QUESTIONS
for the exam in
―Advanced Mathematics (Financial Mathematics)‖
created by Prof. A. A. Mitsel
Questions on the lecture course
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Accumulation. Interest and interest rates. Accumulation at simple and compound rates.
Variable rates.
Discounting. Discounting at simple and compound rates.
Determining the term and interest rates.
Nominal and effective interest rates. Inflation accounting at interest accumulation.
Continuous accumulation and discounting (continuous interest).
Change of contract terms. Funding debts.
Discounting and accumulation at simple and compound discount rates.
Discounting at a nominal discount rate. Continuous discounting and accumulation at a
compound discount rate. Variable discount rate.
Payment and annuity streams. Basic definitions.
Accumulated sum of annual annuity. Interest calculations m times a year.
Accumulated sum of p -due annuity. Interest calculations m times a year.
Present value of ordinary annuity. Interest calculations m times a year.
Present value of p -due annuity. Interest calculations m times a year.
Present value of p -due annuity. Interest calculations m times a year.
Determining parameters of financial contracts (annuities).
Annuity conversion.
Financial transaction yield. Tax and inflation accounting.
Payment stream and its yield.
Credit calculations. Balance of a financial and credit transaction.
Determining total yield of loan operations with commission fees (loans with regular interest
payments).
Determining total yield of loan operations with commission fees (loans with recurrent
expenses).
Determining total yield of loan operations with commission fees (loans with an irregular
payment stream).
Analyzing contracts based on the method of payment capitalization.
Debt service expenses. Forming a repayment fund.
Discharging the debt through repayment by equal instalments of the principal balance of debt (a
differentiated model).
Discharging the debt through repayment by instalments with equal due payments (annuity
model) .
Investments. Net present value.
Internal rate of return of an investment project. Payback period. Profitability index.
Model of human capital investment.
Analysis of fixed income securities. General notions.
Determining the total yield of a bond (bonds without interest payment, bonds without mandatory
redemption with regular interest payments, bonds with interest payments at the term end, bonds
with regular interest payments to be discharged at the term end).
Determining the total yield of bonds in the general case.
Bond yield adjusted for taxation. Bond portfolio return.
Intrinsic value of bonds. Formulas for bond evaluation.
Risk valuation. Average term. Duration.
Connection of duration with bond price change.
The first two properties of bond duration.
The third property of duration and factor of bond convexity.
39.
40.
41.
42.
43.
44.
The fourth property of duration and factor of bond convexity.
The fifth property of duration and factor of bond convexity.
The sixth property of bond duration.
Time dependence of the value of investment in the bond.
Properties of planned and actual values of investment.
The theorem on the immunization property of bond duration.
Problems
Topic 1. Accumulation and Discounting
1. A deposit A = 5000 rubles is placed in a bank. The interest calculated on this deposit
will be i1 = 10% per annum in the first year , i2 = 12% per annum in the second, i3 = 15% per
annum in the third, i4  i5 = 16% per annum in the fourth and the fifth. How much will there be
on the account at the end of the fifth year? How much should have been placed on the account at
the constant interest rate of i = 13% to secure an equal sum? Make calculations at the simple and
compound interest rates.
2. You are asked to lend P = 10000 rubles with a promise of repaying A = 2000 rubles
within N =6 years. You have another way of using this money: placing it in a bank at 7% per
annum and yearly withdrawing A = 2000 rubles from the account. Which of the financial
operations will be of higher profit for you? Make calculations at the simple and compound interest
rates.
3. You have an opportunity of investing in a project with a cost of A = 10000 rubles. In a
year, P1 = 2000 rubles will be refunded, P2 = 4000 rubles – in two years, P3 = 7000 rubles – in
three years. An alternative variant is to place the money in a bank at the i interest per annum. What
annual interest rate will it make it more profitable for you to place money in the investment
project? Make calculations at the simple and compound interest rates.
