the current yield on interest on coupons, yield on sale and yield on securities to maturity.
The coupon rate is the base percentage of the face value of the bond, which is also called coupon yield . The issuer announces this rate in advance and periodically pays it in due time. Coupon period most Russian bonds - half a year or a quarter. An important nuance is that the coupon yield on the bond is calculated 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 since the date of the last coupon payments. The sum of these percentages is called accumulated coupon income(NKD) and added to the current market price of the bond. At the end of the coupon period, the buyer will receive the entire coupon and thus compensate for his expenses related to the reimbursement of ACI to the previous owner of the bond.
Exchange quotes of bonds at many brokers show the so-called net price of the bond, excluding ACI. However, when an investor issues a buy order, the net price will add to the ACI, and the value of the bond may suddenly be higher 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 ACI. After that, evaluate the final costs of buying in a particular brokerage company, taking into account all costs, and find out how much money will be written off from your account if you buy securities.
As the cumulative coupon yield (ACY) rises, the value of the bond rises. After the coupon is paid, the value is reduced by the amount of the ACI.
NKD- accumulated coupon income
FROM(coupon) - the amount of coupon payments for the year, in rubles
t(time) - number of days since the beginning of the coupon period
Example: the investor bought a bond with a face value of 1,000 ₽ with a semi-annual coupon rate of 8% per year, which means a payment of 80 ₽ per year, the transaction took place on the 90th day of the coupon period. His surcharge to the previous owner: ACI = 80 * 90 / 365 = 19.7 ₽
Not really. Each coupon period the investor receives an amount of certain interest in relation to face value bonds to the account that he indicated when concluding an agreement with a broker. However, the real interest that the investor receives on the invested funds depends on purchase price of a bond.
If the purchase price was higher or lower than par, 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 the real return on investment- 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 the yield and price are inversely related to each other. An investor receives a lower yield to maturity than was set by the coupon when he buys a bond at a price higher than par.
CY
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 face value of 1000 ₽ at a net price of 1050 ₽ or 105% of the face value and a coupon rate of 8%, i.e. 80 ₽ per year. Current yield: CY = (80 / 1050) * 100% = 7.6% per annum.
And there is. However, for novice investors who are not very clear on the difference between yield to sell and yield to maturity, this is often a difficult moment. If we consider bonds as a portfolio of investment assets, then its yield to sale in the event of a price increase, like that of stocks, of course, will increase. But the yield of bonds to maturity will change differently.
The whole point is that a bond is a debt, which can be compared with a deposit. In both cases, when buying 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: yield to maturity falls when the price rises.
Beginners who evaluate returns in bonds on the basis of comparison with stocks may come to another erroneous conclusion. For example: when the price of a bond has grown, say, up to 105% and has become more than the face value, then it is not profitable to buy it, because only 100% will be returned upon repayment of the principal debt.
In fact, it is not the price that matters, but bond yield- a key parameter for assessing its attractiveness. Market participants, when trading for a bond, only agree on its yield. Bond price is a derivative of profitability. In fact, he adjusts the fixed coupon rate to the level of the rate of return agreed upon by the buyer and seller.
How yield and price of a bond are related, see the video of the Khan Academy, an educational project created with money from Google and the Bill and Melinda Gates Foundation.
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 the simple yield, which includes a discount or premium to the face value upon purchase:
Y(yield) - simple yield to maturity / offer
CY(current yield) - current yield, from coupon
N
P(price) - purchase price
t(time) - time from purchase to maturity/sale
365/t- multiplier for converting price changes into percentages per annum.
Example 1: the investor purchased a two-year bond with a face value of 1000 ₽ at a price of 1050 ₽ 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 was upgraded 90 days after the purchase of the bond, after which the price of the paper rose to 1070 ₽, so the investor decided to sell it. Let's replace in the formula the face value of the bond with the price of its sale, and the term to maturity - with the holding period. Get simple yield to sell: 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 for it, more than it was worth 90 days ago. Since the price of the bond has risen, the simple yield to maturity for a 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 price of a bond increased by 1.9% in 90 days. In terms of annual yield, this amounted to a serious increase in interest payments on the coupon - 7.72% per annum. With a relatively small change in price, bonds for a short period of time can show a sharp jump in profits for the investor.
