chapter 4 interest rates and present valuectaber/100/interestrates.pdf · present value • present...

INTEREST RATES AND PRESENT VALUE

C H A P T E R 7

Thursday, October 27, 11

INTEREST RATES

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INTEREST RATES

• We have thought about people trading fish and hamburgers lets think about a different type of trade

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INTEREST RATES

• We have thought about people trading fish and hamburgers lets think about a different type of trade• Joe would like to trade coconuts that will be ripe

tomorrow in order to get ripe coconuts today

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INTEREST RATES


tomorrow in order to get ripe coconuts today• Bill is willing to trade some of the coconuts he has today

in order to get coconuts tomorrow.

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INTEREST RATES



in order to get coconuts tomorrow.• At what rate will they trade in a competitive market?

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INTEREST RATES



in order to get coconuts tomorrow.• At what rate will they trade in a competitive market?• For quite a few reasons we think if Bill is going to give up

10 coconuts today he is going to want more than 10 coconuts back tomorrow

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INTEREST RATES



in order to get coconuts tomorrow.• At what rate will they trade in a competitive market?• For quite a few reasons we think if Bill is going to give up

10 coconuts today he is going to want more than 10 coconuts back tomorrow• This is called interest

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WHY DO WE WANT MORE TOMORROW?

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• People are impatient-all else equal I would rather have a candy bar today than wait for it until tommorow

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• People are impatient-all else equal I would rather have a candy bar today than wait for it until tommorow• There is some risk that Joe will die or renege on his

promise (the larger the risk of this the more interest we will want)

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• People are impatient-all else equal I would rather have a candy bar today than wait for it until tommorow• There is some risk that Joe will die or renege on his

promise (the larger the risk of this the more interest we will want)• I might be able to do something useful with the coconuts

(or money) in the meantime• Use the coconuts to build a shelter that I can live in (and if this is

money I could buy a house today, live in it, and then sell the house tomorrow)

• Use it to buy an ice cream truck, make money selling ice cream, and then sell the ice cream truck at the end of the day

• Or if some emergency comes up and I need the money right away, I don’t have it and need to wait until I get paid back 3


INTEREST RATE

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INTEREST RATE

• The interest rate is the percentage, usually expressed in annual terms, of a balance that is paid by a borrower to a lender that is in addition to the original amount borrowed or lent.

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INTEREST RATE


• If I trade 100 coconuts today for 108 coconuts next year, the interest rate is 8%

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INTEREST RATE


• If I trade 100 coconuts today for 108 coconuts next year, the interest rate is 8%• As a practical matter it doesn’t usually work this way:

Banks act as intermediaries• Joe borrows coconuts from the bank• Bill saves his coconuts in the bank• While they never actually meet each other-Joe is essentially

borrowing from Bill4


NOMINAL VS. REAL INTEREST RATES

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• Nominal Interest Rate: the advertised rate of interest

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• Nominal Interest Rate: the advertised rate of interest• Real Interest Rate: the rate of interest after inflation

expectations are accounted for; the compensation for waiting on consumption

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• As an example suppose the inflation rate is 10% and the nominal interest rate is 10%

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• As an example suppose the inflation rate is 10% and the nominal interest rate is 10%• I put $100 into my account this year (2011)

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• As an example suppose the inflation rate is 10% and the nominal interest rate is 10%• I put $100 into my account this year (2011)• I get $110 next year (2012)

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• As an example suppose the inflation rate is 10% and the nominal interest rate is 10%• I put $100 into my account this year (2011)• I get $110 next year (2012)• However, that $110 in 2012 is worth the same amount as $100

in 2011, so I didn’t really earn any interest

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in 2011, so I didn’t really earn any interest• In that case the real interest rate would be 0

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in 2011, so I didn’t really earn any interest• In that case the real interest rate would be 0• I would probably want to spend that $100 now before inflation

erodes its value

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in 2011, so I didn’t really earn any interest• In that case the real interest rate would be 0• I would probably want to spend that $100 now before inflation

erodes its value• In economics we usually care about the real interest rate 5


PRESENT VALUE

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PRESENT VALUE

• Present Value is the interest adjusted value of future payment streams.

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PRESENT VALUE


• To think of this consider two different scenarios:1. I pay you $300 this year 2. I pay you $100 this year, $100 in 2012, and $100 in 2013

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PRESENT VALUE



• The interest rate is 5%

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PRESENT VALUE



• The interest rate is 5%• Are these the same?