4. At what compound interest rate will the sum increase k times within 9 years, if k = 2?
5. A grandmother placed A = $3000 at 3% per annum in a bank on the birthday of her
grandson. What sum will it be by the grandson’s seventeenth birthday? Make calculations at the
simple and compound interest rates.
6. What rate should a bank quote so that with an annual inflation of h = 12% the real rate
equals 6%?
7. Under agreement, a payment of 1000 currency units has been recorded after 3 years. After
a year the interest rate increased. Who is it profitable for: the one who receives payments, or the one
who pays?
8. Compound interest at an annual rate of 8 % is charged on a deposit. The interest for the 6th
year of the deposit ( N1 =6) equals I 6 =117.546 currency units. What is the value of interest for the
3rd ( N 2 =3) and 8th ( N3 =8) years of the deposit? What sum of the deposit will it be by the end of
the N3 =8th year?
9. Compare the accumulation rates of the amount of debt at simple interest rates i and d ,
considering them equal. Show the comparison result on a figure in form of accumulation curves.
Show the profit value of the creditor on the figure, considering the life of the loan defined. For each
of the interest rates i and d , make calculations of dischargeable debt sum in the following credit
transaction: the loan of A =10 thousand currency units was issued at a rate of i =12 % per annum
with monthly simple interest calculations. Calculate the creditor's total income as the function of
time and monthly income. Consider the maximum life of the loan 18 months. Compare the
creditor’s income for every month and the whole life of the loan for the rates i and d . Do the
calculation results correspond the curves built? What conclusion can be made?
10. Compare the discounting rates at simple rates i and d . Draw discount curves.
Considering the life of the loan defined, show the discount value on a figure. Compare the results of
the promissory note discount with the maturity sum of N =300 thousand currency units by the
methods of mathematical and bank discounting at a simple interest rate of i =6 % per annum. For
what sum would the promissory note be discounted by each of the methods within different terms
before its maturity? Calculate the creditor's total income as the function of time and monthly
income. Take 18 months for the maximum life of the loan. Do the calculation results correspond the
curves built?
Topic 2. Payment Streams
1. Calculate the amount of payment of an n - year loan for buying an apartment for A
rubles at an annual rate of i percent and a down payment of q percent. Make calculations for
monthly and annual payments.
Make calculations with the following data: n = 20 years; A = 1 400 000 rubles; i = 18%;
q = 30%.
Make calculations at the compound interest rate.
2. A family wants to buy a country retreat (dacha) after n = 6 years for $20 000. What
(equal) amount must it add annually within these 6 years to its bank account in order to save
$120 000, if the annual interest rate equals 10% in the bank?
Make calculations at the compound interest rate.
3. A publishing house transfers a sum of R rubles p times a year on an author’s bank
account; the bank calculates a compound interest on the sum at a rate of i % m times a year. How
much will there be on the account after n years?
Make calculations with the following data: p = 2; R = 2000 rubles; m =2; i = 7%; n =4
years.
4. It was discovered at a court session that Mr. N had been underpaying taxes of R =1000
rubles per month. The tax service wants to recover the taxes underpaid within the last n =2 years
with interest ( i =3% per month). What sum must Mr. N pay? Make calculations at the compound
interest rate.
5. Determine the interest rate for an n -year loan of A rubles with an annual payment of R
rubles.
Solve the problem with the following given data: n = 10 years, A = 100 000 rubles, R =
16 981 rubles. Make calculations at the compound interest rate.
6. A son had A =500 000 rubles at a bank account that had a monthly interest calculation of
i =0.8%. The son left for a ten-year business trip abroad and authorized his father to spend his
whole account within n =10 years. How much will the father receive per month? Make calculations
at the compound interest rate.
7. A customer offered two variants of payment for the country retreat (dacha) purchase: 1)
R1 =$5000 immediately and then by R2 =$1000 within n =5 years; 2) R3 =$8000 immediately and
then by R4 =$300 within n =6 years. What variant is more profitable at an annual interest rate of:
a) i1 =8% , b) i2 = 3% . Make calculations at the compound interest rate.