After a bond is sold, an investor may not receive the same 1.9% return every three months for a year. Nonetheless, yield converted into annual percentages, is an important indicator characterizing current cash flow investor. With its help, you can make a decision on the early sale of a bond.
Consider the reverse situation: when the yield increases, the price of a bond decreases slightly. In this case, the investor may receive a loss in case of 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 least risk of loss of invested funds in case of early sale is bonds of reliable companies with a short term to maturity or redemption under an offer. Strong fluctuations in them can be observed, as a rule, only during periods of economic crisis. However, their market value recovers fairly quickly as the economy improves or the maturity date approaches.
Deals in safer bonds mean less risk for the investor, but also yield to maturity or offer they will be lower. This is a general rule of risk-reward ratio, which also applies to the purchase and sale of bonds.
So, as the price rises, the yield on a bond falls. Therefore, in order to get the maximum benefit from rising prices in case of early sale, you need to choose bonds, the yield on which can decrease the most. Such dynamics, as a rule, show papers of issuers that have the potential to improve their financial position and credit ratings.
Large changes in yield and price can also be shown by bonds with long maturity. In other words, long bonds are more volatile. The thing is that long-term bonds form a larger cash flow for investors, which has a stronger effect on price changes. How this happens is best illustrated by the example of the same deposits.
Suppose a contributor a year ago deposited money at a rate of 10% per annum for three years. And now the bank accepts money for new deposits already 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 rise to approximately 104% of the face value. The longer the term, the higher the premium 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 go down. If interest rates, on the contrary, rise, then it becomes unprofitable to hold long bonds. In this case, it is better to pay attention to fixed-coupon securities that have short term to maturity, or bonds with floating rate .
Effective yield to maturity- is the total income of the investor from investing in bonds, taking into account the reinvestment of coupons at the rate of initial investment. To estimate the total 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 in different periods of time. In other words, this return on investment in bonds.
You can independently calculate the estimated effective yield 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 par, and slightly less if it was below par.
YTM op (Yield to maturity) - yield to maturity, estimated
C g (coupon) - the amount of coupon payments for the year, in rubles
P(price) - current market price of the bond
N(nominal) - face value of the bond
t(time) - years to maturity
Example 1: the investor purchased a two-year bond with a face value of 1000 at a price of 1050 ₽ 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 was upgraded 90 days after the purchase of the bond, and its price rose to 1070 ₽, after which the investor decided to sell the bond. Let's replace in the formula the face value of the bond with the price of its sale, and the term to maturity - with the holding period. Let's get the estimated effective yield to sell (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 for it, more than it was worth 90 days ago. Since the price of the bond has risen, the effective yield to maturity for a 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 on a particular bond is to use bond calculator on the Rusbonds.ru website. An accurate calculation of the effective return can also be obtained using financial calculator or the program "Exel" 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 - by the method of automatic selection of numbers.
How to find out the yield of a bond, see the video of 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. Conversely, as yields rise, the price of a bond falls.
✔ You can compare comparable things. For example, the net price without ACI - with the net price of the bond, and the full price with ACI - with the full price. This comparison will help you make a decision when choosing a broker.
✔ Short one-two-year bonds are more stable and less dependent on market fluctuations: investors can wait for the maturity date or the redemption by the issuer under the offer.
✔ Long fixed-coupon bonds with lower rates in the economy allow you to earn more on their sale.
✔ A successful rentier can receive three types of income in bonds: from coupon payments, from a change in the market price when sold, or from recovering face value at maturity.
Intelligible dictionary of terms and definitions of the bond market. Reference base for Russian investors, depositors and rentiers.
