2. The Time Value of MoneyThe Time Value of Money
Compounding andCompounding and
Discounting Single SumsDiscounting Single Sums
3. We know that receiving P1 today is worthWe know that receiving P1 today is worth
moremore than P1 in the future. This is duethan P1 in the future. This is due
toto opportunity costsopportunity costs..
The opportunity cost of receiving P1 inThe opportunity cost of receiving P1 in
the future is thethe future is the interestinterest we could havewe could have
earned if we had received the P1earned if we had received the P1
sooner.sooner.
Today Future
4. If we can measure this opportunityIf we can measure this opportunity
cost, we can:cost, we can:
5. If we can measure this opportunityIf we can measure this opportunity
cost, we can:cost, we can:
Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future
(compounding)(compounding)..
6. If we can measure this opportunityIf we can measure this opportunity
cost, we can:cost, we can:
Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future
(compounding)(compounding)..
Today
?
Future
7. If we can measure this opportunityIf we can measure this opportunity
cost, we can:cost, we can:
Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future
(compounding)(compounding)..
Translate P1 in the future into its equivalent todayTranslate P1 in the future into its equivalent today
(discounting)(discounting)..
Today
?
Future
8. If we can measure this opportunityIf we can measure this opportunity
cost, we can:cost, we can:
Translate P1 today into its equivalent in the futureTranslate P1 today into its equivalent in the future
(compounding)(compounding)..
Translate P1 in the future into its equivalent todayTranslate P1 in the future into its equivalent today
(discounting)(discounting)..
?
Today Future
Today
?
Future
10. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 1 year?much would you have in the account after 1 year?
11. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 1 year?much would you have in the account after 1 year?
0 1
PV =PV = FV =FV =
12. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 1 year?much would you have in the account after 1 year?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1N = 1 PV = -100PV = -100
FV =FV = P106P106
00 11
PV = -100PV = -100 FV =FV =
13. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 1 year?much would you have in the account after 1 year?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1N = 1 PV = -100PV = -100
FV =FV = P106P106
00 11
PV = -100PV = -100 FV =FV = 106106
14. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 1 year?much would you have in the account after 1 year?
Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .06,1.06,1 ) (use FVIF table, or)) (use FVIF table, or)
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)11
== P106P106
00 11
PV = -100PV = -100 FV =FV = 106106
15. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 5 years?much would you have in the account after 5 years?
16. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 5 years?much would you have in the account after 5 years?
00 55
PV =PV = FV =FV =
17. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 5 years?much would you have in the account after 5 years?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5N = 5 PV = -100PV = -100
FV =FV = P133.82P133.82
00 55
PV = -100PV = -100 FV =FV =
18. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 5 years?much would you have in the account after 5 years?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5N = 5 PV = -100PV = -100
FV =FV = P133.82P133.82
00 55
PV = -100PV = -100 FV =FV = 133.133.8282
19. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6%, howIf you deposit P100 in an account earning 6%, how
much would you have in the account after 5 years?much would you have in the account after 5 years?
Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .06,5.06,5 ) (use FVIF table, or)) (use FVIF table, or)
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)55
== PP133.82133.82
00 55
PV = -100PV = -100 FV =FV = 133.133.8282
20. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
quarterly compoundingquarterly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
21. 0 ?
PV =PV = FV =FV =
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
quarterly compoundingquarterly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
22. Calculator Solution:Calculator Solution:
P/Y = 4P/Y = 4 I = 6I = 6
N = 20N = 20 PV =PV = -100-100
FV =FV = P134.68P134.68
00 2020
PV = -100PV = -100 FV =FV =
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
quarterly compoundingquarterly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
23. Calculator Solution:Calculator Solution:
P/Y = 4P/Y = 4 I = 6I = 6
N = 20N = 20 PV =PV = -100-100
FV =FV = P134.68P134.68
00 2020
PV = -100PV = -100 FV =FV = 134.134.6868
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
quarterly compoundingquarterly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
24. Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .015,20.015,20 )) (can’t use FVIF table)(can’t use FVIF table)
FV = PV (1 + i/m)FV = PV (1 + i/m) mxnmxn
FV = 100 (1.015)FV = 100 (1.015)2020
== P134.68P134.68
00 2020
PV = -100PV = -100 FV =FV = 134.134.6868
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
quarterly compoundingquarterly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
25. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
monthly compoundingmonthly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
26. Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
monthly compoundingmonthly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
0 ?