Note. Calculations are to be made at the i1 rate in two variants, and at the i2 rate in two
variants.
8. Let us consider an annual annuity for n = 10 years, i = 10%. What will increase the
accumulated value of the annuity: increasing the duration per 1 year ( n =1 year) or increasing the
interest rate per 1% ( i =1%)? Make calculations at the compound interest rate.
9. What must be the payment of the finite annual annuity with a duration of n =8 years, so
that its present value be A =16 000 rubles at a rate of i =10%? Make calculations at the compound
interest rate.
Topic 3. Credit Calculations
1. A loan was taken at i1 =16% per annum; an amount of 500 currency units ( R =500
currency units) must be paid quarterly within n =2 years. Due to a change of situation in the
country, the interest rate was decreased to i2 =6% per annum. The bank agreed with the necessity of
recalculating quarterly payments. What size must the new amount of payment be?
Make calculations at the compound interest rate.
2. A consumer credit of D =40 000 currency units was taken for n =8 years at an annual
compound rate of i =8% for a purchase of a country house. It must be discharged through equal
quarterly payments. Find the amount of such a payment. Find the sum that the bank may obtain if
the incoming payments are placed in another bank at the same rate of i =8% per annum.
3. A shop sells TV-sets by instalments for 1 year. i = 10% is added to the TV-set price
D =$400 immediately, and the whole sum must be discharged within one year; besides, the cost of
the TV-set is discharged uniformly, and the premium – in accordance with the Rule of 78s. Find the
monthly payments.
Note. Maturity according to the Rule of 78s. With this method the principal balance
of the debt D is repaid in equal amounts d  D / n , where n is the number of months ( n =
12). Interest money of D  i amount is paid according to the following rule. Let us combine
the numbers of all the twelve months N  (1  2  ...  12)  78 (hence comes the name of
D i
. In the first month, the sum (d  12 g ) is paid, in
78
the second – the sum (d  11g ) etc., while in the last month – the sum (d  g ) .
this rule) and calculate the value g 
4. A loan of D =$5000 is taken for n =8 years at i =8% per annum. It will be discharged by
equal annual payments of the principal balance of the debt (a differentiated model). Find the annual
payments.
Make calculations at the compound interest rate.
5. A loan of D =20000 currency units is taken for n =8 years at i =8% per annum. It will be
discharged by annual equal payments (an annuity model). Find the amount of such a payment.
Make calculations at the compound interest rate.
6. A loan of D =20 000 currency units is taken for n =10 years at i =8% per annum. It will
be discharged starting from the end of the n1 =6th year by annual equal payments. Find the amount
of such a payment. Make calculations at the compound interest rate.
7. The period to maturity of a loan is n =10 years. At the credit issue, a compound discount
rate of d = 4 % per annum was used. The discount value for the 6th year of the life of the loan
equaled D6 =339 currency units. What is the discount value for the 3rd and 8th years in the life of
the loan? What is the credit amount?
8. A loan commitment at an amount of 50 000 currency units should be accounted 4 before
its maturity. For the commitment accounting the bank uses a compound interest rate of 5 %. The
interest may be calculated 1, 2 or 4 times a year. Indicate the terms of the contract on which this
commitment may be accounted.
9. A commitment of paying R1 = 8000 currency units at the moment t1 = 01.03.2011 and
R2 = 12 000 currency units at the moment t2 = 30.09.2011 was reviewed so that the first
repayment at an amount of R3 = 6000 currency units will be made on the t3 = 01.02.2011, and the
remainder of the debt R4 will be discharged at the moment t4 = 15.11.2011. For the commutation
of the obligation a compound interest rate i = 6 % per annum was used. The financial year has 365
days. Determine the amount of the discharging remainder R4 . Comprise an equivalency equation
relating to t1 = 01.03 and relating to t3 = 01.02.