Discount bonds- discount to the face value of the bond. A bond that is priced below par is said to be selling at a discount. This occurs if the seller and the buyer of the bond have agreed on a higher rate of return than is set by the coupon 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 issuance of securities. Coupon income is accrued daily and is calculated at the rate of the face value of the bond. The coupon rate can be constant, fixed or floating.
Coupon period of a bond- the period of time after which investors receive interest accrued on the nominal value of the security. The coupon period of most Russian bonds is a quarter or half a year, less often a month or a year.
Bond premium- increase to the face value of the bond. A bond that is priced above par is said to be selling at a premium. This occurs if the seller and the buyer of the bond have agreed on a lower rate of return than that set by the coupon issuer.
Simple yield to maturity/offer- is 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 the investor the return on investment without coupon reinvestment.
Simple yield to sell- is calculated as the sum of the current yield from the coupon and the yield from the discount or premium to the bond's selling price, as a percentage per annum. Since this yield depends on the bond's selling price, it can be very different 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 we use the purchase price of a bond, then the resulting figure will show the investor the annual return on his cash flow from coupons on invested funds.
Bond full price- the sum of the market price of the bond as a percentage of the face value and the accumulated coupon yield (ACI). This is the price an investor will pay when buying a paper. The investor compensates for the costs of paying the ACI at the end of the coupon period, when he receives the entire coupon.
Bond net price- the market price of the bond as a percentage of the face value, excluding accumulated coupon income. It is this price that the investor sees in the trading terminal, it is used to calculate the profitability received by the investor on the invested funds.
Effective yield to maturity / offer- the average annual yield on initial investments in bonds, taking into account all payments to the investor in different periods of time, the redemption of the face value and income from the reinvestment of coupons at the rate of initial investments. To calculate the yield, the investment formula of the cash flow internal rate of return is used.
Effective yield to sell- the average annual yield on initial investments in bonds, taking into account all payments to the investor in different periods of time, proceeds from the sale and income from reinvestment of coupons at the rate of initial investments. The effective yield to sell measures the profitability of investing in bonds over a given period.
Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.
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Ministry of Education and Science of the Russian Federation
Federal State Budgetary Educational Institution
higher professional education
"PERM NATIONAL RESEARCH
POLITECHNICAL UNIVERSITY"
Test
in the discipline "Theoretical foundations of financial management"
Option number 73
Completed by a student
Faculty of Humanities
Correspondence department
Profile: Finance and credit
group FK-12B
Flywheel Ksenia Vitalievna
Checked by teacher:
Ageeva Valeria Nikolaevna
Date of delivery ____________________
Perm - 2014
Task #1
Task #2
Task #3
Task #4
Task number 5
Task #6
Task #7
Task #8
Task #9
Task #10
Bibliography
Option exercise period - t = 3 months.
The current price of the underlying asset - S = 35 rubles.
The exercise price of the option-K = 80 rubles.
Risk free rate of return - r = 3%
Underlying asset risk - x = 20%
S = (V)(N(d1)) - ((D)(e-rt))(N(d2)),
where N(d1) and N(d2) are cumulative normal distribution functions,
e is the base of the logarithm (e = 2.71828);
V=S+K=35+80=115 rub.
y 2 \u003d (0.2) 2 \u003d 0.04
d1 = (ln(V/K) +(r + y 2/2) t)/(y)(t 1/2)
d1 = (ln(115/80) + (0.03 + 0.04/2) 0.25)/(0.2)(0.251/2) = 3.75405
N(3.75405) = N(3.75) + 0.99(N(3.8) - N(3.75)) = 0.9999 + 0.00 = 0.9999
d2 \u003d d1 - (y) (t 1/2) \u003d 3.75405-0.2 * 0.251 / 2 \u003d 3.65405
N(3.65405)=N(3.65)+0.99(N(3.7)-N(3.65))=0.9999+0.00=0.9999
S \u003d 115 * 0.9999 - ((80) (2.71828 -0.03 * 0.25))
(0.9999) \u003d 114.99-79.39 \u003d 35.6 rubles.
