Chapter 2. Theory of interest

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Archibald Osborn Golden
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1 Chapter 2 Theory of iterest

2 Tie alue of oey Cash flow ( 現金流 ) aout of oey receied (+) or paid out (-) at soe tie poit Tie alue of oey whe aluig cash flows i differet tie periods, the iterest-earig capacity of oey ust take ito accout.

3 Tie alue of oey Pricipal ( 本金 ) Iitial aout of iestets Accuulated alue Value of iestet after a period of tie Accuulatio fuctio a(t) a(t)=accuulated alue at tie t for $ pricipal a(0)= a(t) is icreasig a(t) is cotiuous Iterest The aout eared=acc. alue pricipal Iterest rate i Accuulated Value - Pricipal a( t) a(0) i Pricipal a(0)

4 Siple/Copoud iterest Siple iterest A sigle iterest is paid oer the whole period of iestet a(t)=+it Copoud iterest Iterest is paid oer a fixed period ad reiested to ear additioal iterest a(t)=(+i) t

5 Discout factor/fuctio/rate Discout factor, (for period) If I wat to hae $ at tie, how uch should I iest right ow? ( i) i Discout fuctio (for t periods) If I wat to hae $ at tie t, how uch should I iest right ow? Discout fuctio = a - (t) Discout rate d =-d

6 Exercises What is the relatioship betwee i, d ad? i = f(d) i = f() d=f(i) d=f() =f(i) =f(d)

7 Exercises What is the relatioship betwee i, d ad? i = d/(-d) i = (-)/ d= i/(+i) d=- =/(+i) =-d

8 Exercises 2 Is 6% Iterest rate = 6% discout rate?

9 Exercises 2 Is 6% Iterest rate = 6% discout rate? i=d/(-d) d=i/(+i) 6% iterest rate discout rate = 0.06/.06=5.66% 6% discout rate Iterest rate = 0.06/0.94 = 6.38% The absolute differece are the sae (0.06), but the ratios are differet

10 Effectie rate durig the -th period Effectie rate of iterest durig -th period a( ) a( ) i a( ) Effectie rate of discout durig -th period a( ) a( ) d a( ) The differece b/w iterest & discout is o the deoiator

11 Noial rate of iterest Soeties iterest is paid ties per year =2 (seiaually) =4 (quarterly) =2 (othly) =360 (daily) Noial rate of iterest payable ties per year: i () i ( ) is paid at the ed of each -th subiteral durig a year

12 Noial rate of iterest Exaple What is the accuulated alue for a 4% quarterly payable oial iterest rate for a iestet of $500 for 3 years? What is the equialet effectie iterest rate per year?

13 Noial rate of iterest Exaple What is the accuulated alue for a 4% quarterly payable oial iterest rate for a iestet of $500 for 3 years? i () = Accu.Value $500a(3) $500 $ What is the equialet effectie iterest rate per year? i i 4.06% 4x3

14 Cotiuous rate of iterest Cotiuous = iterest is paid ties per year =2 (othly) =360 (daily) = (cotiuously) Basic Matheatics result t e = Force of iterest = cotiuous rate of iterest t

15 Tie alue of oey Preset alue = the su of the total alues of the future cash flows discouted at the preset tie poit PV a ( t) c( t) t0

16 Tie alue of oey Accuulated alue = the su of the total alues of the future cash flows discouted at the ed of the whole period. AV t0 a( t) c( t)

17 Tie alue of oey - exaple 0 ) ( ) ( t t c t a PV What is the preset ad accuulated alue for the followig cash flow if i=0.02 for the st year? t t c t a AV 0 ) ( ) (

18 Tie alue of oey - exaple What is the preset ad accuulated alue for the followig cash flow if i=0.02 for the st year ad i=0.03 thereafter? PV AV t0 a ( t) c( t) 5.02 a( t) c( t) 5(.02)(.03) t ( Note that (.02)(.03 5 )= )

19 Auities ( 年金 ) Auities=A series of payets ade at equal tie iterals Types of auities based o ucertaity Auity-certai Nuber of payet is fixed ad kow e.g. Mortgage ( 按揭 ) payet o a house Life-Auity Nuber of payets depeds o the legth of life e.g. Mothly beefit fro a pesio pla ( 退休金 )

