interest ratesslide 1 interest rate determination the rate of interest is the price of money to...

Interest rates slide 1

INTEREST RATE DETERMINATIONINTEREST RATE DETERMINATION

The rate of interest is the price of money to borrow and lend.

Rates of interest are expressed as decimals or as percentages. For example, the rate of interest of 5 percent per year(5%) could be written as i=.05.


One theory views the rate of interest as the price in the market for loanable funds.

Loanable funds are monies borrowed by firms from consumers in order to undertake investment projects.

[NOTE: Investment = additions to capital stock, such as factories, houses, inventories, etc. Investment is not buying stocks and bonds.]


THE MARKET FOR LOANABLE FUNDS

interestrate

supply of loanable funds

demand for loanable funds

QLOANABLE FUNDS

QE

iE


Note that the demand curve for loanable funds is negatively sloped (like every other demand curve).Why would a reduction in the interest rate increase the quantity demanded of loanable funds?

This is a question with a complicated answer.

We begin with the idea of compound interest.


BASICS OF COMPOUND INTERESTBASICS OF COMPOUND INTEREST

Suppose I put on deposit today $1,000 at a rate of interest of 5 percent (i = .05).

After one year my balance becomes

$1,000 + .05($1,000) = (1 + .05)$1,000

If interest is compounded annually, after two years my balance will be

(1 + .05)((1 + .05)$1,000)) = (1 + .05)2$1,000.


THE FORMULA FOR FUTURE VALUETHE FORMULA FOR FUTURE VALUE

In general, a current balance of P(0) placed on deposit for t years at a rate of interest i (compounded annually) becomes

P(t) = P(0) (1 + i)t.

P(t) is called the future value of the current balance.


NOW WE SET A DIFFERENT QUESTION

Suppose I want to have a fixed amount of money available to me in the future.

How much money would I have to put aside today to get the future amount? Remember that what I put aside today accumulates at a compound annual rate of interest, i.


For example, suppose I want to have $25,000 available 5 years from now to buy a new car.

How much would I have to put on deposit today, if the rate of interest is 6 percent, so that I will have the $25,000 when I need it?


The answer to the question can be found in the basic formula for compound interest:

P(t) = P(0) (1 + i)t

We know P(t), the amount we want in the future, and we know i and t.

We need to find P(0), the amount to put on deposit today that will become P(t), t years in the future if the rate of interest is i.


In the example, P(t) = $25,000, t = 5, and i = .06.

So we have

$25,000 = P(0) (1 + .06)5

Therefore, P(0) = $18,681.45.


P(0) is the called the present value of $25,000., 5 years hence, at 6 percent.


Present Value Defined

The Present Value of a future amount is the amount of money I would have to put on deposit today so that today’s deposit would eventually become the future amount at the going compound rate of interest.

Here’s another way to say it:

The Present Value of P(t) dollars t years in the future is the amount that must be put on deposit today at a rate of interest, i, so that the deposit equals P(t) after t years.


P(0) = P(t) / (1 + i)t

Note that the present value of P(t) dollars falls with increases in the rate of interest, i.

This is just another way of saying that if the rate of interest is higher, you don’t have to put away as much today to reach your goal.


Note also that the present value of P(t) falls with increases in t.

This is just another way of saying that the farther in the future you want the money, the less you have to put aside today.

Interest rates

WHAT’S THE PV OF $10,000 t YEARS HENCE?

t 5% 10%

0 10000 100001 9524 90912 9070 82643 8638 75134 8227 68305 7835 62096 7462 56457 7107 51328 6768 46659 6446 424110 6139 385511 5847 350512 5568 3186


Example: Your friend will give you $200 two years from today. What is the present value of the gift?


Example: Your aunt Alice offers you the choice between two gifts. The first is a cash gift today of $5,000 to cover your college costs. The second is a cash gift 5 years from now of $8,000 to help you buy a new car. Which gift do you choose? [Hint: Choose the one with the greater present value.]


EXAMPLE

A rich alumnus decides to leave funds for an endowed chair to the university. The gift will be made when he dies, which is predicted to be in 20 years. His gift at that time will be $5 million.

In order to assure that the funds will be paid the alum sets up a trust. If the interest rate is 7%, what is the PRESENT VALUE of $5 million 20 years hence? That is to say, how much money must he deposit in the trust today?


EXAMPLE

You buy a bond that promises to pay you $100 (in interest) in each of the next 3 years ($100 one year from now, $100 two years from now, etc.)

At the time you get the third interest payment you receive the principal on the bond of $1,000.

How much do you pay for the bond?