Conclusion: the price of the call option was 35.36 rubles.
Task #2
The current share price of the company "ABC" is equal to S = 80 rubles. In a year, the share will cost or Su = 90 rubles. or Sd = 50 rubles. Calculate the actual value of a call option using the binomial model, if the exercise price of the call option K = 80 rubles, term t = 1 year, risk-free rate r = 3%
According to the binomial model, the price of a call option at the time of exercising the option can take strictly two values: it either increases to the value of Su , or falls to the value of Sd . Then, in accordance with the binomial model, the theoretical price of the call option will be equal to:
S - today's price of the underlying asset on which the option is concluded;
K - option strike price
r - risk-free interest rate in the financial market (% per annum);
t - time in years until the option is exercised
It can be seen from this formula that the option price is always a certain fraction (percentage) of the current price of the underlying asset, determined in the binomial model by the multiplier
0.098 * 80 \u003d 7.86 rubles.
Conclusion: the cost of the call option was 7.86 rubles.
r cf. = (35+33+27+14+20)/5 = 26%
Dispersion
(y2) = ((35-26)2+(33-26)2+(27-26)2+(14-26)2+(20-26)2)/5 = 62
The risk of an asset is the standard deviation of return
(y) = v62 = 8%
Conclusion: the risk of the asset was 8%
Task #4
Determine the internal yield of the coupon bond.
Price = 2350 rubles.
Coupon rate - 14%
Maturity =2 years
Number of coupon periods per year - 4 per.
The nominal value of the bond is 2500 rubles.
A bond is called a coupon bond if regular payments of a fixed percentage of the face value, called coupons, are made on this bond, and the par value is paid when the bond is redeemed. The last coupon payment is made on the maturity date of the bond.
We will use the following notation:
A is the face value of the bond;
f - annual coupon rate;
m is the number of coupon payments per year;
q - the amount of a separate coupon payment;
t = 0 - the moment of purchase of the bond or the moment when it is supposed to invest in the bond;
T(in years) - maturity of the bond from the moment t = 0;
The time elapsed from the last coupon payment before the sale of the bond to the purchase of the bond (until the moment t = 0).
The period of time measured in years is called the coupon period. At the end of each coupon period, a coupon payment is made. Since the bond can be purchased at any time between coupon payments, then φ varies from 0 to. If the bond is purchased immediately after the coupon payment, then
means buying a bond just before the coupon payment. Since the purchase of a bond is made only after the payment of the next coupon, φ does not take a value. In this way,
If the bond is sold in time after the coupon payment, and n coupon payments remain before maturity, then the maturity of the bond is equal to
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where n is a non-negative integer. Consequently,
if Tm is an integer, then
if Tm is not an integer, then
Let P be the market value of a bond at time t = 0 with coupons paid m times a year. Suppose a bond is sold after a time period after the coupon payment, when n coupon payments remain to maturity. Formula (1) for a coupon bond has the form:
The annual internal yield r of a coupon bond can be determined from equality (1). Since r is usually small, then
Then the last equality can be rewritten as:
Calculating the sum of n terms of a geometric progression and taking into account that
we get another formula for calculating the internal yield of a coupon bond:
For an approximate estimate of the internal yield of a coupon bond, the "merchant" formula is used:
In our example:
Here the values of the bond parameters are as follows: A = 2500 rubles, f = 0.14, m = 4,
T = 2 years, P = 2350 rubles Let us find the number of coupon payments n remaining until the bond is redeemed, as well as the time φ elapsed from the last coupon payment before the sale of the bond to the purchase of the bond.
Since the work
n=T*m=2*4=8
Is a whole then
To calculate the internal yield of a bond using formula (2), it is necessary to solve the equation
Using the linear interpolation method, we find r 17.4%.
