1. Financial and investment mathematics RNDr. Petr Budinský, CSc.

2. VŠFS2 FINANCIAL MATHEMATICS Future value – different types of compounding

3. VŠFS3 Example: Assume FV = 100.000 CZK and interest rate = 12 %. Calculate future value in 3 years time. … … Example: Assume FV = 100.000 CZK and interest rate = 12 %. Calculate future value in 3 years time. … …

4. VŠFS4 Present value calculated from future value

5. VŠFS5 Example: Assume cash flows given by following table and interest rate Example: Assume cash flows given by following table and interest rate r = 6 %, compounded r = 6 %, compounded a) Once yearly year1234 cash flows0100200300

6. VŠFS6 Example: Assume cash flows given by following table and interest rate Example: Assume cash flows given by following table and interest rate r = 6 %, compounded r = 6 %, compounded b) Continously

7. VŠFS7 Yield calculated in case of fixed cash flows

8. VŠFS8 Equalities and inequalities among yields

9. VŠFS9 Example: Assume an investment P = 10.000 Kč for 5 years, after 5 years you earn an amount FV = 21.000 CZK. Calculate the yield. Example: Assume an investment P = 10.000 Kč for 5 years, after 5 years you earn an amount FV = 21.000 CZK. Calculate the yield.

10. VŠFS10 Example: Assume a loan 1.000.000 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. Example: Assume a loan 1.000.000 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. C can be splitted to the interest rate payment - 80.000 CZK and to the amount 69.029,49 CZK by which the loan will be decreased to 930.970,51 CZK. C can be splitted to the interest rate payment - 80.000 CZK and to the amount 69.029,49 CZK by which the loan will be decreased to 930.970,51 CZK. 1.000.000 = C (1/(1+ y) + 1/(1+ y) 2 +... +1/(1+ y) 10 ) 1.000.000 = C[1-1/(1 + y) 10 ]/y C = 1.000.000 ⋅ 0,08/[1 −1/1,08 10 ] = 149.029,49 Kč

11. VŠFS11 Table of payments InstallmentInterest rate partPrincipal paymentRemaining part

12. VŠFS12 Bonds zero-coupon bond: annuity: perpetuity:

13. VŠFS13 Closed formula for bond price

14. VŠFS14 Rules for bonds Rule 1: Rule 1: If the yield y is equal to the coupon rate c, then the bond price P is equal to face value FV, if yield y is higher, resp. less than the coupon rate c, then the bond price P is smaller, resp. greater than the face value FV. Rule 2: Rule 2: If the price of the bond increases, resp. decreases, this results in a decrease, resp. increase of the yield of the bond. Reverse: decrease, resp. rise in interest rates (yields) results in an increase, resp. decrease in bond prices.

15. VŠFS15 Rule 3: If the bond comes closer to its maturity, then the bond price comes closer to the face value of bond. Rule 3: If the bond comes closer to its maturity, then the bond price comes closer to the face value of bond. Rule 4: Rule 4: The closer is the bond to its maturity the higher is the The closer is the bond to its maturity the higher is the velocity of approaching the face value by the price of the velocity of approaching the face value by the price of the bond. bond. Rules for bonds

16. VŠFS16 Rule for bonds Rule 5: Rule 5: The decrease in a bond yield leads to an increase in bond price by an amount higher than is the amount corresponding to the decrease (in absolute value) in the price of the bond if the yield increases by same percentage as previously decreased. Example: Assume FV = 1.000 CZK, coupon rate c = 10 % = 14 %. Example: Assume a 5-year bond with a face value FV = 1.000 CZK, coupon rate c = 10 % and yield y = 14 %. Yield12 %13 %14 %15 %16 % Price927,90894,48862,68832,39803,54 Price change65,2231,800-30,29-59,14

17. VŠFS17 : Example: Rule for bonds CZK

18. VŠFS18 Relationship of the bond price and time to maturity of the bond

19. VŠFS19 Bond pricing – general approach

20. VŠFS20 A + B = 360 A + B = 360

21. VŠFS21 Example: Example: Assume a 5-year bond with a face value FV = 10.000 CZK issued at 6. 2. 1998 with maturity 6. 2. 2003 and with coupon rate c = 14.85%. The yield of this bond was y = 7% on 9. 11. 1999. Calculate the clean price P CL of the bond. CZK

22. VŠFS22 The sensitivity of bond prices to changes in interest rates (yields) Modified duration D mod is a positive number expressing the increase (in %), resp. decrease (in %) of the bond price if the yield decreases, resp. increases by 1%. Modified duration D mod is a positive number expressing the increase (in %), resp. decrease (in %) of the bond price if the yield decreases, resp. increases by 1%.

23. VŠFS23 Macaulay duration

24. VŠFS24 Macaulay duration zero-coupon bond: annuity: perpetuity:

25. VŠFS25 Example: : FV = 1.000 CZK, n = 5, c = 10 %, y = 14 %. Example: Bond parameters are as follows : FV = 1.000 CZK, n = 5, c = 10 %, y = 14 %. CZK

26. VŠFS26 and y The dependence of duration on c and y 1. 1. 2. 2.

27. VŠFS27 Dependence of duration D on time to maturity n n

28. VŠFS28 Estimate of changes in bond prices Example : Example : a) a) b) b) CZK

29. VŠFS29 Bond convexity Convexity is sometimes called the "curvature" of the bond.

30. VŠFS30 Calculation of convexity CX = 2/y 2 CX = 2/y 2 Zero-coupon bond: Perpetuity:

31. VŠFS31 INVESTMENT MATHEMATICS INVESTMENT MATHEMATICS Risks associated with the bond portfolios When investing in bonds investor must take into account the two risks: 1. risk of capital loss (if yields increase ) 2. risk of loss from reinvestment (if yields decrease )

