The Time Value of Money

The Timeline

Suppose you are investing $1,000 today and the payoff will come in two payments. The timeline looks like this:

The first cash flow at date 0 (today) is represented as a negative amount because it is a cash outflow (from your perspective). The others are positive as they represent cash inflows.

Timelines represent cash flows that occur at the end of any period. (This is the convention in your textbook, only consistency is required.)

We want to ask, is this investment a good idea?

0 21

-$1,000 +$500 +$550

“The 1st Rule of Time Travel” A dollar today and a dollar in one period are

not equivalent. $1,050 > $1,000 is not relevant for several reasons.

It is therefore only possible to compare (or combine) values for the same point in time. To determine if this investment is a good idea you need

to compare it with a relevant alternative. What is a relevant alternative?

One factor to consider: How many periods “later” do you receive the cash flows?

Another is: What is the price of moving money across a period of time?

“The 2nd Rule of Time Travel”

To move a cash flow forward (i.e., to a later date) in time, you must compound it.

The future value of a cash flow in n periods, when the periodic interest rate is r, is: (1 )nFV C r

The 2nd Rule of Time Travel

Suppose you have a choice between receiving $5,000 today or $10,000 in five years. You can earn 10% per year on savings. What the $5,000 would be worth in five years if you saved it. Clearly the current interest rate, here the

10%, is an important consideration as is the five year horizon.

The time line looks like this:0 3 4 521

$5,000 ???

The 2nd Rule of Time Travel

0 3 4 521

$5,000 $5, 500 $6,050 $6,655 $7,321 $8,053x 1.10 x 1.10 x 1.10 x 1.10 x 1.10

In five years, $5,000 becomes: $5,000 × (1.10)5 = $8,053.55 The future value of $5,000 at 10% for five years is $8,053.55

You would be better off forgoing the gift of $5,000 today and taking the $10,000 in five years. $5,000 today is equivalent to $8,053.55 in 5 years. The $10,000 in 5 years can be compared to the $8,053.55. What if you need money today, or if you believe the dollar will

be significantly devalued in 5 years, does that change your decision?

“The 3rd Rule of Time Travel”

To move a cash flow back (i.e., to an earlier date) in time you must discount it.

The present value of a future cash flow n periods in the future, when the periodic interest rate is r, is:(1 )

(1 )n

n

CPV C r

r

The 3rd Rule of Time Travel

What is today’s value of your opportunity to receive $10,000 in five years when the interest rate is 10%?

$10,000 received in five years is worth: $10,000 ÷ (1.10)5 = $6,209.21 today this “present

value” can be compared to the alternative of $5,000 today.

Alternatively: $6,209.21×(1.10)5 = $10,000 so if you had $6,209.21

today it would become $10,000 in 5 years if the interest rate is 10%. Thus today’s value of $10,000 to be received in 5 years must be exactly $6,209.21. Or, if you will receive $10,000 in 5 years you may borrow $6,209.21 today against it.

Valuing a Stream of Cash Flows

Based on the three rules of time travel we can derive a general formula for valuing a stream of cash flows: the present value of a stream of cash flows is simply the sum of the present values of each cash flow.

The Present Value of a Cash Flow Stream

0 0

( ) (1 )

N Nn

n nn n

CPV PV C

r

Example

Suppose you have the opportunity to purchase a claim to a series of cash flows such that you would receive $100 in one year, $200 in two years and $300 in three years. The current interest rate is 10%.

What is the present value of these payments?

How much would you be willing to pay to purchase this claim?

If you are able to purchase the claim for $420 are you better off than you were without this opportunity? By how much?

Solution

The solution is to simply find the total present value of all the future cash flows (the sum of the present values of the individual future cash flows).

Now we compare this value to the proposed cost today:

The NPV tells us how much better off we are in terms of dollars today from the purchase. Borrow $481.59 today and pay $420 for the claim. The

claim will just repay the loan and you keep the $61.59.

2 3

$100 $200 $300$481.59

(1.10) (1.10) (1.10)PV

$481.59 $420 $61.59NPV PV Cost

Example

You have the opportunity to purchase a security that promises a series of cash flows such that you will receive $100 in one year, and $100 each year after that for the following six years (a total of seven payments).

What is the present value of these payments if the current interest rate is 4%?

Example

You have the opportunity to purchase a security that promises a series of cash flows such that you will receive $100 in one year, and $100 each year after that for the following six years (a total of seven payments).