Conclusion: the internal yield of the coupon bond was 17.4%
Task number 5
Determine forward rates one-year after 1 year, after 2 years and two-year after 1 year.
rf (n-1),n = [(1+r n) n /(1+r n-1) n-1] -1
rf (n-1),n-- one-year forward rate for period n -- (n -1);
r n -- spot rate for period n;
r n-1 -- spot rate for period (n -1)
Forward rate after 1 year
rf1,1 = [(1+r 2) 2 /(1+r 2-1) 2-1] -1 = [(1+r 2) 2 /(1+r 1) 1] -1 = [( 1+0.05) 2 /(1+0.035) 1] -1 = = - 1 = 6.5%
Forward rate after 2 years
rf1,2 = [(1+r 3) 3 /(1+r 3-1) 3-1] -1 = [(1+r 3) 3 /(1+r 2) 2] -1 =
= [(1+0,09) 3 /(1+0,05) 2] -1 = - 1 = 17,5 %
2-year forward rate after 1 year
rf2.1 = v (1.05)2 / (1.035)1 - 1 = 3.2%
Task #6
Determine the optimal portfolio structure if:
covAB \u003d cAB * yA * yB \u003d 0.50 * 35 * 30 \u003d 525
WA = (yB2-covAB) / (y2A+y2B-2covAB)
WA = (302-525) / (352 + 302- 2*525) = 0.349 = 34.9%
Conclusion: to minimize the risk, 34.9% of cash should be placed in asset A and 65.1% in asset B.
Task #7
Determine the risk of the portfolio if it consists of two securities A and B.
WB = 100%-35% = 65%
y2AB \u003d W2A * y2A + W2B * y2B + 2WA * WB * cAB * QA * QB
y2AB \u003d 0.352 * 502 + 0.652 * 182 + 2 * 0.35 * 0.65 * 0.50 * 50 * 18
y2AB = 647.89
Conclusion: portfolio risk was 25.5%
Task #8
Determine the intrinsic value of a stock if:
Number of dividend growth periods with gT-(T) = 5
Growth rate of dividends in the first phase of the company's life (gT-) = 5.0%
Dividend growth rate in the second phase of the life of the company (gT+) = 3.0%
Dividend in the period preceding the start of income growth (D0) = 18 rubles.
Required return (r) = 10%
Determine the intrinsic value of a stock using the formula:
PV = 17.18 + 16.4 + 240.47 = 274.05
Conclusion: the intrinsic value of the share was 274.05 rubles.
Task #9
Determine the intrinsic value of the bond.
Cost of Debt (ri) = 3.5%
Coupon payment (CF) = 90 rubles.
Bond maturity (n) = 2 years
Number of coupon payments per year (m) = 12
The nominal value of the bond (N) = 1000 rubles.
Task #10
Determine the required return on a portfolio of two stocks A and B if:
Risk-free yield (rf) = 6%
Market Portfolio Return (rm) = 35%
Paper weight ratio A (A) = 0.65
Paper light factor B (B) = 1.50
Share of paper A in the portfolio (wА) = 48%
ri = rf + bi(rm-rf);
c \u003d 0.90 * (-0.5) + 0.10 * 1.18 \u003d -0.332
ri = 3.5 + (-0.332)(50-3.5) = -11.9%
Bibliography
option bond value
1. Chetyrkin E.M. Financial mathematics: a textbook for universities. - 7th ed., Rev. - M .: Delo, 2007 .-- 397 p.
2. Gryaznova A. G. [et al.] Business valuation: a textbook for universities; Financial Academy under the Government of the Russian Federation; Institute for Professional Evaluation; Ed. A. G. Gryaznova.-- 2nd ed., Revised. and add.-- M. : Finance and statistics, 2008 .-- 734 p.
3. Brigham Yu., Gapensky L. Financial management: Full course: textbook for universities: Per. from English. in 2 volumes - St. Petersburg: School of Economics,. 2-668 p.
4. Kovaleva, A. M. [et al.] Financial management: a textbook for universities; State University of Management; Ed. A. M. Kovaleva.-- M. : Infra-M, 2007 .-- 283 p.
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