ECO 120 Macroeconomics

Week 5

Investment and Savings LecturerDr. Rod Duncan

News

• Mid-term exam in the first hour of class next week. We will go over the answers in class after the exam.

• Bring a pen or pencil and a calculator!

Topics

• A firm’s investment decision

• Present value of $1

• Net present value in the investment decision

• Investment demand

Special Reading: Petty et al. Chapters 4 & 10

Why are we studying investment?

• Investment (I) is a component of aggregate expenditure: – AE = C + I + G + NX

• Changes in I will cause changes in AE.• But any changes in the AE curve, will

cause a shift in the aggregate demand (AD) curve.

• So any changes in I will lead to a shift in the AD curve.

Investment

• Investment can refer to the purchase of new goods that are used for future production. Investment can come in the form of machines, buildings, roads or bridges. This is called “physical capital”.

• Another type of investment is called “human capital”. This is investment in education, training and job skills.

• Usually when we talk about investment, we mean investment in physical capital, but investment should include all forms of capital.

Investment decision-making

• What determines investment?– Businesses or individuals make an investment if they

expect the investment to be profitable.

• Imagine we have a small business owner who is faced with an investment decision.

• The small business owner will make the investment as long as the investment is profitable.

• How to determine profitability of investment?

Profitability of an investment

• Example: – An investment involves the current cost of investment (I). – The investment will pay off with some flow of expected future

profits. – The future stream of profits is R1 in one year’s time, R2 in two

year’s time, … up to Rn at the nth year when the investment ends.

• Imagine you are the business owner. How do we decide whether to make the investment? Can we simply add up the benefits (profits) and subtract the costs (investment)?

Profits today = R1 + R2 + … + Rn – I?

• What is wrong with this calculation?

Present value concept

• Imagine our rule about future values was simply to add future costs and benefits to costs and benefits today.

• Scenario: A friend offers you a deal: – “Give me $10 today, and I promise to give you $20 in 1 years

time.”• If we subtract costs ($10) from benefits ($20), we get a

positive value of $10. Does this seem like a sensible decision?

• Scenario: A friend offers you a deal: – “Give me $10 today, and I promise to give you $20 in 100 years

time.”• If we subtract costs ($10) from benefits ($20), we get a

positive value of $10. Does this seem like a sensible decision?

Present value concept

• Not really. The problem is that a $1 today is not the same as a $1 in a year’s time or 100 years’ time.

• We can not directly add these $1s together since they are not the same things. We are adding apples and oranges.

• We need a way of translating future $1s into $1s today, so that we can add the benefits and costs together.

• The conversion is called “present value”. • In making the decision about our friend’s deal, we would

compare $10 today to the present value of the $20 in a year or 100 years.

Present value concept

• An investment is about giving up something today in order to get back something in the future.

• So an investment decision will always involve comparing $1s today to $1s in the future.

• Investment decisions will always involve present values. If we subtract the present value of future profits from costs today, we get net present value.Net Present Value (NPV) = Present Value of Future

Profits (PV) – Investment (I)

Net present value

• The investment rule will be to invest if and only if:

NPV ≥ 0

• Or

Present Value of Future Profits (PV) – Investment (I) ≥ 0

Interest rates

• To measure present value we will have to use interest rates.

• Interest rates are a general term for the percentage return on a dollar for a year:– that you earn from banks for saving– that you pay banks for borrowing or investing

• For example, the interest rate might be 10%, so if you put $1 in the bank this year, it will become $(1+i) in one year’s time.

• Or if you borrow $100 today, you will have to repay $(1+i)100 next year.

Interest Rates

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

Jan-

70

Jan-

73

Jan-

76

Jan-

79

Jan-

82

Jan-

85

Jan-

88

Jan-

91

Jan-

94

Jan-

97

Jan-

00

Jan-

03

Bank Interest Rates

Discounting future values

• What is the PV of $1 in a year? How do we place a value today on $1 in t years’ time?

• This is called “discounting” the future value. One way to think about this question is to ask:– “How much would we have to put in the bank now to

have $1 in t years’ time?”– Money in the bank earns interest at the rate at the

rate i, i>0. If I put $1 in the bank today, it will grow according to the rate of interest.

– We can construct a chart of our bank account over time.

Bank account

• If we start with $1 in our bank account, what happens to our bank account over time?

Year Value i=.10

0 $1 $1

1 $1(1+i) $1.10

2 $1(1+i)(1+i) $1.21

3 $1(1+i)3 $1.33

… … …

n $1(1+i)n $(1.1)n

How much is a future $1?

