How to Calculate Present Values
Principles of Corporate FinanceBrealey and Myers Sixth Edition
Slides by
Matthew Will Chapter 3
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 2
Topics Covered
Valuing Long-Lived Assets PV Calculation Short Cuts Compound Interest Interest Rates and Inflation Example: Present Values and Bonds
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 3
Present Values
Discount Factor = DF = PV of $1
Discount Factors can be used to compute the present value of any cash flow.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 4
Present Values
Discount Factor = DF = PV of $1
Discount Factors can be used to compute the present value of any cash flow.
DFr t
1
1( )
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 5
Present Values
Discount Factors can be used to compute the present value of any cash flow.
DFr t
1
1( )
1
11 1 r
CCDFPV
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 6
Present Values
Replacing “1” with “t” allows the formula to be used for cash flows that exist at any point in time.
t
tt r
CCDFPV
1
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 7
Present Values
Example
You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 8
Present Values
Example
You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
PV 30001 08 2 572 02
( . )$2, .
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 9
Present Values
PVs can be added together to evaluate multiple cash flows.
PV C
r
C
r
1
12
21 1( ) ( )....
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 10
Present Values
Given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 11
Present Values
Given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%.
87.
83.
2
1
)07.1(00.1
2
)20.1(00.1
1
DF
DF
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 12
Present Values
Example
Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.
000,300000,100000,150
2Year 1Year 0Year
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 13
Present Values
Example - continued
Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.
400,18$
900,261000,300873.2
500,93000,100935.1
000,150000,1500.10Value
Present
Flow
Cash
Factor
DiscountPeriod
207.11
07.11
TotalNPV
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 14
Short Cuts
Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tolls allow us to cut through the calculations quickly.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 15
Short Cuts
Perpetuity - Financial concept in which a cash flow is theoretically received forever.
PV
Cr
luepresent va
flow cashReturn
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 16
Short Cuts
Perpetuity - Financial concept in which a cash flow is theoretically received forever.
r
CPV 1
ratediscount
flow cash FlowCash of PV
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 17
Short Cuts
Annuity - An asset that pays a fixed sum each year for a specified number of years.
trrrC
1
11annuity of PV
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 18
Annuity Short Cut
Example
You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 19
Annuity Short Cut
Example - continuedYou agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?
10.774,12$
005.1005.
1
005.
1300Cost Lease 48
Cost
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 20
Compound Interest
i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate
1 6% 6% 1.06 6.000%
2 3 6 1.032 = 1.0609 6.090
4 1.5 6 1.0154 = 1.06136 6.136
12 .5 6 1.00512 = 1.06168 6.168
52 .1154 6 1.00115452 = 1.06180 6.180
365 .0164 6 1.000164365 = 1.06183 6.183
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 21
Compound Interest
02468
1012141618
Number of Years
FV
of
$1
10% Simple
10% Compound
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 22
Inflation
Inflation - Rate at which prices as a whole are increasing.
Nominal Interest Rate - Rate at which money invested grows.
Real Interest Rate - Rate at which the purchasing power of an investment increases.
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 23
Inflation
1 real interest rate = 1+nominal interest rate1+inflation rate
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 24
Inflation
1 real interest rate = 1+nominal interest rate1+inflation rate
approximation formula
Real int. rate nominal int. rate - inflation rate
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 25
InflationExample
If the interest rate on one year govt. bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate?
Savings
Bond
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 26
InflationExample
If the interest rate on one year govt. bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate?
1
1
+
+
real interest rate =
real interest rate = 1.025
real interest rate = .025 or 2.5%
1+.0591+.033 Savings
Bond
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 27
InflationExample
If the interest rate on one year govt. bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate?
1
1
+
+
real interest rate =
real interest rate = 1.025
real interest rate = .025 or 2.5%
Approximation =.059-.033 =.026 or 2.6%
1+.0591+.033 Savings
Bond
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 28
Valuing a Bond
Example
If today is October 2000, what is the value of the following bond? An IBM Bond pays $115 every Sept for 5 years. In Sept 2005 it pays
an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%).
Cash Flows
Sept 0102 03 04 05
115 115 115 115 1115
©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill
3- 29
Valuing a Bond
Example continuedIf today is October 2000, what is the value of the following bond?
An IBM Bond pays $115 every Sept for 5 years. In Sept 2005 it pays an additional $1000 and retires the bond.
The bond is rated AAA (WSJ AAA YTM is 7.5%).
84.161,1$
075.1
115,1
075.1
115
075.1
115
075.1
115
075.1
1155432
PV