32. VŠFS32 Example: Example: Assume 5 year zero-coupon bond with face value a) for 3 years FV = 1.000 CZK and yield y = 4%. This bond is an investment a) for 3 years CZK

33. VŠFS33 Example b) for 7 years Example b) for 7 years Investment horizon CZK

34. VŠFS34 Investment horizon X Duration When you invest in a particular bond and if our investment horizon is: Short - you suffer a loss in case yields increase („capital loss"> „input of reinvestment“) Long - you suffer a loss in case yields decrease („loss of reinvestment "> „capital gain") If the investment horizon is equal to (Macaulay) duration of the bond, then the "capital loss", resp. „loss of reinvestment" is fully covered by "reinvestment income", resp. by „return on capital", in case of both increase and decrease of yields.

35. VŠFS35 Example: Example: Assume 8-year bond, which has a face value FV = 1.000 CZK with coupon rate c = 9,2 % and the yield y = 9,2 %. T1 year, 2 years, y = 9,2 %. This bond is an investment for 1 year, 2 years, 3 years, …, 8 years - we assume 8 investment strategies. Further assume 5 3 years, …, 8 years - we assume 8 investment strategies. Further assume 5 scenarios of development of the yields: 8,4 %, 8,8 %, 9,2 % (), 9,6 % and 10 %.. 8,4 %, 8,8 %, 9,2 % (unchanged yield ), 9,6 % and 10 %. Combination of the chosen investment strategy with a particular yield scenario will provide 40 different options. For each of these options is calculated the realized yield. All results are summarized in the table. The price of a bond P = 1.000 CZK.

36. VŠFS36 Investment strategies Scenarios

37. „1. line“ – „1. line“ – income from couponsnC; „2. line“ – „2. line“ – income from reinvestment of coupons after deduction of the coupons „3. line“ – „3. line“ – capital gain (the difference between the sale and repurchase price of the bond) „4. line“- the total return (in CZK) the sum of 1., 2. and 3. line „4. line“- the total return (in CZK) the sum of 1., 2. and 3. line „5. line“ –the total return y n in % (p.a.): „5. line“ –the total return y n in % (p.a.): so so VŠFS37

38. Bond portfolio duration The duration of a coupon bond is a weighted average of durations (time to maturities) of the individual cash flows represented by coupons and face value, the weights correspond to the share of individual discounted cash flow as a proportion of the total price of the bond. The duration of a coupon bond is mean lifetime of the bond. The duration of a portfolio consisting of bonds is the weighted average of durations of individual bonds, where the weights correspond to investments in individual bonds as proportions of the total investment in the bond portfolio. VŠFS38

39. VŠFS39 Example: Example: Assume an investment 1.000.000 CZK for 4 years, we have zero-coupon bonds with maturities of 1 year, 2 years,..., 7 years with uniform yields y = 8% (assuming a flat yield curve). Create portfolios A, B, C, D as follows (n is the time to maturity of each bond) CZK

40. VŠFS40 Change in the value V 0 in case of change of the yield Scenarios Realized amounts V 0 (CZK)

41. VŠFS41 Bond portfolio convexity Bond portfolio convexity is the weighted average of convexities of individual bonds, where the weights correspond to investments in individual bonds as proportions of the total investment in the bond portfolio..

42. VŠFS42 The effect of convexity on the behavior of bond portfolios

43. VŠFS43

44. VŠFS44 Change in the value V 4 in case of change in the yield Realized income Scenarios

45. VŠFS45 CZK

46. VŠFS46

47. VŠFS 47 Example: Example: Assume an investment of 2.800.000 CZK for 5 years, and we have available two zero-coupon bonds A, B: Create a portfolio hedged against interest rate risk and calculate yield to the investment horizon, provided that day after the purchased of the portfolio yield increased, resp. decreased by 1%. CZK

48. VŠFS48 Derivative contracts 1.forward contracts 2.option contracts (options) Forward contract is an obligation, option contract is the right, to buy or sell -agreed number of shares -at agreed price -on the agreed date Call option is right to buy. Put option is right to sell.

49. VŠFS49 Forward contract profit Short position Long position

50. VŠFS50 Option contract profit Call option Put option Call option will be exercised only if S T ˃ X and profit of this option is equal to max {S T – X, 0}. Put option will be exercised only if X ˃ S T and profit of this option is equal to max {X - S T, 0}.

51. VŠFS51 Graphs of profits and losses when using options Call option, long positionCall option, short position Put option, long position Put option, short position

52. VŠFS52 Portfolio consisting of options A.Combination of only one option type, eg. call options - portfolio is called "spread". All call options included in the portfolio have the same underlying stock and the same time to expiration, they differ in an exercise price – it is called "vertical spread". „Horizontal spread“ arises if, on the contrary same exercise price at different time expirations. B.Combination of both types of options - portfolio is made up of call options and put options. Portfolio is called "combination". C.Combination of call options or put options with underlying stocks. We speak about "hedging„.

53. VŠFS53 Bullish Spread

54. VŠFS54 Bearish Spread

55. VŠFS55 Butterfly Spread

56. VŠFS56 Condor Spread

57. VŠFS57 Bottom Straddle

58. VŠFS58 Bottom Strangle

59. VŠFS59 Hedging

60. VŠFS60 Example: Example: NumberPrice Excercise price Position Share Call option Put option

61. VŠFS61