Interest ratesslide 1 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.

Interest ratesslide 2 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.]

Interest ratesslide 3 THE MARKET FOR LOANABLE FUNDS interest rate supply of loanable funds demand for loanable funds Q LOANABLE FUNDS QEQE iEiE

Interest ratesslide 4 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.

Interest ratesslide 5 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, ($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.

Interest ratesslide 6 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.

Interest ratesslide 7 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.

Interest ratesslide 8 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?

Interest ratesslide 9 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.

Interest ratesslide 10 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,

Interest ratesslide 11 P(0) is the called the present value of $25,000., 5 years hence, at 6 percent.

Interest ratesslide 12 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.

Interest ratesslide 13 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.

Interest ratesslide 14 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 ratesslide 15 WHAT’S THE PV OF $10,000 t YEARS HENCE?

Interest ratesslide 16 Example: Your friend will give you $200 two years from today. What is the present value of the gift?

Interest ratesslide 17 The discounted present value of $200 two years hence is: $200/ (1+i) 2. a) If the rate of interest, i, is 7% (.07), then the present value is $ b) If the interest rate were 10%, the present value would be $

Interest ratesslide 18 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.]

Interest ratesslide 19 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/(1+i) 5. At i=.07 the second gift is worth $5,704., but at i=.10 it’s worth only $4,967.

Interest ratesslide 20 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?

Interest ratesslide 21 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.

Interest ratesslide 22 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?

Interest ratesslide 23 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/(1+i) + $100/ (1+i) 2 + $1,100/ (1+i) 3 At 5% this equals $1,136. At higher interest rates it would be worth less.

Interest ratesslide 24 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.

Interest ratesslide 25 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.

Interest ratesslide 26 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.

Interest ratesslide 27 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?

Interest ratesslide 28 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 shouldn’t buy the car in this case.

Interest ratesslide 29 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%.

Interest ratesslide 30 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.

Interest ratesslide 31 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.

Interest ratesslide 32 In general, a firm should undertake investments that have a positive net discounted present value.

Interest ratesslide 33 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

Interest ratesslide 34 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

Interest ratesslide 35 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

Interest ratesslide 36 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] = =.11 (approx..) or about 11.1 percent.

Interest ratesslide 37 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.

Interest ratesslide 38 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.

Interest ratesslide 39 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.

Interest ratesslide 40 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.

Interest ratesslide 41 Incomes over your life if you have a high school education. age annual income $ 1865

Interest ratesslide 42 If you go to college you might get a higher income. age annual income $ HS COLLEGE

Interest ratesslide 43 But the higher income also entails a cost: Tuition Books and supplies Forgone income from the job you would have had

Interest ratesslide 44 annual income age THE SHADED AREA REPRESENTS THE COSTS OF A COLLEGE EDUCATION

Interest ratesslide 45 annual income age THIS AREA REPRESENTS THE GAINS IN INCOME FROM A COLLEGE EDUCATION

Interest ratesslide 46 annual income age WE WANT TO COMPARE THESE AREAS TO SEE IF THE BENEFITS EXCEED THE COSTS.

Interest ratesslide 47 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.

Interest ratesslide 48 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.

Interest ratesslide 49 An investment in college education is now one of the best investments around.