The Time Value of Money
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Transcript of The Time Value of Money
Slide 1
The Time Value of Money Time Value of Money Concept Future and Present Values of single payments Future and Present values of periodic payments
(Annuities) Present value of perpetuity Future and Present values of annuity due Annual Percentage Yield (APY)
Slide 2
The Time Value of Money Concept We know that receiving $1 today is worth more
than $1 in the future. This is due to opportunity costs
The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner
Slide 3
The Future Value Future Value equation:
nn iPVFV
nformulatio general Value Future
iiiPVFV
periods three for Value Future
iiPVFV
periods two for Value Future
iPVFV
period one for Value Future
1
111
11
1
0
03
02
101
Slide 4
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:
FVn = $1 x (1 + i)n
FVn = $100 x (1 + 0.06)1
FVn = $106$106
Slide 5
Future Value – Single Sums (Continued)
N I/Y P/Y PV PMT FV MODE
1 6 1 -100 0 106
Calculator Solution (TI BA II PLUS)
N Number of periods
I/Y Interest per Year
P/Y Payment per Year
C/Y Compounding per Year
PV Present Value
PMT PayMenT (Periodic and Fixed)
FV Future Value
MODE END for ending and BGN for beginning
Slide 6
Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6%,
how much would you have in the account after 5 years?
Mathematical Solution:
FVn = $1 x (1 + i)n
FVn = $100 x (1 + 0.06)5
FVn = $133.82$133.82
N I/Y P/Y PV PMT FV MODE
5 6 1 -100 0 133.82
Slide 7
Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have in the account after 5 year?
Mathematical Solution:
FVn = $1 x (1 + i)n
FVn = $100 x (1 + 0.06/4)5x4
FVn = $134.69$134.69
N I/Y P/Y PV PMT FV MODE
20 6 4 -100 0 134.69
Slide 8
Future Value – Single Sums (Continued) If you deposit $100 in an account earning 6% with
monthly compounding, how much would you have in the account after 5 year?
Mathematical Solution:
FVn = $1 x (1 + i)n
FVn = $100 x (1 + 0.06/12)5x12
FVn = $134.89$134.89
N I/Y P/Y PV PMT FV MODE
60 6 12 -100 0 134.89
Slide 9
Future Value – Single Sums (Continued) If you deposit $1,000 in an account earning 8%
with daily compounding, how much would you have in the account after 100 year?
Mathematical Solution:
FVn = $1 x (1 + i)n
FVn = $1,000 x (1 + 0.08/365)100x365
FVn = $2,978,346.07$2,978,346.07
N I/Y P/Y PV PMT FV MODE
36,500 8 365 -1000 0 2,978,346.07
Slide 10
The Present Value Present Value equation:
together. them add
canyou future the in made payments
of value present the finding after
is that additive are Values Present
i
FVPV
:(payment) flow cash futureany of (today) Value Present
nn
10
Slide 11
Present Value – Single Sums (Continued) If you receive $100 one year from now, what is
the PV of that $100 if your opportunity cost is 6%?
Mathematical Solution:
PV0 = $1 / (1 + i)n
PV0 = $100 / (1 + 0.06)1
PV0 = -$94.34-$94.34
N I/Y P/Y PV PMT FV MODE
1 6 1 -94.37 0 100
Slide 12
Present Value – Single Sums (Continued) If you receive $100 five year from now, what is
the PV of that $100 if your opportunity cost is 6%?
Mathematical Solution:
PV0 = $1 / (1 + i)n
PV0 = $100 / (1 + 0.06)5
PV0 = -$74.73-$74.73
N I/Y P/Y PV PMT FV MODE
5 6 1 -74.73 0 100
Slide 13
Present Value – Single Sums (Continued) If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate of return?
Mathematical Solution:
N I/Y P/Y PV PMT FV MODE
5 19 1 -5,000 0 11,933
1
1
n
PV
FVi %191
000,5
933,11 5
1
i
Slide 14
Present Value – Single Sums (Continued) Suppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long will it take for your account to grow to $500?