PV =PV = FV =FV =
27. Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 I = 6I = 6
N = 60N = 60 PV =PV = -100-100
FV =FV = P134.89P134.89
00 6060
PV = -100PV = -100 FV =FV =
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
monthly compoundingmonthly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
28. Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 I = 6I = 6
N = 60N = 60 PV =PV = -100-100
FV =FV = P134.89P134.89
00 6060
PV = -100PV = -100 FV =FV = 134.134.8989
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
monthly compoundingmonthly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
29. Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .005,60.005,60 )) (can’t use FVIF table)(can’t use FVIF table)
FV = PV (1 + i/m)FV = PV (1 + i/m) mxnmxn
FV = 100 (1.005)FV = 100 (1.005)6060
== P134.89P134.89
00 6060
PV = -100PV = -100 FV =FV = 134.134.8989
Future Value - single sumsFuture Value - single sums
If you deposit P100 in an account earning 6% withIf you deposit P100 in an account earning 6% with
monthly compoundingmonthly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
30. Future Value - continuous compoundingFuture Value - continuous compounding
What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with
continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
31. Future Value - continuous compoundingFuture Value - continuous compounding
What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with
continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
0 ?
PV =PV = FV =FV =
32. Mathematical Solution:Mathematical Solution:
FV = PV (eFV = PV (e inin
))
FV = 1000 (eFV = 1000 (e .08x100.08x100
) = 1000 (e) = 1000 (e 88
))
FV =FV = P2,980,957.P2,980,957.9999
00 100100
PV = -1000PV = -1000 FV =FV =
Future Value - continuous compoundingFuture Value - continuous compounding
What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with
continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
33. 00 100100
PV = -1000PV = -1000 FV =FV = P2.98mP2.98m
Future Value - continuous compoundingFuture Value - continuous compounding
What is the FV of P1,000 earning 8% withWhat is the FV of P1,000 earning 8% with
continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
Mathematical Solution:Mathematical Solution:
FV = PV (eFV = PV (e inin
))
FV = 1000 (eFV = 1000 (e .08x100.08x100
) = 1000 (e) = 1000 (e 88
))
FV =FV = P2,980,957.P2,980,957.9999
35. Present Value - single sumsPresent Value - single sums
If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the
PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
36. 0 ?
PV =PV = FV =FV =
Present Value - single sumsPresent Value - single sums
If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the
PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
37. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1N = 1 FV =FV = 100100
PV =PV = -94.34-94.34
00 11
PV =PV = FV = 100FV = 100
Present Value - single sumsPresent Value - single sums
If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the
PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
38. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1N = 1 FV =FV = 100100
PV =PV = -94.34-94.34
PV =PV = -94.-94.3434
FV = 100FV = 100
00 11
Present Value - single sumsPresent Value - single sums
If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the
PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
39. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 100 (PVIF .06,1.06,1 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)11
== P94.34P94.34
PV =PV = -94.-94.3434
FV = 100FV = 100
00 11
Present Value - single sumsPresent Value - single sums
If you receive P100 one year from now, what is theIf you receive P100 one year from now, what is the
PV of that P100 if your opportunity cost is 6%?PV of that P100 if your opportunity cost is 6%?
40. Present Value - single sumsPresent Value - single sums
If you receive P100 five years from now, what isIf you receive P100 five years from now, what is
the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
41. 0 ?
PV =PV = FV =FV =
Present Value - single sumsPresent Value - single sums
If you receive P100 five years from now, what isIf you receive P100 five years from now, what is
the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
42. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5N = 5 FV =FV = 100100
PV =PV = -74.73-74.73
00 55
PV =PV = FV = 100FV = 100
Present Value - single sumsPresent Value - single sums
If you receive P100 five years from now, what isIf you receive P100 five years from now, what is
the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
43. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5N = 5 FV =FV = 100100
PV =PV = -74.73-74.73
Present Value - single sumsPresent Value - single sums
If you receive P100 five years from now, what isIf you receive P100 five years from now, what is
the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
00 55
PV =PV = -74.-74.7373
FV = 100FV = 100
44. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 100 (PVIF .06,5.06,5 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)55
== P74.73P74.73
Present Value - single sumsPresent Value - single sums
If you receive P100 five years from now, what isIf you receive P100 five years from now, what is
the PV of that P100 if your opportunity cost is 6%?the PV of that P100 if your opportunity cost is 6%?