• In order to have $1 next year, we would have to put x in today:

$1 = (1+ i) $x$x = 1/(1+i) < 1

• $1 next year is worth 1/(1 + i) today. Since i>0, $1 next year is worth less than $1 today.

• In order to have $1 in n years’ time, we would have to put x in today:

x = 1/(1+i)n = (1+i)-n

• $1 in n years’ time is worth 1/(1+i)n < 1 today.

PV of $1

Year i=0.01 i=0.05 i=0.10 i=0.20

0 1 1 1 1

1 0.99 0.95 0.91 0.83

2 0.98 0.91 0.83 0.69

3 0.97 0.86 0.75 0.58

10 0.91 0.61 0.39 0.16

n (1.01)-n (1.05)-n (1.10)-n (1.20)-n

Investment decision

• Imagine we are the small business owner we were discussing before. We have a new project which we might invest in:– An investment involves the current cost of

investment (I). – The investment will pay off with some flow of

expected future profits. – The future stream of profits is R1 in one year’s

time, R2 in two year’s time, … up to Rn at the nth year when the investment ends.

Investment decision

Year Benefit Cost PV

0 0 I -I

1 R1 0 R1/(1+i)

2 R2 0 R2/(1+i)2

3 R3 0 R3/(1+i)3

… … … …

n Rn 0 Rn/(1+i)n

Net present value

• The NPV of the investment is the sum of the values in the far-right column- the PVs.NPV = R1/(1+i) + R2/(1+ i)2 + … + Rn/(1+ i)n – I

• If NPV ≥ 0, then go ahead and make the investment. If NPV < 0, then the investment is not worthwhile.

• Let’s look at a more concrete example that we can put some numbers to.

Example of NPV

• Example: A small business in Bathurst that owns photo store is considering installing a state-of-the-art developing machine for digital photographs.– Cost = $12,000 (after selling current machine)– Future benefits = $2,000 per year in extra

business every year for 10 year life-span of machine (assume benefits start next year)

Example of NPV

Year Benefit Cost PV

0 0 I -$12,000

1 $2,000 0 $2,000/(1+i)

2 $2,000 0 $2,000/(1+i)2

3 $2,000 0 $2,000/(1+i)3

… … … …

10 $2,000 0 $2,000/(1+i)10

Example of NPV

• NPV = -$12,000 + $2,000/(1+i) + $2,000/(1+i)2 + $2,000/(1+i)3 + … + $2,000/(1+i)10

• Our NPV then depends upon the interest rate, i, facing the small business.

• For a small business, the relevant interest rate would be the rate that it cost raise the money, say by taking out a bank loan.

• So the interest rate would be the bank small business loan rate.

Example of NPV

• The NPV varies with the interest rate:– At i=0.05, NPV = $3,443, so go ahead with

investment.– At i=0.08, NPV = $1,420, so go ahead with

investment.– At i=0.10, NPV = $289, so go ahead with investment.– At i=0.12, NPV = -$700, so don’t go ahead with the

investment.

• Somewhere between a 10% and a 12% interest rate, NPV = 0. NPV < 0 for all interest rates greater than 12%.

Example of NPV

• Another way of thinking about this problem is to ask “Can I repay the loan and still make money?”

• The small business owner borrows $12,000 from the bank and uses the $2,000 in extra business each year to repay the loan.

• Would the business owner repay the loan before the machine needs to be replaced?

Example of NPV- bank loanYear 0.05 0.08 0.1 0.12

0 -12000 -12000 -12000 -12000

1 -10600 -10960 -11200 -11440

2 -9130 -9836.8 -10320 -10812.8

3 -7586.5 -8623.74 -9352 -10110.3

4 -5965.83 -7313.64 -8287.2 -9323.58

5 -4264.12 -5898.74 -7115.92 -8442.41

6 -2477.32 -4370.63 -5827.51 -7455.49

7 -601.19 -2720.28 -4410.26 -6350.15

8 1368.75 -937.91 -2851.29 -5112.17

9 3437.19 987.06 -1136.42 -3725.63

10 5609.05 3066.03 749.94 -2172.71

Present Value 3443.47 1420.16 289.13 -699.55

Example of a NPV- bank loan

• So for interest rates of 10% and below, the bank loan is repaid before the machine wears out, so the investment is worthwhile.

• For interest rates of 12% and above, the bank loan is not repaid by the time the machine needs to be replaced, so the investment is not worthwhile.

