Costs and Benefits

Interest Rates and the Time Value of Money• Time Value of Money

▫ Imagine a simple investment opportunity with the following cash flows (which are certain to occur). Cost: $1,000 today Benefit: $1,030 in one year Can we judge the opportunity as valuable by noting

that $1,o3o > $1,000?

▫The difference in value between money today and money in the future is due to the time value of money.

The Interest Rate: An Exchange Rate Across Time• The rate at which we can exchange money today for

money in the future is determined by the current interest rate.

▫Suppose the current annual interest rate is 5%. By investing or borrowing at this rate, we can exchange (trade) $1.05 in one year for each $1 today.

Risk–Free Interest Rate (Discount Rate), rf: The interest rate at which money can be borrowed or lent without risk.

Interest Rate Factor = 1 + rf = 1.05 ($s tomorrow/$ today)

Discount Factor = 1 / (1 + rf) = 1/1.05 ($s today/$ tomorrow)

▫ Investors are compensated for surrendering capital.

Problem

•The cost of replacing a fleet of company trucks with more energy efficient vehicles was $100 million in 2010.

•The cost is estimated to rise by 8.5% in 2011.

•If the interest rate was 4%, what was the cost of a delay in terms of dollars in 2011?

Solution• If the project were delayed, it’s cost in 2011

would be:▫$100 million × (1.085) = $108.5 million in 2011

• Compare this amount to the cost of $100 million in 2010 using the interest rate of 4%:▫$100 million × ($1.04 in 2011/$ in 2010) =

$104 million in 2011

• The cost of a delay of one year would be:▫$108.5 million – $104 million = $4.5 million in

2011

Problem

•The cost of replacing a fleet of company trucks with more energy efficient vehicles was $100 million in 2010.

•The cost is estimated to rise by 8.5% in 2011.

•If the interest rate was 4%, what was the cost of a delay in terms of dollars in 2010?

Solution•We know delaying the project costs $108.5

million in 2011.

•Compare this amount to the cost of $100 million in 2010 using the interest rate of 4%:▫$108.5 million ÷ ($1.04 in 2011/$ in 2010) =

$104.33 million in 2010

•The cost of a delay of one year would be:▫$104.33 million – $100 million = $4.33 million

in 2010 (Why are the results different?)

Discussion• This example is the same as the example of trading

gold for silver presented in the text.• The price today of receiving $1.04 in one year is $1.• The price today of receiving $1 today is of course $1.• After we convert the values using these prices we are

not wondering about whether to trade $X today for $Y in one year but rather: should we trade $V today for $W today.

• In these terms the choice is clear: V W or V W.• We will discuss and extend this idea as we go forward.

Present Value and the NPV Rule

•The net present value (NPV) of a project or investment is the difference between the present value of its benefits and the present value of its costs.▫Net Present Value (Benefits) (Costs) NPV PV PV

(incremental project cash flows)NPV PV

The NPV Decision Rule• When making an investment decision, take

the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today.

• Accept all projects with a positive NPV. Accepting them is equivalent to receiving their NPV in cash today.

• Reject projects with a negative NPV. Accepting these projects is equivalent to reducing current wealth by their NPV.

Consider Three Projects (rf = 20%)Project Cash Today ($) Cash in one year

($)

A 42 42

B -20 144

C -100 225

Project Cash Today PV of Cash in one year

NPV ($ today)

A 42 42 ÷ 1.20 = 35 42 + 35 = 77

B -20 144 ÷ 1.20 = 120

-20 + 120 = 100

C -100 225 ÷ 1.20 = 187.5

-100 + 187.5 = 87.5

NPV•Although Project B has the highest NPV,

what if we do not want to (can’t) spend the $20 for the cash outlay? Would Project A be a better choice? Should this affect our choice of projects?

•As long as we are able to borrow and lend at the risk-free interest rate, Project B is superior whatever our preferences regarding the timing of the cash flows.

•Again, the same as the gold for silver trade.

NPV and Individual Preferences

Project B and borrowing

Cash today Cash in one year

Project B -20 144

Borrow 62 -62 × 1.20 = -74.4

Total 42 (compare A: 42) 69.6 (A: 42)

Project B and saving

Cash today Cash in one year

Project B -20 144

Save -80 80 × 1.20 = 96

Total -100 (compare C: -100)

240 (C: 225)

NPV and Individual Preferences

Project B and borrowing

Cash today Cash in one year

Project B -20 144

Borrow 120 -144

Total 100 0

• As we said, accepting a positive NPV project is equivalent to receiving that much cash today.• This arrangement is equivalent to selling the right to project B.

Separation Principle•The only arbitrage free price for a security is

for price to equal the present value of the cash flows paid by the security.

•This means that in an arbitrage free market the NPV of buying or selling securities is zero.▫Every trade has a buyer and a seller, if the price

is not such that the NPV is not zero one will object.

•This means that any investment project can be evaluated independently of the financing necessary to pursue it.