00 55
PV =PV = -74.-74.7373
FV = 100FV = 100
45. Present Value - single sumsPresent Value - single sums
What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
46. 00 1515
PV =PV = FV =FV =
Present Value - single sumsPresent Value - single sums
What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
47. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 7I = 7
N = 15N = 15 FV =FV = 1,0001,000
PV =PV = -362.45-362.45
Present Value - single sumsPresent Value - single sums
What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
00 1515
PV =PV = FV = 1000FV = 1000
48. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 7I = 7
N = 15N = 15 FV =FV = 1,0001,000
PV =PV = -362.45-362.45
Present Value - single sumsPresent Value - single sums
What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
00 1515
PV =PV = -362.-362.4545
FV = 1000FV = 1000
49. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 100 (PVIF .07,15.07,15 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.07)PV = 100 / (1.07)1515
== P362.45P362.45
Present Value - single sumsPresent Value - single sums
What is the PV of P1,000 to be received 15 yearsWhat is the PV of P1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
00 1515
PV =PV = -362.-362.4545
FV = 1000FV = 1000
50. Present Value - single sumsPresent Value - single sums
If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5
years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of
return?return?
51. 00 55
PV =PV = FV =FV =
Present Value - single sumsPresent Value - single sums
If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5
years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of
return?return?
52. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 N = 5N = 5
PV = -5,000PV = -5,000 FV = 11,933FV = 11,933
I =I = 19%19%
00 55
PV = -5000PV = -5000 FV = 11,933FV = 11,933
Present Value - single sumsPresent Value - single sums
If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5
years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of
return?return?
53. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
5,000 = 11,933 (PVIF5,000 = 11,933 (PVIF ?,5?,5 ))
PV = FV / (1 + i)PV = FV / (1 + i)nn
5,000 = 11,933 / (1+ i)5,000 = 11,933 / (1+ i)55
.419 = ((1/ (1+i).419 = ((1/ (1+i)55
))
2.3866 = (1+i)2.3866 = (1+i)55
(2.3866)(2.3866)1/51/5
= (1+i)= (1+i) i =i = .19.19
Present Value - single sumsPresent Value - single sums
If you sold land for P11,933 that you bought 5If you sold land for P11,933 that you bought 5
years ago for P5,000, what is your annual rate ofyears ago for P5,000, what is your annual rate of
return?return?
54. Present Value - single sumsPresent Value - single sums
Suppose you placed P100 in an account that paysSuppose you placed P100 in an account that pays
9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?will it take for your account to grow to $500?
00
PV =PV = FV =FV =
55. Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 FV = 500FV = 500
I = 9.6I = 9.6 PV = -100PV = -100
N =N = 202 months202 months
Present Value - single sumsPresent Value - single sums
Suppose you placed P100 in an account that paysSuppose you placed P100 in an account that pays
9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long
will it take for your account to grow to P500?will it take for your account to grow to P500?
00 ??
PV = -100PV = -100 FV = 500FV = 500
56. Present Value - single sumsPresent Value - single sums
Suppose you placed P100 in an account that paysSuppose you placed P100 in an account that pays
9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long
will it take for your account to grow to P500?will it take for your account to grow to P500?
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
100 = 500 / (1+ .008)100 = 500 / (1+ .008)NN
5 = (1.008)5 = (1.008)NN
ln 5 = ln (1.008)ln 5 = ln (1.008)NN
ln 5 = N ln (1.008)ln 5 = N ln (1.008)
1.60944 = .007968 N1.60944 = .007968 N N = 202 monthsN = 202 months
57. Hint for single sum problems:Hint for single sum problems:
In every single sum present value andIn every single sum present value and
future value problem, there are fourfuture value problem, there are four
variables:variables:
FVFV,, PVPV,, ii andand nn..