What is the present value of these payments if the current interest rate is 4%?

There must be an easier way!

2 3 4 5 6 7

$100 $100 $100 $100 $100 $100 $100$600.21

(1.04) (1.04) (1.04) (1.04) (1.04) (1.04) (1.04)PV

Perpetuity

A stream of equal payments, starting in one period, and made each period, forever.

Please, please remember, this gives the value of this stream of cash flows as of time 0, one period before the first payment is made.

…

0 1 2 3

$C $C $C

r

CPV

Growing Perpetuity

A growing perpetuity is a stream of periodic payments that grow at a constant rate and continue forever.

The present value of a perpetuity that pays the amount $C1 in one period, payments grow at the rate g per period indefinitely, and the periodic discount rate is r is:

…

0 1 2 3

$C1 $C1(1+g) $C1(1+g)2

gr

CPV

1

Perpetuity Examples

Perpetuity, C = $100, r = 10%, then P0 = $1,000. If P0 = $1,000 then having $1,000 today generates

the infinite stream of payments. Does it? What will the perpetuity be worth right after the first

payment is made? What is it worth just before the first payment is

made? Growing perpetuity, C1 = $100, r = 10%, and g =

2%, then P0 = $1,250. What is this worth right after the first payment is

made? What is it worth just before the first payment?

Annuities

An annuity is a series of equal payments, starting next period, and made each period for a specified number (3) of periods.

If payments occur at the end of each period (the first is one period from now) it is an ordinary annuity or an annuity in arrears.

If the payments occur at the beginning of each period (the first occurs now) it is an annuity in advance or an annuity due.

0 1 2 3

C C C

0 1 2 3

C C C

Valuing Annuities

We can do a lot of grunt work or we can notice that a T period annuity is just the difference between a standard perpetuity and one whose first payment comes at date T+1.

The present value of a T period annuity paying a periodic cash flow of C, when the discount rate is r, is:

If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment (and the sum) is higher by a factor of (1+r).

Or we can add C to the value of a T-1 period annuity.

TT rr

C

r

rC

r

CPV

)1(

11

)1(

/

Example

Imagine you take a loan from the bank. For simplicity you borrow the money Dec 31st of 2012 and equal payments are due at the end of every year for the next five years. The interest rate is 4% and you borrow $65,000. What are the 5 equal payments that must be made? How can you be sure you repay the bank appropriate

interest and principle with these payments? How much principle and how much interest was in the

1st payment? What is the state of the loan on Jan 1st 2014 (Dec 30th)?

Annuity Example

Compute the present value of a 3 year ordinary annuity with payments of $100 at r = 10%.

or,

2 3

$100 $100 $100$248.69

1.1 1.1 1.1PV

3

1 1$100 248.69

0.1 0.1(1.1)PV

Annuity Due Example

What if the last example had the payments at the beginning of each period not the end?

or,

or,

55.273$1.1

100$

1.1

100$100$

2PV

55.273$)1.1(68.248$ PV

55.273$)1.1(1.0

1

1.0

1100$100$

2

PV

Growing Annuities

A stream of payments each period for a fixed number of periods where the payment grows each period at a constant rate.

…

0 1 2 T-1 T

C1 C1(1+g) C1(1+g)T-2 C1(1+g)T-1

PV Cr g r g

g

r

T

1

1 1 1

1

T

r

g

gr

CPV

1

111

The Effective Annual Rate (EAR) Indicates the total amount of interest (as a

percent of the initial investment) that will be earned at the end of one year. Conceptually, this is what we have been

working with.

The EAR considers the effect of compounding within the year Also referred to as the effective annual yield

(EAY) or annual percentage yield (APY)

Interest Rate Quotes and Adjustments

Adjusting the Discount Rate to Different Time Periods Suppose 4% is the effective annual rate, however

interest is compounded every six months, what is the 6-month rate?

Earning a 4% return annually is not the same as earning 2% every six months.

(1.04)0.5 – 1= 1.0198 – 1 = .0198 = 1.98% Note: n = 0.5 since we are solving for the six month

(or ½ year) rate Note: this implicitly assumes that the six month rate is the

same for both six month periods in the year.

Interest Rate Quotes and Adjustments

Equivalent -Period Discount Rate (1 ) 1 nn r

Problem Suppose your bank account pays interest

monthly with an effective annual rate of 5%. What is the interest you will earn in one month? What is the interest you will earn in two years?

