公司理财第二章
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12%
p.a. compounded monthly
1,000 (1 + 0.12/12)12 = 1,126.83 EAIR = 12.683%
12%
p.a. compounded daily
1,000 (1 + 0.12/360)360 = 1,127.47 EAIR = 12.747%
FV1 = $9,500×(1.05) = $9,975 < $10,000.
Corporate Finance Steven S. Wang 13
Annual Interest Rates and Compounding Periods
Compounding periods (m)
How
often is interest computed the bank usually quotes
Time line: there are only two points, t = 0, and t = 1. Cash flow: at t = 0, PV0; at t = 1, FV1. Interest rate: r FV1=PV0(1+r); or C1=C0(1+r) PV0=FV1/(1+r); or C0=C1/(1+r)
Steven S. Wang 5
Corporate Finance
Future Value and Compounding
Compounding: How much will $100 invested today at 5% be worth in two years?
(The Time Line) Year 0 1 2
How Long is the Wait?
If we deposit $100,000 today in an account paying 10%, how long do we have to wait for it to grow to $1,000,000? Solve for T: FVT = PV0 x (1 + r)T $1,000,000 = $100,000 x (1.10)T (1.10)T = 10 T = ln(10) / ln(1.10) = 24.16 years
r Effective Annual Interest Rate: EAIR 1 1 m
Corporate Finance Steven S. Wang 15
r is constant over time. Starting with initial investment of C0, proceeds from the first period’s investment are reinvested for second period and so on (i.e. investment at compound interest).
Corporate Finance
Steven S. Wang
14
Effective Annual Interest Rates and Compounding Periods
$1,000
invested at 12% p.a. for one year
1,000 (1.12) = 1,120 EAIR = 12%
Single Future Cash Flow
Time line: there are T points, t = 0, t= 1,…, t = T. Cash flow: at t = 0, C0 ; at t = T, CT. No cash flows received between time T and time 0. Interest rate: rt = r. Assumptions:
1
2
Steven S. Wang
3
4
5
7
Present Value and Discounting
Discounting: How much is $100 that we will receive
in two years worth today (r = 5%)?
Year
0
1
2 $100
$90.70
Cash flow at t =1: FV1 Time
Cash flow at t = 0: PV0
Corporate Finance
Βιβλιοθήκη Baidu
Steven S. Wang
3
Example (One period case)
Suppose you have one ounce of gold and you want to sell it. The Bank of China offers you US$ 1,000 immediately. The Citi Bank offers you US$ 1,080 in one year’s time. In either case you hand over the gold immediately. The interest rate for one-year term now is 3%. Make you investment decision.
$10,000 NPV $9,500 $23.81 1.05
If we had not undertaken the project and instead invested our $9,500 elsewhere at 5%, our FV would be less than the $10,000 and we would be unambiguously worse off in FV terms as well:
T = ln(FVT/PV0) / ln(1+ r)
Corporate Finance Steven S. Wang 9
What Rate Is Enough?
Assume the total cost of a college education will be $200,000 when your child enters college in 18 years. You have $20,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education?
Solve for r : FVT= PV0 x (1 + r)T 200,000 = 20,000 x (1 + r)18 (1 + r)18 = 10 (1 + r) = 10(1/18) r = 0.13646 = 13.646% per year
Corporate Finance Steven S. Wang 10
By comparing future values of the two offers. or Comparing the present values of the two offers
Steven S. Wang 4
Corporate Finance
FV and PV: Multiple-Period,
Steven S. Wang 12
Corporate Finance
NPV: Example
Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?
$100
$110.25
Future value (FV2) = $100 x 1.052 = $110.25 1.052 = 1.1025 is the compound factor (FVIF0.05, 2).
Corporate Finance Steven S. Wang 6
Future Value and Compounding
real opportunity cost inflationary impact uncertainty and risk
Steven S. Wang 2
Corporate Finance
Present Value and Future Value: One Period Case
Lecture 2: Discounted Cash Flow Valuation
The Time Value of Money Present Value and Future Value Net Present Value Effective Interest Rates Simplifications
Present value = $100 / 1.052 = $90.70
The interest rate (5%) is also called the discount rate.
1/1.052 = 0. 9070 is the discount factor (PVIF0.05, 2).
Corporate Finance Steven S. Wang 8
CT Time
PV0=?
Corporate Finance
C1
C2
Steven S. Wang
11
NPV and The NPV Rule
NPV, the net present value is the difference between the present value of cash flows and the present value of the investment cost.
FV and PV: Multiple-Period, Multiple Future Cash Flows
Time line: there are T points, t = 0, t= 1,…, t = T. Cash flow: at t = 1, C1 ; … , at t = T, CT. Receiving cash flows in each period. Assume r is constant over time. What is C0 or PV0?
NPV I 0 PV (Future Cashflows) c1 c2 I0 2 1 r (1 r ) ct I0 t (1 r ) t 1
T
cT (1 r )T
An investment is worth making if it has a positive NPV. If an investment’s NPV is negative, it should be rejected.
Stated (nominal) Annual Interest Rate (SAIR, r)
What Annual
percentage rate Should have the compounding interval followed r = 12% p.a. compounded quarterly
Time Value of Money
Time Value of Money: relationship between present value and future value of money “A dollar today is worth more than a dollar tomorrow.” Three factors:
$1.00 (1.05) $1.00 (1.05) 2 $1.00 (1.05)
FVT= PV0 x (1 + r)T
$1.00 (1.05)5 $1.00 (1.05) 4
3
$1.00 $1.05 $1.1025 $1.1576 $1.2155 $1.2763
0
Corporate Finance
p.a. compounded monthly
1,000 (1 + 0.12/12)12 = 1,126.83 EAIR = 12.683%
12%
p.a. compounded daily
1,000 (1 + 0.12/360)360 = 1,127.47 EAIR = 12.747%
FV1 = $9,500×(1.05) = $9,975 < $10,000.