When doing problems, you will be givenWhen doing problems, you will be given
three variables and you will solve for thethree variables and you will solve for the
fourth variable.fourth variable.
Keeping this in mind makes solving timeKeeping this in mind makes solving time
value problems much easier!value problems much easier!
58. The Time Value of MoneyThe Time Value of Money
Compounding and DiscountingCompounding and Discounting
Cash Flow StreamsCash Flow Streams
0 1 2 3 4
59. AnnuitiesAnnuities
Annuity:Annuity: a sequence ofa sequence of equalequal cashcash
flowsflows, occurring at the, occurring at the endend of eachof each
period.period.
60. Annuity:Annuity: a sequence ofa sequence of equalequal cashcash
flows, occurring at the end of eachflows, occurring at the end of each
period.period.
0 1 2 3 4
AnnuitiesAnnuities
61. Examples of Annuities:Examples of Annuities:
If you buy a bond, you willIf you buy a bond, you will
receive equal semi-annual couponreceive equal semi-annual coupon
interest payments over the life ofinterest payments over the life of
the bond.the bond.
If you borrow money to buy aIf you borrow money to buy a
house or a car, you will pay ahouse or a car, you will pay a
stream of equal payments.stream of equal payments.
62. If you buy a bond, you willIf you buy a bond, you will
receive equal semi-annual couponreceive equal semi-annual coupon
interest payments over the life ofinterest payments over the life of
the bond.the bond.
If you borrow money to buy aIf you borrow money to buy a
house or a car, you will pay ahouse or a car, you will pay a
stream of equal payments.stream of equal payments.
Examples of Annuities:Examples of Annuities:
63. Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
64. 0 1 2 3
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
65. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
FV =FV = P3,246.40P3,246.40
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
0 1 2 3
10001000 10001000 10001000
66. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
FV =FV = P3,246.40P3,246.40
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
0 1 2 3
10001000 10001000 10001000
67. Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
68. Mathematical Solution:Mathematical Solution:
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
69. Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
70. Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or)
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
71. Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
72. Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33
- 1 =- 1 = P3246.40P3246.40
.08.08
Future Value - annuityFuture Value - annuity
If you invest P1,000 each year at 8%, how muchIf you invest P1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
73. Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
74. 0 1 2 3
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
75. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
PV =PV = P2,577.10P2,577.10
0 1 2 3
10001000 10001000 10001000
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
76. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
PV =PV = P2,577.10P2,577.10
0 1 2 3
10001000 10001000 10001000
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
77. Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
78. Mathematical Solution:Mathematical Solution:
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
79. Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
80. Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08, 3.08,3 ) (use PVIFA table, or)) (use PVIFA table, or)
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
81. Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08, 3.08,3 ) (use PVIFA table, or)) (use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
Present Value - annuityPresent Value - annuity
What is the PV of $1,000 at the end of each of theWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
82. Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08, 3.08,3 ) (use PVIFA table, or)) (use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33
== P2,577.10P2,577.10
.08.08
Present Value - annuityPresent Value - annuity
What is the PV of P1,000 at the end of each of theWhat is the PV of P1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
83. Other Cash Flow PatternsOther Cash Flow Patterns
0 1 2 3
The Time Value of Money
84. PerpetuitiesPerpetuities
Suppose you will receive a fixedSuppose you will receive a fixed
payment every period (month, year,payment every period (month, year,
etc.) forever. This is an example ofetc.) forever. This is an example of
a perpetuity.a perpetuity.
You can think of a perpetuity as anYou can think of a perpetuity as an
annuityannuity that goes onthat goes on foreverforever..
85. Present Value of aPresent Value of a
PerpetuityPerpetuity
When we find the PV of anWhen we find the PV of an annuityannuity,,
we think of the followingwe think of the following
relationship:relationship:
86. Present Value of aPresent Value of a
PerpetuityPerpetuity
When we find the PV of anWhen we find the PV of an annuityannuity,,
we think of the followingwe think of the following
relationship:relationship:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
90. Mathematically,Mathematically,
(PVIFA i, n ) =(PVIFA i, n ) =
We said that a perpetuity is anWe said that a perpetuity is an
annuity where n = infinity. Whatannuity where n = infinity. What
happens to this formula whenhappens to this formula when nn
gets very, very large?gets very, very large?