• The bottom line shows that the remainder in the bank account at the end of 10 years is the NPV of the investment decision.

• So another way to think of NPV is as the money left in an account at the end of a project.

Investment demand

• Instead of thinking about a single small business, think of a whole economy of businesses and individuals making investment decisions.

• Some of these investment decisions will be very good ones and some will be very poor ones. There is a whole range.

• As i rises, the PV of future profits will drop, so the NPV will fall. If we imagine that there are thousands of potential investments to be made, as i rises, fewer of these potential investments will be profitable, and so investment will fall.

Investment demand

• If we graphed the investment demand for goods and services (I) against interest rates, it would be downward-sloping in i. The higher is i, the lower is investment demand.

• What can shift the I curve? Factors that affect current and expected future profitability of projects:– New technology– Business expectations– Business taxes and regulation

Shifts in investment demand

• Example: An increase in business confidence/expectations raises the expected future profits for businesses.

• At the same interest rates as before, since the Rs are higher, the NPVs of all investment projects will be higher.

• The investment demand curve is shifted to the right. I is higher for all interest rates.

Uses of PV concept

• Housing valuation: We can use the PV concept to estimate what house prices should be.

• What do you have when you own a home? You have the future housing services of that home plus the right to sell the home.

• Value of housing services should be the price people pay to rent an equivalent home. Rent is the price of a week of housing services.

• Let’s say your home rents for $250 per week.

Housing valuation

• If you stayed in your home for 50+ years, your house is worth the PV of 50 years of 52 weekly $250 payments plus any sale value at 50 years. How do we calculate the PV of such a long stream of numbers?

• Trick: For very long streams, the sum:• PV = ($250 x 52) + ($250 x 52)/(1+i) + …• Is very close to:• PV = ($250 x 52) / i = $13,000 / i

Housing valuation

• So we get the house values:– At i=0.02, PV House = $650,000– At i=0.03, PV House = $433,000– At i=0.05, PV House = $260,000– At i=0.06, PV House = $217,000– At i=0.07, PV House = $186,000

• At a house price above this price, you are better off selling your house and renting for 50 years. At a house price below this price, you are better off owning a house.

Housing valuation

• You can also see how sensitive house prices are to the interest rate. When i rose from 6% to 7%, the value of the house dropped $31,000.

• You can see why home owners care so much about the home loans rates.

• But what about the resale price at 50 years? – The PV of the house sale in 50 years time is (Sale

Price) / (1+i)50, which for most values of i is going to be a very small number- 8% of Sale Price at 5% interest and 3% of Sale Price at 7% interest.

Housing price bubbles

• Sometimes the price of housing can vary from this PV of housing services price. Some analysts argue that today’s housing prices is one case- these periods are called “bubbles”.

• Example: At 6% interest rates our house was worth $217,000. Let’s say Sam bought the house for $300,000 in order to sell the house one year from now.

• In order to be able to repay the $300,000, Sam has to gain $18,000 (6% of $300,000) by holding the house for a year.

Housing price bubbles

• Since Sam gets $13,000 worth of housing services from the house, the value of the house has to rise $5,000 to $305,000 in next year’s sale for a total gain of $18,000.

• Even though the house is unchanged, the “overpayment” for the house has to rise- the house is still only worth $217,000 in housing services- but it now sells for $305,000.

• So in a “bubble”, if people are overpaying for a house, the overpayment has to keep rising. Eventually people realize that the house only generates $217,000 in services.

Housing price bubbles

• Example: In Holland in 1636, the price of some rare and exotic tulip bulbs rose to the equivalent of a price of an expensive house. People paid that much in plans to resell at even higher prices.

• In 1637, prices for tulips crashed and by 1639, tulip bulbs were selling for 1/200th of the peak prices.

• Bubbles tend to crash fast and dramatically.

Example question

Question 2 (5 marks)

You can save money at the bank and earn a 10% yearly return on your savings. What is the most you would be willing to pay for (include your calculations and explain carefully):

a. a promise of a $1 in one year’s time (assume that this promise will not be broken);

Example question

b. a 10 year $100 savings bond (the bond will pay you $100 in the year 2015, where 2015 is known as the ‘maturity date’) and do a graph of the value of the 10 year $100 maturity in 2015 savings bond as we get closer to the maturity date; and

Example question

c. a 10 year $100 savings bond that also pays you $5 per year for every year that you hold the bond (including the 10th year).