The Time Value of Money

Problem #3

Congratulations!

What is today’s value of a $1,000 face value bond with a 5% coupon rate (interest is paid semi-annually) which has three years remaining to maturity. The bond is priced to yield 8%.

You obviously understand this material. Now try the next problem.

Problem #2

How to solve a time value of money problem.

The Time Value of Money

Comparing PV to FV

What is the value of $100 per year for four years, with the first cash flow one year from today, if one is earning 5% interest, compounded annually? Find the value of these cash flows four years from today.

Which would you rather have - $1,000 today or $1,000 in 5 years?

• The “value four years from today” is a future value amount.
• The “expected cash flows of $100 per year for four years” refers to an annuity of $100.
• Since it is a future value problem and there is an annuity, you need to solve for a FUTURE VALUE OF AN ANNUITY.

• What is the “Time Value of Money”?
• Compound Interest
• Future Value
• Present Value
• Frequency of Compounding
• Annuities
• Multiple Cash Flows
• Bond Valuation

Remember, both quantities must be present value amounts or both quantities must be future value amounts in order to be compared.

Obviously, $1,000 today.

Problem #1

Valuing a Bond

Money received sooner rather than later allows one to use the funds for investment or consumption purposes. This concept is referred to as the

TIME VALUE OF MONEY!!

Future Value (Graphic)

If you invested $2,000 today in an account that pays 6% interest, with interest compounded annually, how much will be in the account at the end of two years if there are no withdrawals?

You must decide between $25,000 in cash today or $30,000 in cash to be received two years from now. If you can earn 8% interest on your investments, which is the better deal?

Compound Interest

• The interest payments represent an annuity and you must find the present value of the annuity.
• The maturity value represents a future value amount and you must find the present value of this single amount.
• Since the interest is paid semi-annually, discount at HALF the required rate of return (4%) and TWICE the number of years to maturity (6 periods).

Future Value (Formula)

Why TIME?

FV = future value, a value at some future point in time

PV = present value, a value today which is usually

designated as time 0

i = rate of interest per compounding period

n = number of compounding periods

Future Value Example

John wants to know how large his $5,000 deposit will become at an annual compound interest rate of 8% at the end of 5 years.

TIME allows one the opportunity to postpone consumption and earn INTEREST.

NOT having the opportunity to earn interest on money is called OPPORTUNITY COST.

When interest is paid on not only the principal amount invested, but also on any previous interest earned, this is called compound interest.

FV = Principal + (Principal x Interest)

= 2000 + (2000 x .06)

= 2000 (1 + i)

= PV (1 + i)

Note: PV refers to Present Value or Principal

Future Value Solution

Calculation based on general formula:

How can one compare amounts in different time periods?

One can adjust values from different time periods using an interest rate.

Remember, one CANNOT compare numbers in different time periods without first adjusting them using an interest rate.

Multiple Cash Flows Example

Bond Valuation Problem

Suppose an investment promises a cash flow of $500 in one year, $600 at the end of two years and $10,700 at the end of the third year. If the discount rate is 5%, what is the value of this investment today?

Find today’s value of a coupon bond with a maturity value of $1,000 and a coupon rate of 6%. The bond will mature exactly ten years from today, and interest is paid semi-annually. Assume the discount rate used to value the bond is 8.00% because that is your required rate of return on an investment such as this.

Solution

Interest = $30 every six months for 20 periods

Interest rate = 8%/2 = 4% every six months

Double Your Money!!!

Present Value

Finding “n” or “i” when one knows PV and FV

Example of an Ordinary Annuity - FVA

Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?

Example of an Ordinary Annuity - FVA

Discounting is the process of translating a future value or a set of future cash flows into a present value.

If one invests $2,000 today and has accumulated $2,676.45 after exactly five years, what rate of annual compound interest was earned?

We will use the "Rule-of-72"

Present Value (Graphic)

Present Value (Formula)

Present Value Example

Joann needs to know how large of a deposit to make today so that the money will grow to $2,500 in 5 years. Assume today’s deposit will grow at a compound rate of 4% annually.

Assume that you need to have exactly $4,000 saved 10 years from now. How much must you deposit today in an account that pays 6% interest, compounded annually, so that you reach your goal of $4,000?

Present Value Solution

Calculation based on general formula:

Frequency of Compounding

If one agrees to repay a loan by paying $1,000 a year at the end of every year for three years and the discount rate is 7%, how much could one borrow today?

If one saves $1,000 a year at the end of every year for three years in an account earning 7% interest, compounded annually, how much will one have at the end of the third year?

Example

Suppose you deposit $1,000 in an account that pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals?

PV = $1,000

i = 12%/4 = 3% per quarter

n = 8 x 4 = 32 quarters

n: Number of Years

m: Compounding Periods per Year

i: Annual Interest Rate

FVn,m: FV at the end of Year n

PV : PV of the Cash Flow today

0