1 -1 -
11
(1 + i)(1 + i)nn
ii
91. When n gets very large,When n gets very large,
92. When n gets very large,When n gets very large,
1 -
1
(1 + i)n
i
93. When n gets very large,When n gets very large,
this becomes zero.this becomes zero.
1 -
1
(1 + i)n
i
94. When n gets very large,When n gets very large,
this becomes zero.this becomes zero.
So we’re left with PVIFA =So we’re left with PVIFA =
1
i
1 -
1
(1 + i)n
i
95. So, the PV of a perpetuity is verySo, the PV of a perpetuity is very
simple to find:simple to find:
Present Value of a Perpetuity
96. PMT
i
PV =
So, the PV of a perpetuity is verySo, the PV of a perpetuity is very
simple to find:simple to find:
Present Value of a Perpetuity
97. What should you be willing to pay inWhat should you be willing to pay in
order to receiveorder to receive P10,000P10,000 annuallyannually
forever, if you requireforever, if you require 8%8% per yearper year
on the investment?on the investment?
98. What should you be willing to pay inWhat should you be willing to pay in
order to receiveorder to receive P10,000P10,000 annuallyannually
forever, if you requireforever, if you require 8%8% per yearper year
on the investment?on the investment?
PMT P10,000PMT P10,000
i .08i .08
PV = =PV = =
99. What should you be willing to pay inWhat should you be willing to pay in
order to receiveorder to receive P10,000P10,000 annuallyannually
forever, if you requireforever, if you require 8%8% per yearper year
on the investment?on the investment?
PMT P10,000PMT P10,000
i .08i .08
= P125,000= P125,000
PV = =PV = =
101. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
102. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7
103. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7
PVPV
inin
ENDEND
ModeMode
104. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7
PVPV
inin
ENDEND
ModeMode
FVFV
inin
ENDEND
ModeMode
105. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8
106. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8
PVPV
inin
BEGINBEGIN
ModeMode
107. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8
PVPV
inin
BEGINBEGIN
ModeMode
FVFV
inin
BEGINBEGIN
ModeMode
108. Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
109. Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
0 1 2 3
10001000 10001000 10001000
110. Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
Using an interest rate of 8%, weUsing an interest rate of 8%, we
find that:find that:
0 1 2 3
10001000 10001000 10001000
111. Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
Using an interest rate of 8%, weUsing an interest rate of 8%, we
find that:find that:
TheThe Future ValueFuture Value (at 3) is(at 3) is
P3,246.40P3,246.40..
0 1 2 3
10001000 10001000 10001000
112. Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
Using an interest rate of 8%, weUsing an interest rate of 8%, we
find that:find that:
TheThe Future ValueFuture Value (at 3) is(at 3) is
P3,246.40P3,246.40..
TheThe Present ValuePresent Value (at 0) is(at 0) is
P2,577.10P2,577.10..
0 1 2 3
10001000 10001000 10001000
113. What about this annuity?What about this annuity?
Same 3-year time line,Same 3-year time line,
Same 3 P1000 cash flows, butSame 3 P1000 cash flows, but
The cash flows occur at theThe cash flows occur at the
beginningbeginning of each year, ratherof each year, rather
than at thethan at the endend of each year.of each year.
This is anThis is an “annuity due.”“annuity due.”
0 1 2 3
10001000 10001000 10001000
114. 0 1 2 3
Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
115. Calculator Solution:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = -1,000PMT = -1,000
FV =FV = P3,506.11P3,506.11
0 1 2 3
-1000-1000 -1000-1000 -1000-1000
Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
116. 0 1 2 3
-1000-1000 -1000-1000 -1000-1000
Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Calculator Solution:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = -1,000PMT = -1,000
FV =FV = P3,506.11P3,506.11
117. Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
118. Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
119. Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or)
120. Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
(1 + i)(1 + i)
121. Future Value - annuity dueFuture Value - annuity due
If you invest P1,000 at the beginning of each of theIf you invest P1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33
- 1 =- 1 = P3,506.11P3,506.11
(1 + i)(1 + i)
(1.08)(1.08)
122. Present Value - annuity duePresent Value - annuity due
What is the PV of P1,000 at the beginning of eachWhat is the PV of P1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?
