Chapter 5

Discount

Calculate the present value of some future amount

Future value tables are not as common anymore

Due to: -Calculators -tables only available for minimal rates -tables needed for 3 to 4 decimal places would get quite large

What would our $100 be worth after five years?

FV = (1 + .10)^5 FV = 1.6105 1.6105 x $100 = $161.05

PV is the reciprocal of FV

FV = (1 + r)^t PV = 1 / (1 + r)^t

Future Value of C invested at r Percent for t Periods

FV = C x (1 + r)^t The term (1 + r)^t is called the future value factor

FV =

Future Value, what cash flows are worth in the future

Present Value of more than one period =

$1 / (1 + r)^t

Future Value =

$1 x (1 + r)^t

PV for a Single Period

$1 x [1/(1 + r)] = $1 / (1 + r) *The PV of $1 to be received in one period*

Investing for a Single Period

$100 deposit into a savings account 10% interest rate per year annually ?? how much will you have in one year (1+r) per dollar invested = 1+.10 = 1.1 dollars per dollar invested = 1.1 x $100 = $110 = FV

What is the PV of $1000 in two years if the relevant rate is 7%?

$1000 = PV x 1.07 x 1.07 $1000 = PV x (1.07)^2 $1000 = PV x 1.1449 PV = $1000/1,1449 PV = $873.44

For a given length of time, the higher the discount rate is ...

... The lower the present value

Not taking the time value of money into account when computing growth rates or rates of return often leads to ...

... misleading numbers in the real world

The discount rate is called the ...

... rate of return, or the return on investment

The interest earned in each year is equal to ...

... the beginning amount x the interest rate

Discount Factor AKA PVIF

1/(1+r)^t

Investing for More than One Period

If you leave the entire $110 in the bank, you will earn $110 x .10 = $11 in interest during the second year, so you will have a total of $110 + $11 = $121 or One year from now you are effectively investing $110 at 10% for a year. This is a single-period problem, so you'll end up with $1.10 for every dollar invested, or $110 x 1.1 = $121

Present values and discount rates are inversely related

Increasing the discount rate decreases the PV and vice versa

PV is thus just the reverse of FV

Instead of compounding the money forward into the future, we discount it back to the present

r =

Interest rate, rate of return, or discount rate per period - typically, but not always, one year

t = OR r =

Number of periods - typically, but not always, the number of years

Present Value of C to be Received in t Periods at r Percent per Period

PV = C / (1 + r)^t The term 1 / (1 + r)^t is called the present value factor

The Basic Present Value Equation Giving the Relationship between Present and Future Value

PV = FV / (1 + r)^t

Determining the Discount Rate

PV = FV / (1 + r)^t PV - Present Value FV - Future Value r - Discount rate t - Life of the investment Given any three of these, we can always find the fourth

PV =

Present value, what future cash flows are worth today

Future Value

Refers to the amount of money an investment will grow to over some period of time at some given interest rate -The cash value of an investment at some time in the future

Future values depend critically on the assumed interest rate, particularly for long-lived investments

The effect of compounding is not great over short time periods, but it really starts to add up as the horizon grows

Compounding

The process of accumulating interest on an investment over time to earn more interest

Discount Rate

The rate used to calculate the present value of future cash flows

Basic Present Value Equation

PV x (1 + r)^t = FV PV = FV/(1+r)^t PV = FV x [1/(1 + r)^t]

How much do we have to invest today at 10% to get $1 in a year?

PV x 1.10 = $1 PV = $1 / 1.10 PV = $.909

Future value question: "What will my $2000 investment grow to if it earns a 6.5% return every year for the next six years?"

Present value question: Suppose you need to have $10,000 in 10 years, and you can earn 6.5% on your money: "How much do you have to invest today to reach your goal?"

As the length of time until payment grows ...

Present values decline -PV tends to become small as the time horizon grows -They will always approach zero

Time Value of Money

Refers to the fact that money received today is worth more than money received next year or the year after -You could earn interest while you waited; So a dollar today would grow to more than a dollar later -Depends on the rate you can earn by investing

Simple interest is constant each year, but the amount of compound interest you earn gets bigger every year

The amount of compound interest keeps increasing because more and more interest builds up and there is thus more to compound

Present Value

The current value of future cash flows discounted at the appropriate discount rate

Rule of 72

The number of years it takes for a certain amount to double in value is equal to 72 / r% Ex: 72/r = 8 years, implying that r = 9% *Accurate for discount rates in the 5% to 20% range*

Over the five-year span of this investment, the simple interest is $100 x .10 = $10 per year, so $50 after 5 years

The other $11.05 is from compounding

Suppose you need $1000 in three years and you can earn 15% on your money, how much do you need to invest today?

1 / (1 + .15)^3 = 1 / 1.5209 = .6575 $1000 x .6575 = $657.50 $657.50 is the present (or discounted) value of $1000 to be received in three years at 15%

Ex: $100 at 10% for five years n = 5 i = 10 PV = -$100 FV=?

1. Enter 100 - press PV key 2. Enter 10 - press I/YR key 3. Enter 5 - press N key 4. Press FV key to obtain answer FV = $161.05

1. What is the basic present value equation? 2. What is the Rule of 72?

1. PV = 1/(1 + r) ^ t 2. for reasonable rate of return, the time it takes to double your money is given approximately by 72/r%

1. What do we mean by the future value of an investment? 2. What does it mean to compound interest? How does compound interest differ from simple interest? 3. In general, what is the future value of $1 invest at r per period for t periods?

1. The cash value of an investment at some time in the future 2. Compounding interest means to leave your money and any accumulated interest in an investment for more than one person (reinvesting the interest). Compound interest differs from simple interest because with simple interest, the interest is not reinvested 3. Future value formula = $1 x (1 + r) ^t

1. What do we mean by the present value of an investment? 2. The process of discounting a future amount back to the present is the opposite of doing what? 3. What do we mean by discounted cash flow (DCF) valuation? 4. In general, what is the present value of $1 to be received in t periods, assuming a discount rate of r per period?

1. The reverse of future value —> instead of compounding money forward into the future, we discount it back to the present. Ex: What amount, if invested today, will grow to x amount of money in one year if the interest rate is 10 percent? —> The amount invested today is the present value. 2. Compounding or calculating the future value 3. Discounted cash flow (DCF) - Calculating the present value of a future cash flow to determine its worth today. Discount rate - the rate used to calculate the present value of future cash flow 4. PV = $1 / (1 + r)^t

Interest on Interest

Interest earned on the reinvestment of previous interest payments

Simple Interest

Interest earned only on the original principal amount invested

Discounted Cash Flow (DCF) Valuation

Calculating the present value of a future cash flow to determine its value today

C =

Cash amount

n = Periods i = Interest Rate Pmt = Payment PV = Present Value FV = Future Value

Cash outflows = Negative Cash inflows = Positve

Compound Interest

interest earned on both the principal amount and the interest reinvested from prior periods

What is the PV of $1000 to be received in three years at 15%?

n = 3 i = 15 FV = $1000 PV = ? PV = -$657.50 The answer has a negative sign because it represents an outflow today in exchange for the $1000 inflow later