Simple Interest
Interest is earned on principalInterest is earned on principal $100 invested at 6% per year$100 invested at 6% per year 11stst year year interest is $6.00interest is $6.00 22ndnd year year interest is $6.00interest is $6.00 33rdrd year year interest is $6.00interest is $6.00 Total interest earned:$18Total interest earned:$18
Compound Interest
Interest is earned on previously earned interestInterest is earned on previously earned interest $100 invested at 6% with annual compounding$100 invested at 6% with annual compounding 11stst year year interest is $6.00interest is $6.00 Principal is $106Principal is $106 22ndnd year year interest is $6.36interest is $6.36 Principal is $112.36 Principal is $112.36 33rdrd year year interest is $6.74interest is $6.74 Principal is $119.11Principal is $119.11 Total interest earned: $19.10Total interest earned: $19.10
We know that receiving $1 today is worth We know that receiving $1 today is worth moremore than $1 in the future. This is due than $1 in the future. This is due toto opportunity costsopportunity costs..
The opportunity cost of receiving $1 in The opportunity cost of receiving $1 in the future is thethe future is the interestinterest we could have we could have earned if we had received the $1 sooner.earned if we had received the $1 sooner.
Today Future
If we can measure this opportunity cost, we can:
Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..
Translate $1 in the future into its equivalent todayTranslate $1 in the future into its equivalent today (discounting)(discounting)..
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Today Future
Today
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Future
Future Value - single sums
If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?
0 1
PV =PV = FV = FV =
Future Value - single sums
If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1 N = 1 PV = -100 PV = -100
FV = FV = $106$106
00 1 1
PV = -100PV = -100 FV = FV = 106106
Future Value - single sums
If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?
Mathematical Solution:Mathematical Solution:
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)1 1 = = $106$106
00 1 1
PV = -100PV = -100 FV = FV = 106106
Future Value - single sums
If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 PV = -100 PV = -100
FV = FV = $133.82$133.82
00 5 5
PV = -100PV = -100 FV = FV = 133.133.8282
Future Value - single sums
If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?
Mathematical Solution:Mathematical Solution:
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)5 5 = = $$133.82133.82
00 5 5
PV = -100PV = -100 FV = FV = 133.133.8282
Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding, how much would
you have in the account after 5 years?
Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have
in the account after 5 years?
0 ?PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 4P/Y = 4 I = 6I = 6
N = 20 N = 20 PV = PV = -100-100
FV = FV = $134.68$134.68
00 20 20
PV = -100PV = -100 FV = FV = 134.134.6868
Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have
in the account after 5 years?
Mathematical Solution:Mathematical Solution:
FV = PV (1 + i/m) FV = PV (1 + i/m) m x nm x n
FV = 100 (1.015)FV = 100 (1.015)20 20 = = $134.68$134.68
00 20 20
PV = -100PV = -100 FV = FV = 134.134.6868
Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have
in the account after 5 years?
Future Value - single sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have
in the account after 5 years?
Future Value - single sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have
in the account after 5 years?
0 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 I = 6I = 6
N = 60 N = 60 PV = PV = -100-100
FV = FV = $134.89$134.89
00 60 60
PV = -100PV = -100 FV = FV = 134.134.8989
Future Value - single sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have
in the account after 5 years?
Mathematical Solution:Mathematical Solution:
FV = PV (1 + i/m) FV = PV (1 + i/m) m x nm x n
FV = 100 (1.005)FV = 100 (1.005)60 60 = = $134.89$134.89
00 60 60
PV = -100PV = -100 FV = FV = 134.134.8989
Future Value - single sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have
in the account after 5 years?
Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
0 ?
PV =PV = FV = FV =
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
0 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1 N = 1 FV = FV = 100100
PV = PV = -94.34-94.34
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = PV = -94.-94.3434 FV = 100 FV = 100
00 1 1
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)1 1 = = $94.34$94.34
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = PV = -94.-94.3434 FV = 100 FV = 100
00 1 1
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
0 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 FV = FV = 100100
PV = PV = -74.73-74.73
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
00 5 5
PV = PV = -74.-74.7373 FV = 100 FV = 100
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)5 5 = = $74.73$74.73
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
00 5 5
PV = PV = -74.-74.7373 FV = 100 FV = 100
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 7I = 7
N = 15 N = 15 FV = FV = 1,0001,000
PV = PV = -362.45-362.45
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = -362.-362.4545 FV = 1000 FV = 1000
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 1000/ (1.07)PV = 1000/ (1.07)15 15 = = $362.45$362.45
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = -362.-362.4545 FV = 1000 FV = 1000
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
00 5 5
PV = PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 N = 5N = 5
PV = -5,000 PV = -5,000 FV = 11,933FV = 11,933
I = I = 19%19%
00 5 5
PV = -5000PV = -5000 FV = 11,933 FV = 11,933
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
5,000 = 11,933 / (1+ i)5,000 = 11,933 / (1+ i)5 5
.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 sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Hint for single sum problems:
In every single sum future value and In every single sum future value and present value problem, there are 4 present value problem, there are 4 variables: variables:
FVFV, , PVPV, , ii, and , and nn When doing problems, you will be When doing problems, you will be
given 3 of these variables and asked to given 3 of these variables and asked to solve for the 4th variable.solve for the 4th variable.