0 1 2 3
123. Calculator Solution:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = 1,000PMT = 1,000
PV =PV = P2,783.26P2,783.26
0 1 2 3
10001000 10001000 10001000
Present Value - annuity duePresent Value - annuity due
What is the PV of P1,000 at the beginning of eachWhat is the PV of P1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?
124. Calculator Solution:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = 1,000PMT = 1,000
PV =PV = P2,783.26P2,783.26
0 1 2 3
10001000 10001000 10001000
Present Value - annuity duePresent Value - annuity due
What is the PV of P1,000 at the beginning of eachWhat is the PV of P1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?
125. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution:
126. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
127. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
128. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or)
129. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
(1 + i)(1 + i)
130. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33
== P2,783.26P2,783.26
.08.08
(1 + i)(1 + i)
(1.08)(1.08)
131. Is this anIs this an annuityannuity??
How do we find the PV of a cash flowHow do we find the PV of a cash flow
stream when all of the cash flows arestream when all of the cash flows are
different? (Use a 10% discount rate.)different? (Use a 10% discount rate.)
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
132. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
133. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
134. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
135. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
136. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
138. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Which is the better loan:Which is the better loan:
8%8% compoundedcompounded annuallyannually, or, or
7.85%7.85% compoundedcompounded quarterlyquarterly??
We can’t compare these nominal (quoted)We can’t compare these nominal (quoted)
interest rates, because they don’t include theinterest rates, because they don’t include the
same number of compounding periods persame number of compounding periods per
year!year!
We need to calculate the APY.We need to calculate the APY.
141. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Find the APY for the quarterly loan:Find the APY for the quarterly loan:
APY =APY = (( 1 +1 + )) mm
- 1- 1quoted ratequoted rate
mm
142. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Find the APY for the quarterly loan:Find the APY for the quarterly loan:
APY =APY = (( 1 +1 + )) mm
- 1- 1quoted ratequoted rate
mm
APY =APY = (( 1 +1 + )) 44
- 1- 1
.0785.0785
44
143. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Find the APY for the quarterly loan:Find the APY for the quarterly loan:
APY =APY = (( 1 +1 + )) mm
- 1- 1quoted ratequoted rate
mm
APY =APY = (( 1 +1 + )) 44
- 1- 1
APY = .0808, or 8.08%APY = .0808, or 8.08%
.0785.0785
44
144. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Find the APY for the quarterly loan:Find the APY for the quarterly loan:
The quarterly loan is more expensive thanThe quarterly loan is more expensive than
the 8% loan with annual compounding!the 8% loan with annual compounding!
APY =APY = (( 1 +1 + )) mm
- 1- 1quoted ratequoted rate
mm
APY =APY = (( 1 +1 + )) 44
- 1- 1
APY = .0808, or 8.08%APY = .0808, or 8.08%
.0785.0785
44
146. ExampleExample
Cash flows from an investment areCash flows from an investment are
expected to beexpected to be P40,000P40,000 per year at theper year at the
end of years 4, 5, 6, 7, and 8. If youend of years 4, 5, 6, 7, and 8. If you
require arequire a 20%20% rate of return, what israte of return, what is
the PV of these cash flows?the PV of these cash flows?
147. ExampleExample
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
Cash flows from an investment areCash flows from an investment are
expected to beexpected to be P40,000P40,000 per year at theper year at the
end of years 4, 5, 6, 7, and 8. If youend of years 4, 5, 6, 7, and 8. If you
require arequire a 20%20% rate of return, what israte of return, what is
the PV of these cash flows?the PV of these cash flows?
148. This type of cash flow sequence isThis type of cash flow sequence is
often called aoften called a ““deferred annuitydeferred annuity.”.”
