Financial and investment mathematics

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

2. FINANCIAL MATHEMATICSFuture value – different types of compounding

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

4. Present value calculated from future value

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

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

7. Yield calculated in case of fixed cash flows

8. Equalities and inequalitiesamongyields

9. 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. 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. 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 −1/1,0810 ] = 149.029,49 Kč

11. Table ofpayments Installment Interest rate part Principal payment Remaining part

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

13. Closed formula for bond price

14. Rulesforbonds • 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 ishigher, resp. less than the coupon rate c, then the bond price P is smaller, resp. greater than the face value FV. • Rule 2:If the price of the bond increases, resp. decreases, this results in a decrease, resp. increase ofthe yield of the bond. Reverse: decrease, resp. rise in interest rates (yields) resultsin an increase, resp. decreasein bond prices.

15. Rulesforbonds • Rule 3:Ifthe bond comescloser to its maturity, thenthe bond pricecomescloser to the face valueof bond. • Rule 4: Thecloseristhe bond to its maturity thehigheristhe velocityofapproachingthe face value by thepriceofthe bond.

16. Rule forbonds • Rule 5:The decreasein abond yield leadsto an increase in bond priceby an amount higherthanisthe amount corresponding to thedecrease (in absolutevalue) in the price of the bond iftheyieldincreases by samepercentage as previouslydecreased. Example:Assumea 5-year bond with a face value FV = 1.000 CZK, couponrate c = 10 % and yield y= 14 %.

17. Rule forbonds • Example: CZK CZK CZK CZK CZK CZK

18. Relationshipof the bondprice and time to maturity ofthe bond

19. Bond pricing – generalapproach

20. A + B = 360

21. 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 PCL of the bond. CZK CZK CZK CZK CZK

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

23. Macaulayduration

24. Macaulayduration zero-coupon bond: annuity: perpetuity:

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

26. The dependence of duration on cand y • 1. • 2.

27. n Dependence ofduration D on time to maturity n

28. Estimateof changes in bond prices • Example: CZK CZK • a) CZK CZK • b)

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

30. Calculationofconvexity Zero-coupon bond: • CX = 2/y2 Perpetuity:

31. investmentmathematicsRisks associatedwiththebond 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. Example:Assume 5 year zero-coupon bond withface value FV = 1.000 CZK and yield y = 4%. This bond isaninvestmenta) for 3 years CZK CZK CZK

33. Exampleb) for 7 years CZK Investment horizon

34. Investmenthorizon X Duration • When youinvest in a particular bond and if our investment horizon is: Short- you suffer a loss in case yieldsincrease („capital loss"> „input of reinvestment“) Long- you suffer a loss in case yieldsdecrease („lossof reinvestment "> „capital gain") • If the investment horizon is equal to (Macaulay) duration of the bond, then the "capital loss", resp. „lossof reinvestment" isfullycovered by "reinvestment income" , resp. by „returnon capital" , in case ofbothincrease and decreaseofyields.

35. Example: Assume 8-year bond, which has a face value FV = 1.000 CZK withcouponrate c = 9,2 % and theyield y = 9,2 %. This bond isaninvestmentfor1 year, 2 years, 3 years, …, 8 years- weassume 8 investmentstrategies. Furtherassume 5 scenariosofdevelopmentoftheyields: 8,4 %, 8,8 %, 9,2 % (unchanged yield ), 9,6 % and 10 %. Combinationofthe chosen investment strategy with a particular yieldscenario willprovide40 different options. For each of these options is calculated therealizedyield. All results are summarized in the table.The price of a bond P = 1.000 CZK.

36. Investment strategies Scenarios

37. „1. line“ – incomefromcoupons nC; „2. line“ – incomefrom reinvestment of coupons after deduction of the coupons „3. line“ – capital gain(the difference between the sale and repurchase price of the bond) „4. line“- thetotal return (in CZK) the sum of 1., 2. and 3. line „5. line“ –thetotal return yn in % (p.a.): so

38. Bond portfolio duration The duration of a coupon bond is a weighted averageof durations (timeto 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 proportionofthe 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 averageof durations of individual bonds, wherethe weights correspond to investments in individual bonds as proportionsofthe total investment in the bond portfolio.

39. Example: Assumeaninvestment 1.000.000 CZKfor 4 years, we have zero-coupon bonds with maturitiesof 1 year, 2 years, ..., 7 years with uniform yields y = 8% (assuming a flat yield curve). Create portfoliosA, B, C, Das follows (n is the time to maturity of each bond) CZK CZK CZK CZK CZK CZK CZK

40. Change in the value V0in case ofchange oftheyield Realized amounts V0 (CZK) Scenarios

41. 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. The effect of convexity on the behavior of bond portfolios

44. Change inthe value V4in case ofchange in theyield Realized income Scenarios

45. CZK CZK

47. CZK CZK CZK CZK CZK Example: Assumeaninvestment 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 purchasedof the portfolio yieldincreased, resp. decreased by 1%.

48. Derivativecontracts • forward contracts • option contracts (options) • Forward contract is an obligation, option contract is the right, to buy or sell • agreednumberofshares • atagreedprice • on theagreeddate • Call optionisright to buy. • Putoptionisright to sell.

49. Forward contract profit Short position Long position

50. profit Call option Put option Optioncontract Call option will be exercised only if ST ˃ X and profit of this option is equal to max {ST – X, 0}. Put option will be exercised only if X ˃ ST and profit of this option is equal to max {X - ST, 0}.