Discussion 2
Ch. 5 – Time Value of Money
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Understand the time value of money
Value series of cash flows
Understand compounding
Distinguish between nominal and effective interest rates
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Objectives
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Suppose one year CD pay 3% interest. Then, how much would you receive one year from now if you invest $1000 in this type of CD today?
The time value of money
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3
$1 today is more than $1 one year from now!
Money that you have today can be invested and will start earning interest immediately
The time value of money
4
Assuming that the rate of return (interest) on government bonds is 6%, how much would you pay today for a government bond that pays $1000 one year from now?
The time value of money
5
The present value of cash flow received some time in the future is equal to that cash flow multiplied by the discount factor, also called the present value factor
PV(C1)=C1×Discount factor
The discount factor is usually expressed as a rate of return
Discount factor =
Present Value
6
What is the present value of the bond we discussed before?
What is the one-year discount factor for this bond?
Present Value
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7
If $1 today is worth more than $1 one year from today, then it makes sense that $1 one year from today is worth more than $1 two years from today.
How much would you pay for a bond that pays $1,000 two years from now, assuming that the rate of return each year is 5%?
Valuing long-lived assets
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The present value of a cash flow 1 year from today is
The present value of a cash flow 2 years from today is
Valuing long-lived assets
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And, in general, the present value of a cash flow t years from today is
And the discount factor is
Valuing long-lived assets
10
Assuming that the rate of return is 7%, how would you value the following stream of cash flows:
Valuing long-lived assets
$100 one year from now
$200 two years from now
$300 three years from now
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A stream of cash flows over many periods can be valued as
Valuing cash flow over many periods
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This formula is called the discounted cash flow (DCF) formula
A shorthand way to write this formula is
The discounted cash flow formula
13
Suppose you put $1,000 into a savings account today that will pay 11% interest for 5 years. How much will you have at the end of five years?
Future Value
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The future value of a cash flow received some time in the future is equal to that cash flow divided by the discount factor:
Where
C0 is cash flow at date 0,
r is the appropriate interest rate, and
T is the number of periods over which the cash is invested.
Future Value
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Future Value
Suppose that you invest $500 in a savings account that earns annual interest rate of 3%. What will your account be worth in five years?
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FV = 500*(1.03)^5 = 579.64
Emphasis – 4 variables in equation…if we know 3, can always solve for the 4th
Compounding
Compounding an investment m times a year for T years provides for future value of wealth:
If you invest $50 for 3 years at 12% compounded semi-annually, what will your investment grow to?
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Continuous Compounding
General formula:
Where
C0 is cash flow at date 0
r is the stated annual interest rate
T is the number of periods over which the cash is invested
e is a transcendental number approximately equal to 2.718.
{ex is a key on your calculator}
Example: You invest $1,000 at a continuously compounded rate of 10% for 2 years. How much will your investment be worth?
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1000*e^(.1*2) = 1221.40
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e is a transcendental number because it transcends the real numbers.
- not a solution to a polynomial
- in the limit… we go to e
1000*e^(.1*2) = 1221.40
Assuming that the discount rate is 10% per year, what is the present value of $10 paid once a year forever, starting one year from now?
Perpetuity
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A constant stream of cash flows that lasts forever.
The formula for the present value of a perpetuity is:
Perpetuity
20
You just won a lottery that will pay you (and your estate) $100,000 per year forever, starting one year from now. The lottery organizers offered to pay you $1 million in cash today instead of the perpetuity. The discount rate is 8%. Should you accept?
Perpetuity: Example
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21
What if the payments started today instead of one year from now?
Perpetuity
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If the cash flows from a perpetuity begin today (not in the future), its present value is
Perpetuity
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A constant stream of cash flows with a fixed maturity.
An ordinary annuity has the following characteristics:
The payments are always made at the end of each interval;
The interest rate compounds at the same interval as the payment interval.
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Annuity
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The formula for the present value of an ordinary annuity is:
is called annuity factor.
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Annuity
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Assume that the lottery payments would start a year from now at $100,000. They would remain constant each year, but they would stop after the 5th payment. What is the PV of the lottery payments? The discount rate is 8%.
How would we deal with this problem using our knowledge of how to price perpetuities?
Annuity: Example
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An annuity is valued as the difference between two perpetuities:
one perpetuity that starts at time 1
less a perpetuity that starts at time T + 1
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Annuity Intuition
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What would be the PV of our lottery payments if they started today and ended after the 5th payment?
Annuity Due
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Annuity Due: the annuity payments are made at the beginning rather than the end of the period...
Annuity Due
AnnuityDue = AnnuityOrdinary x (1+r)
Annuity Due
Ordinary Annuity
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So far, we assumed that interest is compounded annually. This is not what happens in reality
If we invest $1,000 today at an interest rate of 10% compounded annually, how much will we receive 2 years from today?
What if the interest were compounded semi-annually?
Nominal and effective interest rates
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Effective interest rate is an annually compounded rate equivalent to nominal annual interest rate (also called annual percentage rate , APR) compounded more than once a year
Where m is the number of compound intervals per year
If m becomes very large (infinite), then
Nominal and effective interest rates
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Which investment would you prefer?
An investment paying interest of 12% compounded annually?
An investment paying interest of 11.7% compounded semiannually?
An investment paying interest of 11.5% compounded continuously?
Nominal and effective interest rates
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Time value of money
Present value and future value
Discount factor (present value factor)
Perpetuity
Ordinary annuity and annuity due
Nominal and effective interest rates
Glossary
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