INTEREST RATE DETERMINATION

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INTEREST 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.
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• 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.

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THE MARKET FOR LOANABLE FUNDS
interest rate
supply of loanable funds
iE
demand for loanable funds
Q
QE
LOANABLE FUNDS
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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.

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BASICS 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)21,000.

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THE 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.

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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.

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• 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?

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• 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.

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• 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.

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

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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 todays deposit would eventually
  become the future amount at the going compound
  rate of interest.
• Heres 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.

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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 dont have to put
  away as much today to reach your goal.

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• 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.

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WHATS THE PV OF 10,000 t YEARS HENCE?
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• Example Your friend will give you 200 two
  years from today. What is the present value of
  the gift?

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• The discounted present value of 200 two years
  hence is 200/ (1i)2.
• a) If the rate of interest, i, is 7 (.07), then
  the present value is 174.69.
• b) If the interest rate were 10, the present
  value would be 165.29.

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• 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.

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• Once again, it depends on the rate of interest.
• The DPV of the first gift is 5,000. The DPV of
  the second gift is 8,000/(1i)5.
• At i.07 the second gift is worth 5,704., but at
  i.10 its worth only 4,967.

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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?

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• The answer is
• 5,000,000 / (1 .07)20,
• which equals 1,292,095.
• At a rate of interest of 10, the present value
  is only 743,218.

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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?

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• The bond promises 4 payments
• 1) 100 one year hence.
• 2) 100 two years hence.
• 3) 100 three years hence.
• 4) 1,000 three years hence.
• The present value of the bond is therefore
• 100/(1i) 100/ (1i)2 1,100/ (1i)3
• At 5 this equals 1,136. At higher interest
  rates it would be worth less.

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• 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.

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• The PV of 1,000 today is 1,000
• ( 1,000/(1i)0)
• The PV of 1,200 one year from now is
• 1,200/(1i)
• If i gt .2, take 1,000 today. If ilt.2, take the
  1,200 in one year.

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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.

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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?

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• The present value of the returns at 8 equals
  19,627.
• The present value of the cost is 20,000.
• Therefore the net present value is -373., a bad
  deal.
• You shouldnt buy the car in this case.

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• 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.

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• At 6, the present value of the returns to buying
  the car equals 20,390.
• Therefore, the net present value is 390., and
  the investment is profitable. You should buy the
  car.

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• 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.

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

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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
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• 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
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INTERNAL 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
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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 1,000/9001 - 1 1.111 - 1 .11 (approx..)
  or about 11.1 percent.

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• 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.

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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.

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• 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.

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• 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.

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

annual income
18
65
age
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If you go to college you might get a higher
income.

COLLEGE
annual income
HS
18
65
22
age
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• But the higher income also entails a cost
• Tuition
• Books and supplies
• Forgone income from the job you would have had

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annual income
THE SHADED AREA REPRESENTS THE COSTS OF A
COLLEGE EDUCATION
age
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THIS AREA REPRESENTS THE GAINS IN INCOME FROM
A COLLEGE EDUCATION
annual income
age
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WE WANT TO COMPARE THESE AREAS TO SEE IF THE
BENEFITS EXCEED THE COSTS.
annual income
age
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• 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.

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• 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.

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