Corporate Finance Steven S. Wang 13
Annual Interest Rates and Compounding Periods
Compounding periods (m)
How
often is interest computed the bank usually quotes
Time line: there are only two points, t = 0, and t = 1. Cash flow: at t = 0, PV0; at t = 1, FV1. Interest rate: r FV1=PV0(1+r); or C1=C0(1+r) PV0=FV1/(1+r); or C0=C1/(1+r)
Steven S. Wang 5
Corporate Finance
Future Value and Compounding
Compounding: How much will $100 invested today at 5% be worth in two years?
(The Time Line) Year 0 1 2
How Long is the Wait?
If we deposit $100,000 today in an account paying 10%, how long do we have to wait for it to grow to $1,000,000? Solve for T: FVT = PV0 x (1 + r)T $1,000,000 = $100,000 x (1.10)T (1.10)T = 10 T = ln(10) / ln(1.10) = 24.16 years
r Effective Annual Interest Rate: EAIR 1 1 m
Corporate Finance Steven S. Wang 15
r is constant over time. Starting with initial investment of C0, proceeds from the first period’s investment are reinvested for second period and so on (i.e. investment at compound interest).
Corporate Finance
Steven S. Wang
14
Effective Annual Interest Rates and Compounding Periods
$1,000
invested at 12% p.a. for one year
1,000 (1.12) = 1,120 EAIR = 12%
Single Future Cash Flow
Time line: there are T points, t = 0, t= 1,…, t = T. Cash flow: at t = 0, C0 ; at t = T, CT. No cash flows received between time T and time 0. Interest rate: rt = r. Assumptions:
1
2
Steven S. Wang
3
4
5
7
Present Value and Discounting
Discounting: How much is $100 that we will receive
in two years worth today (r = 5%)?
Year
0
1
2 $100
$90.70
Cash flow at t =1: FV1 Time
Cash flow at t = 0: PV0
Corporate Finance
Βιβλιοθήκη Baidu
Steven S. Wang
3
Example (One period case)
Suppose you have one ounce of gold and you want to sell it. The Bank of China offers you US$ 1,000 immediately. The Citi Bank offers you US$ 1,080 in one year’s time. In either case you hand over the gold immediately. The interest rate for one-year term now is 3%. Make you investment decision.
$10,000 NPV $9,500 $23.81 1.05
If we had not undertaken the project and instead invested our $9,500 elsewhere at 5%, our FV would be less than the $10,000 and we would be unambiguously worse off in FV terms as well:
T = ln(FVT/PV0) / ln(1+ r)
Corporate Finance Steven S. Wang 9
What Rate Is Enough?
Assume the total cost of a college education will be $200,000 when your child enters college in 18 years. You have $20,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education?
Solve for r : FVT= PV0 x (1 + r)T 200,000 = 20,000 x (1 + r)18 (1 + r)18 = 10 (1 + r) = 10(1/18) r = 0.13646 = 13.646% per year
Corporate Finance Steven S. Wang 10
By comparing future values of the two offers. or Comparing the present values of the two offers
Steven S. Wang 4
Corporate Finance
FV and PV: Multiple-Period,
Steven S. Wang 12
Corporate Finance
NPV: Example
Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?
$100
$110.25
Future value (FV2) = $100 x 1.052 = $110.25 1.052 = 1.1025 is the compound factor (FVIF0.05, 2).
Corporate Finance Steven S. Wang 6
Future Value and Compounding
real opportunity cost inflationary impact uncertainty and risk
Steven S. Wang 2
Corporate Finance
Present Value and Future Value: One Period Case
Lecture 2: Discounted Cash Flow Valuation
The Time Value of Money Present Value and Future Value Net Present Value Effective Interest Rates Simplifications
Present value = $100 / 1.052 = $90.70
The interest rate (5%) is also called the discount rate.
1/1.052 = 0. 9070 is the discount factor (PVIF0.05, 2).
Corporate Finance Steven S. Wang 8
CT Time
PV0=?
Corporate Finance
C1
C2
Steven S. Wang
11
NPV and The NPV Rule
NPV, the net present value is the difference between the present value of cash flows and the present value of the investment cost.
FV and PV: Multiple-Period, Multiple Future Cash Flows
Time line: there are T points, t = 0, t= 1,…, t = T. Cash flow: at t = 1, C1 ; … , at t = T, CT. Receiving cash flows in each period. Assume r is constant over time. What is C0 or PV0?
NPV I 0 PV (Future Cashflows) c1 c2 I0 2 1 r (1 r ) ct I0 t (1 r ) t 1
T
cT (1 r )T
An investment is worth making if it has a positive NPV. If an investment’s NPV is negative, it should be rejected.
Stated (nominal) Annual Interest Rate (SAIR, r)
What Annual
percentage rate Should have the compounding interval followed r = 12% p.a. compounded quarterly
Time Value of Money
Time Value of Money: relationship between present value and future value of money “A dollar today is worth more than a dollar tomorrow.” Three factors:
$1.00 (1.05) $1.00 (1.05) 2 $1.00 (1.05)
FVT= PV0 x (1 + r)T
$1.00 (1.05)5 $1.00 (1.05) 4
3
$1.00 $1.05 $1.1025 $1.1576 $1.2155 $1.2763
0
Corporate Finance