10. A loan of D =10 000 currency units must be repaid within n =10 years by a constant
ordinary annuity to be paid monthly. The amount of a monthly payment is calculated on the basis of
the monthly interest rate of i =1%. Find:
a) the amount of the monthly instalment;
b) the value of the discharged principal balance of the debt and repaid interest by the end of
the first year;
c) the number of payment t after which the active debt decreases Dt =5000 currency units.
11. A contract with a term of T = 4 years stipulates the payments in two stages with
charging compound interest at an annual interest rate of r1 = 0.08 at the first stage within the first
t1 = 1.5 years and at an annual interest rate of r2 = 0.1 at the second stage within the following
t2 = 2.5 years. At the first stage the payments of R1 = 5000 currency units are made at the end
of every six months. At the second stage the payments of R2 = 8000 currency units are made at the
end of every quarter (three months). Find the amount of the deposit by the end of the T year of the
contract.
12. A loan of D =3000 currency units is to be discharged by equal monthly payments. The
debt is charged with monthly compound interest at a rate of i =12% per annum. Within what term
will the debt be discharged if the monthly payment is:
a) R1 =50 currency units; b) R2 =100 currency units?
13. In order to buy equipment for D = 200 000 currency units after n = 12 years, a
company annually invests money into its safety fund for charging compound interest at an annual
interest rate of i1 =0.06. The initial payments were R1 = 11855 currency units each. After m = 8
years the bank increased the annual interest rate to i2 = 0.08. What amount were the payments
equal in the remaining period?
14. An interest-free loan belongs to the category of soft loans. Find the relative and absolute
grant-elements for such a loan with D = 1000 currency units, n = 5 years, i = 10%.
Note. Soft credits (loans).
A soft credit (loan) is issued at a reduced rate that is lower than the standard rate. In fact, the
loan debtor thus receives a subsidy that is calculated as a difference between the corresponding
present amounts.
Let the credit of D be issued for n years at a reduced rate g that is lower than the standard
rate i to be discharged by equal payments. These payments form an annual annuity. Let us indicate
the amount of a single payment y , then the present value of the annuity equals y  a(n, g ) . From
here we find y  D / a(n, g ) . However, if the payments were made at the standard rate i , then the
amount of each payment were z  D / a(n, i ) . The difference z  y  D / a(n, i )  D / a(n, g )
implies annual losses by the creditor, and the present annuity value of these losses at the current rate

a ( n, i ) 
i , т.е. ( z  y )  a(n, i )   D / a(n, i )  D / a(n, g ) a(n, i )  D 1 
 is in fact the
 a ( n, g ) 
creditor’s subsidy to the loan debtor. This subsidy is also called the absolute grant-element, and
the value 1 
a ( n, i )
— the relative grant-element. The accumulated sum of the absolute granta ( n, g )
element or, what is the same, the accumulated sum of the subsidy is called total losses by the
creditor.
15. A shop sells goods worth P = 1000 currency units on credit for a term of n = 12
months. The credit is discharged uniformly with a monthly payment of the debt amount. The
monthly bank rate is r = 1.0 %. To attract customers, the shop has the following offers: 1) down
payment – 0 %; 2) for the credit – 0 %. Calculate the monthly payment. How much will the goods
cost if they are bought immediately?
Topic 4. Financial Transaction Yield
1. The capital values at time points 0; 1; 2; 4 are K 0 = 100, K1 = 200, K 2 = 300, K 4 = 400.
Find the absolute and average annual yield for each of the six separate intervals.
2. Let us assume there is a ―cyclic‖ investment project. The factory works in cycles: one
year of n = 10 it undergoes heavy repairs and renovations that requires K = $30 000, and the other
nine years of the cycle the factory brings a profit of R = $10 000 per year. Find the domestic return
of this investment project and the average annual yield. Associate the modernization costs with the
beginning of the period.
3. Let us consider a foreign currency transaction. Let the dollar rate increase from Н = 28
rubles to K = 31 rubles in 2011. The bank bought dollars for rubles at the beginning of the year,
and sold dollars at the end of the year getting rubles. Find the yield of this transaction in the annual
interest. If this year’s inflation was  = 6%, then what is the real yield of the transaction?