20 Auities ( 年金 ) Auities=A series of payets ade at equal tie iterals Types of auities based o st payet tie Auity-iediate Payet ade at the ed of each iteral Preset Value: Accu. Value: a s Auity-due Payet ade at the begiig of each iteral Preset Value: Accu. Value: a s a a s s

21 Basic calculatio of Auities Auity-iediate Auity-due Auities ( 年金 ) s a a 3 2 a s ) ( a s i a 3 2 ) ( a i s

22 Perpetuities = Auities that pay util ifiity Perpetuity-iediate Perpetuity-due Perpetuities ( 永續年金 ) a 2 a 2

23 Auities payable ties a year Auity-iediate Auity-due More o Auities (optioal) ) ( 2 ) ( ) ( i i a ) ( ) ( ) ( a s i ) ( ) ( ) ( a i s ) ( 2 ) ( i a

24 More o Auities (optioal) Cotiuous Auities No eed to distiguish betwee iediate & due Preset Value a 0 t dt t log 2 0 li log f(t)= t Accuulated Value s i ( ) a Cotiuous perpetuities : a

25 Hided R codes for p.20 =0.9;t=(:000)/00;plot(t,^t,type="l",ylab="^t",xaxt="") for (j i :4){lies(c(j,j),c(0,^j))} axis(,at=c(0,,2,3,4,0),labels=c(0,"/","2/","3/","...",""))

26 Auities Exaple Toy has a ortgage pla for his ew house. He borrowed $3 fro HSBC. What is the othly repayet if he wat to clear the debt i 20 years ad the oial iterest rate is 0.02?

27 Auities Exaple Toy has a ortgage pla for his ew house. He borrowed $3 fro HSBC. What is the othly repayet if he wat to clear the debt i 20 years ad the oial iterest rate is 0.02? $3,000,000 P $ Pa (2) 20 3,000,000 $ P x ,

28 Balace ad Resere Balace ( 結餘 ) aout accuulated up to (just before) tie k Balace Resere ( 儲備 ) k j0 c( j) a( k j) +e c(i) iflow -e c(i) outflow aout eeded at (just before) k for future obligatio Resere N jk c ( j) a ( j k)

29 Balace ad Resere Exaple Toy borrows $000 ow ad $2000 at the ed of yr. The loa is repaid by yearly payet of $500 for yrs, begiig 5 years fro ow. For the whole period, i=0.06. What are the balace ad resere at tie 8?

30 Balace ad Resere Exaple Toy borrows $000 ow ad $2000 at the ed of yr. The loa is repaid by yearly payet of $500 for yrs, begiig 5 years fro ow. For the whole period, i=0.06. What are the balace ad resere at tie 8? Balace Balace 000(.06) (.06) 7 500( ) Resere Resere 500( 7 )

31 Balace ad Resere Exaple 2 Toy borrows $000 ow ad $2000 at the ed of yr. The loa is repaid by yearly payet of yrs, begiig 5 years fro ow. Agai, i=0.06. What is the fair repayet? What are the balace ad resere at tie 8?

32 Balace ad Resere Exaple 2 Toy borrows $000 ow ad $2000 at the ed of yr. The loa is repaid by yearly payet of yrs, begiig 5 years fro ow. Agai, i=0.06. What is the fair repayet? What are the balace ad resere at tie 8?.06 Repayet P( Balace Balace 000(.06) 2000(.06) P( Resere 7 Resere P( ) ) P 2.06) If the Total cash flow is 0, Balace=Resere at ay tie

33 Suary Siple/copoud iterest Iterest rate, discout rate/factor Noial rate of iterest Cotiuous rate (Force) of iterest Preset/Accuulated Value Auities/Perpetuities Auity-iediate Auity-due Balace/Resere

Section 3.3 Exercises Part A Simplify the following. 1. (3m 2 ) 5 2. x 7 x 11

Section 3.3 Exercises Part A Simplify the following. 1. (3m 2 ) 5 2. x 7 x 11 123 Sectio 3.3 Exercises Part A Simplify the followig. 1. (3m 2 ) 5 2. x 7 x 11 3. f 12 4. t 8 t 5 f 5 5. 3-4 6. 3x 7 4x 7. 3z 5 12z 3 8. 17 0 9. (g 8 ) -2 10. 14d 3 21d 7 11. (2m 2 5 g 8 ) 7 12. 5x 2