Mathematical Solution:
N I/Y P/Y PV PMT FV MODE
202 9.6 12 -100 0 500
frequency gcompoundin:m
1ln
ln
mi
PVFV
n months. 202
12096.0
1ln
100500
ln
n
Slide 15
Hint for Single Sum Problems In every single sum future value and present value
problem, there are 4 variables: FV, PV, i, and n When doing problems, you will be given 3 of
these variables and asked to solve for the 4th variable
Keeping this in mind makes “time value” problems much easier!
Slide 16
Compounding and Discounting Cash Flow Streams Annuity: a sequence of equal cash flows,
occurring at the end of each period If you buy a bond, you will receive equal semi-
annual coupon interest payments over the life of the bond
If you borrow money to buy a house or a car, you will pay a stream of equal payments
Slide 17
Future Value – Annuity If you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:
N I/Y P/Y PV PMT FV MODE
3 8 1 0 -1000 3,246.40
i
iPMTFV
n
n
11
40.246,3$
08.0
108.01000,1$
3
3
3
FV
FV
Slide 18
Present Value – Annuity What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Mathematical Solution:
N I/Y P/Y PV PMT FV MODE
3 8 1 2,577.10 -1000 0
ii
PMTPVn1
11
0
10.577,2$
08.008.01
11
000,1$
0
3
0
PV
PV
Slide 19
Perpetuities Suppose you will receive a fixed payment every
period (month, year, etc.) forever. This is an example of a perpetuity
You can think of a perpetuity as an annuity that goes on
Slide 20
Perpetuities (Continued)
toreducesequation PV theTherefore
zero. approaches 1
1 term then thelarge getsn If
1
11
:as PVfor equation theknow We
0
n
n
i)(
ii
PMTPV
i
PMTPV
iPMTPV
00 or 1
Slide 21
Perpetuities (Continued) What should you be willing to pay in order to
receive $10,000 annually forever, if you require 8% per year on the investment?
PV = $10,000 / 0.08 = $125,000
Slide 22
Future Value – Annuity DueAnnuity Due: The cash flows occur at the beginning of each
year, rather than at the end of each year If you invest $1,000 at the beginning of each of
the next 3 years at 8%, how much would you have at the end of year 3?
Mathematical Solution:
N I/Y P/Y PV PMT FV MODE
3 8 1 0 -1000 3,506.11 BEGIN
)1(
11i
i
iPMTFV
n
n
11.506,3$
)08.01(08.0
108.01000,1$
3
3
3
FV
FV
Slide 23
Present Value – Annuity DueAnnuity Due: The cash flows occur at the beginning of each
year, rather than at the end of each year What is the PV of $1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%?
Mathematical Solution:
N I/Y P/Y PV PMT FV MODE
3 8 1 2,783.26 -1000 0 BEGIN
)1(
1
11
0 ii
iPMTPV
n
26.783,2$
)08.1(08.0
08.01
11
000,1$
0
3
0
PV
PV
Slide 24
Uneven Cash FlowsHow do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate)
Period CF PVCF
0 -10,000 -10,000.00
1 2,000 1,818.15
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
Total 4,412.95
Slide 25
CF
CF0 -10000 ENTER
C01 2000 ENTER F01 1.00 ENTER
C02 4000 ENTER F02 1.00 ENTER
C03 6000 ENTER F03 1.00 ENTER
C04 7000 ENTER F04 1.00 ENTER
NPV 10 ENTER CPT 4,412.95
Uneven Cash Flows
Slide 26
Annual Percentage Yield (APY) or Effective Annual Rate (EAR)
Which is the better loan: 8.00% compounded annually, or 7.85% compounded quarterly?
We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year!
We need to calculate the APY
Slide 27
Annual Percentage Yield (APY) or Effective Annual Rate (EAR) (Continued)
Find the APY for the quarterly (m = 4) loan:
The quarterly loan is more expensive than the 8% loan with annual compounding!
frequency gcompoundin theis m
1rate quoted
1
m
mAPY
0808014
078501
4
..
APY
Slide 28
Annual Percentage Yield (APY) or Effective Annual Rate (EAR) (Continued)
2nd ICONV NOM 7.85 ENTER (EFF) C/Y 4 ENTER (EFF) CPT 8.08