4. The bank pays i1 = 15% per annum on a one-year ruble time deposit. The forecast of
increase in the dollar rate per year is from Н = 27 rubles to K = 31 rubles. What decision must be
made: bring the rubles to the bank or buy dollars for them and keep them at home?
5. The bank pays i1 = 14% per annum on a one-year ruble time deposit, and i2 = 8% on the
identical foreign currency deposit. The forecast of increase in the dollar rate per year is from H =
29 rubles to K = 31 rubles. What decision must be made: bring the rubles to the bank or buy dollars
for them and deposit them on a foreign currency account?
6. The foreign currency exchange rates in the bank are: US dollar – 29.1/29.8 rubles per
dollar (i.e. Db = 29.1, Ds = 29.8); euro – 40.2/40.9 rubles per euro (i.e. Eb = 40.2, Es = 40.9).
What is the dollar to euro exchange transaction yield for the bank?
7. A car loan of A = 480 000 rubles with an annual rate of i = 13% and the down payment
of q = 20% was taken for n = 3 years. The bank services for credit use make g = 0.2% per
month. Calculate the amount of a monthly payment (the amount of a due payment), average annual
yield and domestic return of the transaction for the bank.
Topic 5. Investment Processes
1. How will the payback period of a project change with a change of the values of
investments, annual returns, interest rate? Build graphs and give explanations.
2. Check the following calculations of an investment project: K = 4000 currency units,
subsequent annual return at i = 8% per annum equals R = 1000 currency units, project duration
n = 6 years and it is found that the net present value is NPV = 623 currency units and the payback
period – t = 5 years.
3. A credit of D = 10000 currency units for a term of n = 10 years at a rate of g = 5%
was taken from a bank to develop an investment project. The credit was taken in one bank, while
the return from the project is placed in another bank at a higher rate of i = 8%. A repayment fund is
created to secure the repayment of debt. Calculate the final characteristics for the following
discharge models:
1) The principal balance of the debt is discharged from the fund at the end of the term by a
single payment. The amount of payments to the fund with their interest must equal the debt at the
moment of its payment. The interest on the debt is paid not from the fund.
2) The conditions of a financial commitment instead of a regular interest payment stipulate
their addition to the amount of the principal balance of the debt.
3) The fund is formed in such a way that a regular payment of the interest on the debt
(from the fund) is secured, as well as the repayment of the principal balance of the debt.
4. Somebody has received inheritance in a form of a fat bank account K = 300 000
currency units and now is ―eating it away‖ by taking a sum of R = 3000 currency units from the
account monthly and spending it within a month. The bank rate equals i =10% per annum. In
essence, it is an ―inverse‖ investment process. What are the investments, payback period, inner rate
of return, net present value here? Calculate these characteristics.
5. An investment project is under consideration. The project stipulates the following stream
of investment payments: the first payment K1 = 160 thousand currency units at the time point
tK 1 = 0, the second payment K 2 = 200 thousand currency units at the time point tK 2 = 0.5 years,
the third payment K3 = 250 thousand currency units at the time point tK 3 = 1.5 years. The
project profit starts after the time
= 0.5 years after the last investment payment. The payment
stream is as follows: R1 = 200 thousand currency units at the time point tR1 = 2 years, R2 = 300
thousand currency units at the time point tR 2 = 3.6 years, R3 = 400 thousand currency units at the
time point tR 3 = 4 years, R4 = 500 thousand currency units at the time point tR 4 = 4.5 years. The
risk-free interest rate is r = 10%. Calculate the characteristics of the investment project (net
present value, internal rate of return, payback period, profitability index).
6. Calculate the annual payment for leasing of the equipment worth P =$20 000 within
n =10 years, if the residual value of the equipment is S =$10 000 by the end of the leasing term.
Calculate the discount rate considering the internal rate of return i equal 13%.
7. Determine if equipment worth P =$20 000 should be bought or rented for n =8 years
with an annual rent payment of R =$3000, if the interest rate is i =6% per annum, and the ratio of
equipment depreciation is h = 15%.