The concept of PRESENT VALUE allows us to compare the values of returns and costs that may accrue at different times in the future.

For example, which would you prefer, $1,000 now or $1,200 one year from now? If you are like most people you will choose the one that has the greatest present value. And which asset has the greater PV depends on the rate of interest.


The PV of $1,000 today is $1,000

(= 1,000/(1+i)0)

The PV of $1,200 one year from now is

1,200/(1+i)

If i > .2, take $1,000 today. If i<.2, take the $1,200 in one year.


NET PRESENT VALUE

The net present value of an investment is the present value of the returns minus the present value of the costs.

As a general rule, it will be best to undertake investments whose net present value is greater than zero.


EXAMPLE:

A new car costs $20,000 today. It yields returns of $7,000 in each of the first three years of operation, and then you can sell it for scrap for $2,000. (Assume the returns occur at the ends of the years in question.)

If the interest rate at which you can borrow is 8 percent, should you buy the car?


But what if you could borrow at 6% instead of 8%?

Certainly at the lower interest rate, the present value of the returns is greater than it was at 8%.


Because lowering interest rates raises the present value of future returns, the demand to make investments tends to increase as the rate of interest falls.

In other words, the demand curve for loanable funds is negatively sloped.


In general, a firm should undertake investments that have a positive net discounted present value.


ANOTHER VIEW:ANOTHER VIEW:

FINDING RATES OF RETURN ON INVESTMENTS.

We know that P(0) dollars put on deposit today will become P(t) after t years if the rate of interest is i.

P(t) = P(0) (1 + i)t


Suppose you know the initial value of an asset, P(0), and you also know the final value after t years, P(t). We can ask what the rate of interest would have to be to turn the initial value into the final value.

This boils down to finding the value of i in the equation:

P(t) = P(0) (1 + i)t


INTERNAL RATE OF RETURNINTERNAL RATE OF RETURN

(Internal) rate of return (IRR): The rate of interest that equates the discounted present value of the returns from an asset and the discounted present value of the costs.

For an asset that returns P(t) dollars t years hence, and costs P(0) dollars today, the IRR is:

i = [P(t)/P(0)](1/t) - 1


EXAMPLE

Suppose a bond costs $900 today and agrees to pay you $1,000 one year from today. What is the (internal) rate of return on the bond?

In other words, what is the rate of interest the bond pays you?

i = [1,000/900]1 - 1 = 1.111 - 1 = .11 (approx..) or about 11.1 percent.


So we can use the formula to compute the rate of return on any promise to pay in the future.

This seems to say that if an asset pays, say 15 percent, and we can borrow money at a lower rate, say 7 percent, then the asset should be bought.

In most cases, this rule for when to invest gives the same decision as the Net Discounted Present Value rule discussed above.


AN APPLICATION TO EDUCATION

Economists often analyze people acquiring education as an investment decision.

Acquiring education requires you to pay the costs quite early in your lifetime in return for what you hope will be higher incomes throughout the remainder of your life.

The decision to acquire human capital can be analyzed using the same rules used to examine the profitability of investment in physical capital.


Suppose you have just graduated from high school, and you are trying to decide whether to get a college education or get a job.

One way to think about this is as an investment decision. One reason for going to college is that you will earn a higher income over your working life than if you did not.


You can compute the discounted value of the costs and benefits of a college education (at current rates of interest) and see whether the extra return exceeds the cost.


Incomes over your life if you have a high school education.

age

annual income

$

18 65


If you go to college you might get a higher income.

age

annual income

$

18 6522

HS

COLLEGE


But the higher income also entails a cost:Tuition

Books and supplies

Forgone income from the job you would have had


annualincome

age

THE SHADED AREA REPRESENTS THE COSTS

OF A COLLEGE EDUCATION


annualincome

age

THIS AREA REPRESENTS THE GAINS IN INCOME

FROM ACOLLEGE EDUCATION


annualincome

age

WE WANT TO COMPARETHESE AREAS TO SEE

IF THE BENEFITS EXCEED

THE COSTS.


Because the benefits and costs occur at different times, we cannot simply compute the graphical areas.

Sensible comparison of benefits and costs requires that we find the present value of the benefits and the present value of the costs.

Or we could compute the internal rate of return on a college education and compare it with the market rate of interest.


It turns out that the returns to a college education in recent years have been rising, and the returns to a high school education have been falling.

The best recent estimates suggest that the rate of return to a college education may be as high as 12 to 15 percent compared to a high school education.


An investment in college education is now one of the best investments around.

interest ratesslide 1 interest rate determination the rate of interest is the price of money to...

Documents