Keeping this in mind makes “time Keeping this in mind makes “time value” problems much easier!value” problems much easier!
The Time Value of Money
Compounding and DiscountingCompounding and Discounting
Cash Flow StreamsCash Flow Streams
0 1 2 3 4
Annuities
Annuity:Annuity: a sequence of a sequence of equalequal cash cash flowsflows, occurring at the , occurring at the endend of each of each period.period.
Annuity:Annuity: a sequence of a sequence of equalequal cash cash flows, occurring at the end of each flows, occurring at the end of each period.period.
0 1 2 3 4
Annuities
Examples of Annuities:
If you buy a bond, you will If you buy a bond, you will receive equal semi-annual coupon receive equal semi-annual coupon interest payments over the life of interest payments over the life of the bond.the bond.
If you borrow money to buy a If you borrow money to buy a house or a car, you will pay a house or a car, you will pay a stream of equal payments.stream of equal payments.
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000 PMT = -1,000
FV = FV = $3,246.40$3,246.40
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
10001000 10001000 1000 1000
Mathematical Solution:Mathematical Solution:
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 = $3246.40$3246.40
.08 .08
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
0 1 2 3
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000 PMT = -1,000
PV = PV = $2,577.10$2,577.10
0 1 2 3
10001000 10001000 1000 1000
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Mathematical Solution:Mathematical Solution:
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 = = $2,577.10$2,577.10
.08.08
Present Value - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Perpetuities
Suppose you will receive a fixed Suppose you will receive a fixed payment every period (month, year, payment every period (month, year, etc.) forever. This is an example of etc.) forever. This is an example of a perpetuity.a perpetuity.
You can think of a perpetuity as an You can think of a perpetuity as an annuityannuity that goes on that goes on foreverforever..
PMT i
PV =
The PV of a perpetuity is very The PV of a perpetuity is very simple to find:simple to find:
Present Value of a PerpetuityPresent Value of a Perpetuity
What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?
What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?
PMT $10,000PMT $10,000 i .08 i .08
PV = =PV = =
What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?
PMT $10,000PMT $10,000 i .08 i .08
= $125,000= $125,000
PV = =PV = =
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 5 6 7
PVPVinin
ENDENDModeMode
FVFVinin
ENDENDModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 6 7 8
PVPVinin
BEGINBEGINModeMode
FVFVinin
BEGINBEGINModeMode
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we Using an interest rate of 8%, we find that:find that:
The The Future ValueFuture Value (at 3) is (at 3) is $3,246.40$3,246.40..
The The Present ValuePresent Value (at 0) is (at 0) is $2,577.10$2,577.10..
0 1 2 3
10001000 10001000 1000 1000
Is this an Is this an annuityannuity?? How do we find the PV of a cash flow How do we find the PV of a cash flow
stream when all of the cash flows are stream when all of the cash flows are different? (Use a 10% discount rate).different? (Use a 10% discount rate).
Uneven Cash Flows
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
periodperiod CF CF PV (CF)PV (CF)
00 -10,000 -10,000 -10,000.00-10,000.00
11 2,000 2,000 1,818.181,818.18
22 4,000 4,000 3,305.793,305.79
33 6,000 6,000 4,507.894,507.89
44 7,000 7,000 4,781.094,781.09
PV of Cash Flow Stream: $ 4,412.95PV of Cash Flow Stream: $ 4,412.95
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Example
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
Cash flows from an investment are Cash flows from an investment are expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?
This type of cash flow sequence is This type of cash flow sequence is often called a often called a ““deferred annuitydeferred annuity.”.”
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
Or,Or,
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
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 = 5 PMT = 40,000; N = 5
PVPV33= = $119,624$119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
Then discount this single sum back to Then 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 = $69,226$69,226
119,624119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
The PV of the cash flow The PV of the cash flow stream is stream is $69,226$69,226..
69,22669,226
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
119,624119,624
Retirement Example
After graduation, you plan to invest After graduation, you plan to invest $400$400 per month per month in the stock market. in the stock market. If you earn If you earn 12%12% per year per year on your on your stocks, how much will you have stocks, how much will you have accumulated when you retire in accumulated when you retire in 3030 yearsyears??
Retirement Example
After graduation, you plan to invest After graduation, you plan to invest $400$400 per month in the stock market. per month in the stock market. If you earn If you earn 12%12% per year on your per year on your stocks, how much will you have stocks, how much will you have accumulated when you retire in 30 accumulated when you retire in 30 years?years?
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
Using your calculator,Using your calculator,
P/YR = 12P/YR = 12
N = 360 N = 360
PMT = -400PMT = -400
I%YR = 12I%YR = 12
FV = FV = $1,397,985.65$1,397,985.65
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
If you borrow If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years years in order to in order to buy a house, what will be your buy a house, what will be your
monthly house paymentmonthly house payment??
House Payment Example
House Payment Example
If you borrow If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years in order to years in order to buy a house, what will be your buy a house, what will be your
monthly house payment?monthly house payment?