Solution We know from above that a 5% EAR is equivalent

to earning (1.05)1/12 -1 = .4074% per month.

We can use the same formula to find the two year rate (1.05)2 – 1 = 10.25% (in particular, not 10%) or (1.004074)24 – 1 = 10.25%.

Example

The annual percentage rate (APR) or stated annual rate, indicates the amount of simple interest earned in one year. Simple interest is the amount of interest earned

ignoring the effect of compounding.

The APR is typically less than the effective annual rate (EAR, or the interest you will actually earn/pay).

APR is simply a communication device, a way to communicate a periodic interest rate where the period is not a year.

Annual Percentage Rates

The APR itself cannot be used as a discount rate (except in one very special case). The APR with k compounding periods each

year is simply a annualized way of quoting the actual interest earned each compounding period:

Annual Percentage Rates

APRInterest Rate per Compounding Period

periods / yeark

Converting an APR to an EAR

The EAR increases with the frequency of compounding (holding the APR fixed). Continuous compounding is compounding

every instant. A very useful mathematical abstraction we will not dwell upon.

Annual Percentage Rates

1 1

kAPR

EARk

If the APR is 6% what are the EARs for different compounding intervals? Annual compounding: (1 + 0.06/1)1 – 1 = 6% Semiannual compounding: (1 + 0.06/2)2 – 1 = 6.09% Monthly compounding: (1 + 0.06/12)12 – 1 = 6.1678% Daily compounding: (1 + 0.06/365)365 – 1 = 6.1831%

A 6% APR with continuous compounding results in an EAR of approximately 6.1837%.

Annual Percentage Rates

You are offered a chance to buy a perpetuity paying $100 per year (beginning in one year) for $1,650 today. You are able to borrow and lend money at a 6% APR with monthly compounding. Should you buy the perpetuity?

First we must find the EAR since the perpetuity makes annual payments. The effective annual rate is (1+0.06/12)12 – 1 = 6.16778%.

The value of the perpetuity is $100/0.0616778 = $1,621.33.

Example

Inflation and Real Versus Nominal Rates Nominal Interest Rate: The rates quoted

by financial institutions and commonly used for discounting or compounding cash flows – growth in money

Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation

Determinants of Interest Rates

The Real Interest Rate

The Nominal Interest Rate

Determinants of Interest Rates

1 Growth of MoneyGrowth in Purchasing Power 1

1 Growth of Prices

r

rr

i

1 1

1 1 r

r r ir r i

i i

1 (1 )(1 ) (1 )(1 ) 1r rr r i or r r i

Problem In the year 2006, the average 1-year Treasury rate

was about 4.93% and the rate of inflation was about 2.58%.

What was the real interest rate in 2006?

Solution Using the equation given above, the real interest

rate in 2006 was: (4.93% − 2.58%) ÷ (1.0258) = 2.29%

Which is approximately equal to the difference between the nominal rate and inflation: 4.93% – 2.58% = 2.35%

Example

Term Structure: The relationship between the investment term and the interest rate

Yield Curve: A graph of the term structure

The Yield Curve and Discount Rates

Term Structure of Risk-Free U.S. Interest Rates, January 2004, 2005, 2006, etc.

Term (Years) Rate Price1 5.06% 951.84$ 2 4.88% 909.11$ 3 4.79% 869.04$

Example

When the yield curve is not flat we must discount each future cash flow at the appropriate rate. Compute the present value of a risk-free,

three-year annuity of $500 per year, given the following yield curve:

2 3

$500 $500 $500

1.0506 1.0488 1.0479 PV

Example

Solution Each cash flow must be discounted by the

corresponding interest rate:

It is very important to note that even though this is a 3-year annuity, we cannot use the annuity formula to find its present value.

$475.92 $454.55 434.52 $1,364.99 PV

The shape of the yield curve is influenced by interest rate expectations. An inverted yield curve indicates that

interest rates are expected to decline in the future. Because interest rates tend to fall in response

to an economic slowdown, an inverted yield curve is often interpreted as a negative forecast for economic growth.

Each of the last six recessions in the United States was preceded by a period in which the yield curve was inverted.

The Yield Curve and the Economy

The shape of the yield curve is influenced by interest rate expectations. A steep yield curve generally indicates that

interest rates are expected to rise in the future. The yield curve tends to be sharply increasing

as the economy comes out of a recession and interest rates are expected to rise.

The Yield Curve and the Economy