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
149. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
150. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
151. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
152. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
153. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
154. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
155. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
Or,Or,
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
156. 2)2) Find the PV of the annuity:Find the PV of the annuity:
PVPV:: End mode; P/YR = 1; I = 20;End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5PMT = 40,000; N = 5
PV =PV = P119,624P119,624
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
157. 2)2) Find the PV of the annuity:Find the PV of the annuity:
PVPV3:3: End mode; P/YR = 1; I = 20;End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5PMT = 40,000; N = 5
PVPV33== P119,624P119,624
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
159. Then discount this single sum back toThen discount this single sum back to
time 0.time 0.
PV: End mode; P/YR = 1; I = 20;PV: End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624;N = 3; FV = 119,624;
Solve: PV =Solve: PV = P69,226P69,226
119,624119,624
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
161. The PV of the cash flowThe PV of the cash flow
stream isstream is P69,226P69,226..
69,22669,226
00 11 22 33 44 55 66 77 88
P0P0 00 00 00 4040 4040 4040 4040 4040
119,624119,624
162. Retirement ExampleRetirement Example
After graduation, you plan to investAfter graduation, you plan to invest
P400P400 per monthper month in the stock market.in the stock market.
If you earnIf you earn 12%12% per yearper year on youron your
stocks, how much will you havestocks, how much will you have
accumulated when you retire inaccumulated when you retire in 3030
yearsyears??
163. Retirement ExampleRetirement Example
After graduation, you plan to investAfter graduation, you plan to invest
P400P400 per month in the stock market.per month in the stock market.
If you earnIf you earn 12%12% per year on yourper year on your
stocks, how much will you havestocks, how much will you have
accumulated when you retire in 30accumulated when you retire in 30
years?years?
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
166. Retirement ExampleRetirement Example
If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
167. Retirement ExampleRetirement Example
If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
Mathematical Solution:Mathematical Solution:
168. Retirement ExampleRetirement Example
If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
169. Retirement ExampleRetirement Example
If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table)
170. Retirement ExampleRetirement Example
If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
171. Retirement ExampleRetirement Example
If you invest P400 at the end of each month for theIf you invest P400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
FV = 400 (1.01)FV = 400 (1.01)360360
- 1 =- 1 = P1,397,985.65P1,397,985.65
172. If you borrowIf you borrow P100,000P100,000 atat 7%7% fixedfixed
interest forinterest for 3030 yearsyears in order toin order to
buy a house, what will be yourbuy a house, what will be your
monthly house paymentmonthly house payment??
House Payment ExampleHouse Payment Example
173. House Payment ExampleHouse Payment Example
If you borrowIf you borrow P100,000P100,000 atat 7%7% fixedfixed
interest forinterest for 3030 years in order toyears in order to
buy a house, what will be yourbuy a house, what will be your
monthly house payment?monthly house payment?
181. Team AssignmentTeam Assignment
Upon retirement, your goal is to spendUpon retirement, your goal is to spend 55
years traveling around the world. Toyears traveling around the world. To
travel in style will requiretravel in style will require P250,000P250,000 perper
year at theyear at the beginningbeginning of each year.of each year.
If you plan to retire inIf you plan to retire in 3030 yearsyears, what are, what are
the equalthe equal monthlymonthly payments necessarypayments necessary
to achieve this goal? The funds in yourto achieve this goal? The funds in your
retirement account will compound atretirement account will compound at
10%10% annually.annually.
182. How much do we need to have byHow much do we need to have by
the end of year 30 to finance thethe end of year 30 to finance the
trip?trip?
PVPV3030 = PMT (PVIFA= PMT (PVIFA .10, 5.10, 5) (1.10) =) (1.10) =
= 250,000 (3.7908) (1.10) == 250,000 (3.7908) (1.10) =
== P1,042,470P1,042,470
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250250 250 250 250 250
184. Now, assuming 10% annualNow, assuming 10% annual
compounding, what monthlycompounding, what monthly
payments will be required for youpayments will be required for you
to haveto have P1,042,466P1,042,466 at the end ofat the end of
year 30?year 30?
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250250 250 250 250 250
1,042,4661,042,466
186. So, you would have to placeSo, you would have to place P461.17P461.17 inin
your retirement account, which earnsyour retirement account, which earns
10% annually, at the end of each of the10% annually, at the end of each of the
next 360 months to finance the 5-yearnext 360 months to finance the 5-year
world tour.world tour.