Note. The residual value of the equipment is S  P(1  n  h) , where P is the cost of the
equipment, and n – operating life.
8. Calculate the characteristics of the investment project whose payment stream is shown on
the figure.
K  2000
0
R1  1000
i1  5%
1
R2  800
i2  8%
2
R3  800
i3  6%
3
R4  600
i4  10%
4
Topic 6. Fixed Income Securities
1. What is good for the security holder: increase or decrease of the current interest rate at
the period of holding the security if it is a: a) bond; b) share; в) certificate of deposit.
2. Find the bond rate without discharging with a regular – once a year – interest payment
with q = 8% , i = 5% . Calculate the yield of such a bond if its rate equals K =120.
3. Find the rate of a zero coupon bond for m =5 years before its maturity with i = 6%.
Calculate the yield of such a bond if its rate equals K =70.
4. Find the rate of a zero coupon bond with interest payment at its discharge 5 years prior to
its maturity with i =4%, if the bond was issued for 10 years and q = 8%. Calculate the yield of
such a bond if its rate equals 100.
5. Find the price of the indefinite share with quarterly dividends of d =200 at an annual rate
of i = 8%.
6. Calculate the transaction yield of a discount of a promissory note at a rate of q = 30%,
m =3 months prior to its payment (the base annual number equals 360 days – a month equals 30
days). During the discount transaction, a commission fee of k =0.5% of the promissory note worth
was retained from the account of the promissory note holder.
7. Quarterly coupon payments are promised to be made on a 6% coupon bond with the face
value of N =200 currency units. Determine the bond price at the moment when the term left before
the bond maturity is: a) 16 months; b) 15 months. The market interest rate is i = 10 %.
8. An 8% coupon bond with the face value of 1000 currency units is considered; coupon
payments are promised to be made on this bond twice a year within 3 years. Risk-free interest rates
are the same for all the intervals and equal 10% per annum.
1) Calculate the bond duration and factor of convexity;
2) Evaluate the relative change in the bond price at the interest rate change by ± 1% using:
a) only bond duration; b) bond duration and factor of convexity. Indicate the role of each of the
factors in evaluating the bond price change. Represent by a diagram the dependency P / P on
r /(1 r ) by formulas (9.9) and (9.10).
9. Two bonds with 10% coupon rates and a face value of 1000 are given. One of them has a
a term before maturity of T1 = 4 years, and the other – T2 = 15 years. Annual interest payments are
made on both bonds. Supposing the domestic return of the bonds increases from r1 = 10% to r2 =
14%, calculate the bond price before and after the interest rate changes. Explain the differences in
the bond price interest changes.
10. Without calculations, range the following bonds on duration (coupon payments are paid
at the end of each year), see Table 6.9
Table 6.9.
Bond
Term before
maturity
Coupon rate
Domestic return
А
30 years
10 %
10%
В
30 years
0%
10 %
С
30 years
10 %
7%
D
5 years
10 %
10 %
11. Two bonds are given, whose payment streams are given in Table 6.10.
Table 6.10.
Payment
moment t1 ,
years
1
2
3
4
Payments, R1
10
10
10
300
Payment
moment t2 ,
years
2
3
4
5
Payments, R2
10
10
10
300
The domestic return of the bonds is r1
factor of convexity of these bonds.
r2 = 8% per annum. Determine the duration and
12. A bond with a payment stream (defined in Table 6.12) is given.
Table 6.12.
Payment
moment t ,
years
0.5
1.0
1.5
2.0
2.5
3.0
Payment R
4
4
5
5
5
100
Risk-free interest rates for all terms are common and equal r = 6% per annum. All the
payments on bonds were deferred by t = 0.5 years. Evaluate the percentage change of the price of
the bond with deferred payments if the risk-free interest rates for all the terms were increased by
r = 1%.
