Financial and Investment Mathematics. Dr. Eva Cipovová Department of Business Management

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1 Financial and Investment Mathematics Dr. Eva Cipovová Department of Business Management

2 Content 1. Interest and annual interest rate. Simple and compound interest, frequency of compounding. Continuous compounding. 2. Interest rate and yield, present and future value, principle of discounting. Calculation of present value of an investment 3. Yield calculation for given present and future value, recalculation of yields for different frequencies of compounding 4. Classification of bonds. The nominal value of the bond, coupon rate, bond yield to maturity. Calculation of bond price for zero-coupon bond, annuity and perpetuity 5. Rules for bonds calculation. Relationship between bond price and yield. Relationship between coupon rate and bond yield. 6. Accrued interest, gross and clean price of the bond. Calculation of clean price of the bond 7. Sensitivity of bond prices to changes in yields. Modified and Macaulay duration, convexity 8. Bond duration and convexity properties. Calculation of bond duration and convexity 9. Investing in the bonds. Reinvestment and capital risk. Duration and investment horizon 10. Bond portfolio, its duration and convexity. Construction of bond portfolio for given duration equal to the investment horizon 11. Derivates Portfolio I forward contracts and options, basic graphs of profits depending on the price of the underlying asset 12. Derivates Portfolio II strategies as spread, combination and hedging

3 Required literature 1. Block S., Hirt G., Danielsen B.: Foundations of Financial Management, McGraw-Hill/Irwin, New York NY, 2009, 13th Edition, ISBN RICHARD A. BREALEY, Richard A. Principles of corporate finance. 10th ed. [international edition]. Boston, Mass: McGraw- Hill, ISBN P. Budinský, P. Záškodný: Finanční a investiční matematika, Praha 2004 Recommended literature 1. J. O. Grabbe: International Financial Markets, Elsevier Sc. Publ., J. Hull: Options, Futures and Other Derivative Securities, Prentice- Hall P. Budinský, P. Záškodný: Ekonomická statistika, Praha D. Blake: Analýza finančních trhů, Grada Publ., Praha 1995

4 Conditions for successful completion of the course Granting of credit: at least 80% participation in seminars home work aimed at calculating the net price of the bond Grating of exam: Pass written test

5 1. lecture

6 The time value management Time value of money applies to many day to day decisions Understanding the effective rate on a business loan, the mortage payment in a real estate transaction, or the true return on an investment depends on understanding the time value of money As long as an investor can make a positive return on idle dollars, distinctions must be made between money received today and money received in the future The investor/lender essentially demands that a financial rent be paid on his or her funds as current dollars are set aside today in anticipation of higher returns in the future

7 Relationship to the capital outlay decision The decision to purchase new plant and equipment or to introduce a new product in the market requires using capital allocating or capital budgeting techniques. Essentially, we must determine whether future benefits are sufficiently large to justify current outlays.

8 Interest rate and yield Financial system is affected by one essential factor, i.e. an interest rate. Interest rate is given in percent per year ( % p.a.) and represent actually price of money. It reflects the fact that $ 1 today had a different value yesterday and will have a different value tomorrow. Interest rate r (p.a., p.s., p.q., p.m., p.d.) Future value FV Present value PV Number of years n Frequency of compounding - m Firstly, we would like to calculate our FV, where we know how much is our present value, interest rate and number of years

9 Simple interest

10 Compound interest is interest added to the principal of a deposit or loan so that the added interest also earns interest from then on. This addition of interest to the principal is called compounding. The interest is calculated from previous interest and so on.. If the interest is added m-time a year (m frequency of compounding), then the formula for compound interest is as follows: Frequency of compounding could be annually (m=1), semiannually (m=2), quarterly (m=4), monthly (m=12)

11 Continous compounding It represents special type of compounding where making the compounding period infinitesimally small. Within the conitnous compounding, infinitely small interests are added infinitely many times, which leads to Euler number (e = 2, ) Reason: in the bank, deposits are remunerated at a fixed interest rate, and interest are added continuously Therefore, when n>1, the following applies:

12 Example Assume FV = CZK and interest rate = 12 %. Calculate future value in 3 years time.

13 Example Assume FV = CZK and interest rate = 12 %. Calculate future value in 3 years time.

14 Effective interest rate It is used to compare the annual interest between loans with different compounding terms (daily, monthly, annually, or other). The effective interest rate is calculated as if compounded annually. The effective rate is calculated in the following way, where r is the effective annual rate, i the nominal rate, and the number of compounding periods per year (for example, 12 for monthly compounding): Example: a nominal interest rate of 6% compounded monthly is equivalent to an effective interest rate of 6.17%. 6% compounded monthly is credited as 6%/12 = every month. After one year, the initial capital is increased by the factor ( )

15 Present value calculated from future value compounding vs. discounting

16 Example: Assume cash flows given by following table and interest rate r = 6 % p.a., compounded a) Once yearly Calculate PV, FV 4, FV 6, FV 3

17 Example: Assume cash flows given by following table and interest rate r = 6 % p.a., compounded a) Once yearly

18 Example Assume cash flows given by following table and interest rate r = 6 % p.a., compounded a) Continously Calculate PV, FV 4, FV 6, FV 3

19 Example Assume cash flows given by following table and interest rate r = 6 % p.a., compounded a) Continously Calculate PV, FV 4, FV 6, FV 3

20 Yield calculated in case of fixed cash flows

21 Example Assume an investment P = Kč for 5 years, after 5 years you earn an amount FV = CZK. Calculate the yield

22 Example Assume an investment P = Kč for 5 years, after 5 years you earn an amount FV = CZK. Calculate the yield

23 Annuity

24 Example We are saving certain amount of money (PMT = 100 CZK) within three years without initial investment. Interest rate is 10 % p.a. Calculate future value (FV(1), FV(2), FV (4), FV(12), FV(*)) of this saving with different frequency of compounding.

25 Example We are saving certain amount of money (PMT = 100 CZK) within three years without initial investment. Interest rate is 10 % p.a. Calculate future value (FV (1), FV (2), FV (4),FV (12), FV(*)) of this saving with different frequency of compounding.

26 Example Assume a loan CZK for 10 years. This loan is paid by same installments C at the end of each year with the yield y (1) = 8 % p.a. Calculate the installment C.

27 Example Assume a loan CZK for 10 years. This loan is paid by same installments C at the end of each year with the yield y (1) = 8 % p.a. Calculate the installment C. Table of payments Payment Interest Principal Balance

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