ECO 120 Macroeconomics Week 5

Investment and Savings

Lecturer Dr. Rod Duncan

Topics

• A firm’s investment decision • Present value of $1 • Net present value in the investment decision • Investment demand

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 R 1 in one year’s time, R 2 in two year’s time, … up to R n ends.

at the nth year when the investment • 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 = R 1 + R 2 + … + R n – 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:

– “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?

A friend offers you a deal: •

Scenario:

– “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?

A friend offers you a deal:

Present value concept

• Not really. The problem is that a $1 today is

not

same as a $1 in a year’s time or 100 years’ time.

the • 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.

18.00

16.00

14.00

12.00

10.00

8.00

6.00

4.00

2.00

0.00

Interest Rates

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 0 1 2 3 … n Value $1 $1(1+i) $1(1+i)(1+i) $1(1+i) 3 … $1(1+i) n i=.10

$1 $1.10

$1.21

$1.33

… $(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.

Year 0 1 2 3 10 n

PV of $1

i=0.01

1 0.99

0.98

0.97

0.91

(1.01) -n i=0.05

1 0.95

0.91

0.86

0.61

(1.05) -n i=0.10

1 0.91

0.83

0.75

0.39

(1.10) -n i=0.20

1 0.83

0.69

0.58

0.16

(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 R time, R 2 1 in one year’s in two year’s time, … up to R n nth year when the investment ends.

at the

Year 0 1 2 3 … n

Investment decision

Benefit 0 R 1 R 2 R 3 … R n Cost I 0 0 0 … 0 PV -I R 1 /(1+i) R 2 /(1+i) 2 R 3 /(1+i) 3 … R n /(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 = R 1 /(1+i) + R 2 /(1+ i) 2 + … + R n /(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)

Year 0 1 2 3 … 10

Example of NPV

Benefit 0 $2,000 $2,000 $2,000 … $2,000 Cost I 0 0 0 … 0 PV -$12,000 $2,000/(1+i) $2,000/(1+i) 2 $2,000/(1+i) 3 … $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 loan

Year 0.05

0.08

0.1

0.12

0 1 2 3 4 5 6 7 8 9 10 -12000 -10600 -9130 -7586.5

-5965.83

-4264.12

-2477.32

-601.19

1368.75

3437.19

5609.05

-12000 -10960 -9836.8

-8623.74

-7313.64

-5898.74

-4370.63

-2720.28

-937.91

987.06

3066.03

-12000 -11200 -10320 -9352 -8287.2

-7115.92

-5827.51

-4410.26

-2851.29

-1136.42

749.94

-12000 -11440 -10812.8

-10110.3

-9323.58

-8442.41

-7455.49

-6350.15

-5112.17

-3725.63

-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:

prices.

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 • In 1637, prices for tulips crashed and by 1639, tulip bulbs were selling for 1/200 th of the peak prices. • Bubbles tend to crash fast and dramatically.