13. An investor considers buying a 20-year bond whose coupon payments are to be paid
every six months. The face value of the bond is N = 1000 currency units, annual coupon rate is g
= 8 %, yield to maturity r = 10 % per annum. The investor expects that s/he will be able to reinvest
coupon payments at an annual rate of i = 6% within m = 3 years. At the end of the m year, the
investor hopes to sell the bond with a yield to maturity of r1 = 7 % per annum. Determine the
average annual yield of investing in this bond for m = 3 years under the same conditions.
14. There is a 9% coupon bond with a face value of 1000 currency units on the market, on
which coupon payments are promised to be made within 5 years. The risk-free interest rates r are
common and equal 9% per annum. Find the planned and actual value of investing in the bond at the
time point that equals the bond duration if after t1 = 0.5 years after the purchase of the bond, the
interest rates decreased to r1 = 8.5 % , and after t2 = 1.5 years after the purchase came back to the
level of r2 = 9 % per annum.
15. A 10% coupon bond with semi-annual coupons is given. The domestic return of the bond
equals 6%. determine the bond duration when there is
n
years left before its maturity if n =
2
1,2,…,10. Show the dependency of the duration on the term before maturity on a figure.
Topic 7. The Optimal Portfolio of Securities
1. Calculate the Markowitz optimal portfolio of the minimum risk for three securities with
yields and risks: (4,10); (10,40); (40,80); the lower boundary of the portfolio return m p is given
equal 15.
2. Calculate the optimal portfolio of maximum effectiveness for the three securities with
yields and risks: (4,10); (10,40); (40,80) (the same securities as in Example 1); the upper boundary
of the risk rp is given equal 50.
3. Having risk-free securities with an effectiveness of m0 =4 and uncorrelated risk-laden
securities with an effectiveness of m1 =8 and m2 =14 and risks r1 =10 and r2 =30, generate an
optimal portfolio with a yield m p that equals 12. The portfolio may contain negative ratios of the
securities. How should the negative ratios of the securities be understood?
4. Form a minimum risk portfolio from two security types: risk-free with a yield of m0 =2
and risk-laden with a yield of m1 =10 and risk of r1 =5. Draw a graph of the dependency of the
portfolio yield on its risk.
5. Form a portfolio with maximum yield and risk not more than the given one from three
security types: the risk-free ones with an effectiveness of m0 =2 and uncorrelated risk-laden ones
of the expected effectiveness of m1 =4 and m2 =10 and risks of r1 =2 and r2 =4. What are the
correlations of the security ratio in the risk-laden part of the optimal portfolio? Draw a graph of the
dependency of the portfolio yield on its risk.
6. Out of two uncorrelated securities with an effectiveness of 2 and 6 and risks of 10 and
20, six portfolios were built on the computer: in the portfolio with the number k the ratio of the
first securities is x  1  0,2k (k  0,1,2,3,4,5) , the ratio of the second ones is ( 1  x ), i.e. the
portfolio containing only the securities of the 1st type gains Number 0, and the portfolio containing
only the securities of the 2nd type gains Number 5. The computer found their effectiveness and
risks (cf. Table 5.6).
Table 5.6.
Effectiveness
2.0
2.8
3.6
4.4
5.2
6.0
Risks
10.0
8.9
10.0
12.6
16.1
20
Portfolios
0
1
2
3
4
5
Check the computer calculations. Then indicate the portfolios as dots on the riskeffectiveness plane and mark the dominated portfolios and the non-dominated ones, i.e. the Paretooptimal portfolios.
7. The portfolio securities with a yield of 5% per annum make 30% in their cost, and the
remaining securities have a yield of 8% per annum. What is the portfolio yield?
8. Let us record the portfolio yield variation V p 
 xi x jVij
in a form of:
i, j




V p   xi   x jVij  and call the value Ri    x jVij  the portfolio yield
 j

 j

i




covariance of the i security. Prove that in the optimal portfolio, this covariance is proportional to
the yield excess of the securities over the risk-free investments (the existence of the latter on the
market is implied).