Financial Mathematics Lecture Dr Wioletta Nowak

Transcription

1 Financial Mathematics Lectue 9-0 Wioletta Nowa

2 Fundamentals of bond valuation Bond a loan between a boowe (issue) and a lende (investo, cedito) The issue pomises to mae eula inteest payments to the investo at a specified ate (the coupon ate) on the amount it has boowed (the face/pa amount) until a specified date (the matuity date). Once the bond matues, the inteest payments stop and the issue is equied to epay the face amount of the pincipal to the investo.

3 Fundamentals of bond valuation Bonds can be piced at a pemium, discount, o at pa. If the bond's pice is hihe than its pa value, it will sell at a pemium because its inteest ate is hihe than cuent pevailin ates. If the bond's pice is lowe than its pa value, the bond will sell at a discount because its inteest ate is lowe than cuent pevailin inteest ates.

4 Fundamentals of bond valuation Bond valuation is the detemination of the fai pice of a bond. The pice of bond is the sum of the pesent values of all expected coupon payments plus the pesent value of the pa value at matuity. Yield to matuity is the intenal ate of etun eaned by investo who buys the bond today at the maet pice, assumin that the bond will be held until matuity.

5 Bond picin coupon bonds C i income fom the owneship bonds at time i, n numbe of payments, YTM yield to matuity, bond pice C C C Cn 0 n YTM YTM YTM YTM C YTM C n i n YTM YTM i YTM C n C i

6 Bond picin coupon bonds Constant coupon ate, C coupon payment, M value at matuity o pa value, n numbe of payments, YTM yield to matuity, bond pice C YTM C YTM YTM C C M YTM YTM n M n YTM YTM n C YTM YTM n M YTM n

7 Example Suppose a 4-yea bond with the value at matuity of 00 LN and a coupon ate of 0%. Time to matuity ice of bond YTM= 9% YTM=0% YTM=% emium iscount ecent of pemium decline ecent of discount decline %.% % 9.9% % 47.9%

8 Example Suppose a -yea bond with the value at matuity of 00 LN. Coupon ate ice of bond YTM = 8% YTM = % ecent of decease 0% % 5% %

9 Example n=0 M=00 00,00 80,00 60,00 40,00 0,00 00,00 80,00 60,00 40,00 0,00-0 0, 0, 0, 0,4 0,5 0,6 YTM c=5% c=0% c=5%

10 Example 4 Suppose a bond with the value at matuity of 00 LN and a coupon ate of 0%. Time to matuity (in yeas) ice of bond YTM = 8% YTM = % ecent of decease % %

11 Example 5 M=00, c=5% n= n=5 n=0 YTM 0 0,05 0, 0,5 0, 0,5 0,

12 Example 6 Calculate the pice of a bond with a pa value of 00 LN to be paid in two yeas (afte and befoe the coupon payment), a coupon ate of 0%, and a equied yield of 9%

13 Example 7 Calculate the pice of a bond with a pa value of 00 LN to be paid in two yeas and six months, a coupon ate of 0%, and a equied yield of 8%. An annual coupon payment

14 Zeo-coupon bonds Zeo-coupon o accual bonds do not pay a coupon. Instead, these types of bonds ae issued at a deep discount and pay the full face value at matuity.

15 Fundamentals of bond valuation bond pice Zeo-coupon bond, M value at matuity, n numbe of peiods, inteest ate, bond pice M 0 n M n

16 Example 8 Calculate the pice of a zeo-coupon bond that is matuin in one and a half yeas, has a pa value of 00 LN and a equied yield of 5%

17 epetual bond picin A bond with no matuity date. Issues pay coupons foeve. C C C C coupon inteest on bond, an expected yield fo maximum tem available C

18 uation of a financial asset that consists of fixed cash flows The weihted aveae of the times until the fixed flows ae eceived w t w t w n t n w i V i Vi V V V V the pesent value of the payment at time ti V n

19 The Macaulay duation n n YTM C YTM C YTM C n peiod in which a coupon is eceived

20 The Macaulay duation The weihted aveae of the time of eceipt of a bond s fixed cash flow payments. The balance point of a oup of cash flows. It helps to compae bonds with diffeent time to matuity and diffeent coupon ates. The hihe a bond s coupon the shote the Macaulay duation. The lone a bond s matuity the eate its duation. n m a fequency of coupon YTM m

21 The Macaulay duation The hihe the YTM the shote the Macaulay duation Hihe fequency of coupon payment the shote the Macaulay duation. Zeo-coupon bonds have duations equal to thei matuities.

22 Example 9 Suppose a -yea bond with a value at matuity of 00 LN, coupon ate of 5%, YTM of 0%. What is the Macaulay duation of the bond?

23 The modified Macaulay duation The modified Macaulay duation measues the pice sensitivity of a bond when thee is a chane in the yield to matuity YTM YTM M YTM M YTM M YTM m m a fequency of coupon

24 Example 0 Suppose a -yea bond with the value at matuity of 00 LN, a coupon ate of 5% and the YTM of 0%. How much will the bond pice chane if the YTM inceases by pecentae point (deceases by pecentae point) M. 59. YTM %.59% YTM % M.59% YTM

25 Shae evaluation models ividend discount model method of estimatin the value of a shae stoc as the pesent value of all expected futue dividend payments. Constant dividend model Constant dividend owth ate model Godon model Two-stae dividend owth model Multistae dividend owth model

26 ividend discount model 0

27 ividend discount model n n n n

28 Constant dividend model

29 Constant dividend owth ate model ividend will ow at a constant owth ate. Fo a nown

30 Constant pepetual owth model Model in which dividends ow foeve at a constant ate, and the owth ate is stictly less than the discount ate.

31 Constant pepetual owth model

32 Example Suppose the cuent dividend is 00 LN. If the discount ate is 0%, what is the value of the stoc? Constant dividend discount model Constant pepetual owth model (suppose dividends ae pojected to ow at 8% foeve)

33 Two-stae dividend owth model ividend ow at a ate duin a fist stae of owth lastin n yeas and theeafte ow at a ate duin a pepetual second stae of owth n n n n n n n n n n n

34 Example Suppose a fim has a cuent dividend of 00 LN which is expected to ow at the ate of 8% fo yeas, and theeafte ow at the ate of %. With a discount ate of 0%, what is the value of stoc? 4

35 Example

36 Example ividend is expected to ow at fo 4 yeas, at fo yeas, at fo yeas, and theeafte at