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PRESENT VALUE



• The interest rate is 5%• Are these the same?• Think about it this way:

1. I get my $300 now, spend $100 and save $2002. Next year I have $210 in the bank, I spend $110 and save $1003. In 2013 I now have $105 to spend

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•Obviously I prefer the $300 today.


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•This is because the present value of a stream of $100 for three years is less than $300.


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•Well how much is it exactly?


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•The present value of the first $100 today is $100


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•The present value of $100 next year must solve the following equation (r is the interest rate)


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•The present value of $100 next year must solve the following equation (r is the interest rate)

PV (1 + r) = $100

That is, if I put PV in the bank today, I will have $100 in the bank tomorrow


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So

PV =$100

(1 + r)=

$100(1.05)

= $95.23


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So

PV =$100

(1 + r)=

$100(1.05)

= $95.23

Now what about the present value of $100 in 2012?

To see where this comes from suppose •I put $Y into the bank in 2011.•This would be give me $Y(1+r) in 2012•If I just kept it in the bank I would get $Y(1+r)(1+r) in 2013 Thus the present value of $100 in 2013 is:


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So

PV =$100

(1 + r)=

$100(1.05)

= $95.23

Now what about the present value of $100 in 2012?

To see where this comes from suppose •I put $Y into the bank in 2011.•This would be give me $Y(1+r) in 2012•If I just kept it in the bank I would get $Y(1+r)(1+r) in 2013 Thus the present value of $100 in 2013 is:

PV =$100

(1 + r)2=

$100(1.05)2

= $90.70


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And the value of the stream of $100 for 3 years is worth:

$100 +$100

(1 + r)+

$100(1 + r)2

= $100 + $95.23 + $90.70

= 285.93More generally the present value of a payment $Y n years from now is worth

today

Rather than doing more calculations lets look at what this means

$Y

(1 + r)n


THE AMOUNT PAYABLE FOR EVERY DOLLAR BORROWED (FOR SEVERAL INTEREST RATES AND

LOAN DURATIONS)

Interest rate -> Years ↓

20% 10% 5% 2% 1%

30 237.38 17.45 4.32 1.81 1.35

10 6.19 2.59 1.63 1.22 1.10

5 2.49 1.61 1.28 1.10 1.05

1 1.20 1.10 1.05 1.02 1.01

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EXAMPLES FROM THIS TABLE

• If you borrow $1 and promise to pay it back in 5 years at 5% interest you will owe $1.28 which is the original $1 plus 28 cents in interest.

• If you borrow $1 and promise to pay it back in 30 years at 20% interest you will owe $237.38 which is the original $1 plus $236.38 in interest.

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BUSINESS VENTURE

• Suppose you have the following business venture:• Invest $100 into the venture for 5 years• receive $100 in proceeds starting in year 6 and lasting for 7 years

• Is this a good investment?• It depends on the interest rate

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A MULTIPLE YEAR EXAMPLE @ 5%

Year Cost Benefit PV Cost @5% PV Benefit @5%1 100 100.00

2 100 95.243 100 90.704 100 86.385 100 82.276 100 78.357 100 74.628 100 71.079 100 67.68

10 100 64.4611 100 61.3912 100 58.47

500 700 454.60 476.05Thursday, October 27, 11



2 100 95.243 100 90.704 100 86.385 100 82.276 100 78.357 100 74.628 100 71.079 100 67.68

10 100 64.4611 100 61.3912 100 58.47

500 700 454.60 476.05

This is a gooddeal




2 100 92.593 100 85.734 100 79.385 100 73.506 100 68.067 100 63.028 100 58.359 100 54.03

10 100 50.0211 100 46.3212 100 42.89

500 700 431.21 382.68Thursday, October 27, 11



2 100 92.593 100 85.734 100 79.385 100 73.506 100 68.067 100 63.028 100 58.359 100 54.03

10 100 50.0211 100 46.3212 100 42.89

500 700 431.21 382.68

Not a good dealat this interest rate


INTERNAL RATE OF RETURN

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• At 5% (or less) this was a good investment

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• At 5% (or less) this was a good investment• At 8% (or more) this is a bad investment

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• Internal rate of return : The interest rate where the present value of costs and benefits are equal.

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• If my interest rate is lower than the internal rate, I should do it

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• If my interest rate is lower than the internal rate, I should do it

• If it is higher than the internal rate, I should not do it

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MORTGAGES, CAR PAYMENTS, ETC.

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• Mortgages are loans taken out to buy homes. Typically you borrow a large sum of money and promise to pay it back in even amounts each month for 10, 15, or 30 years.

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• Mortgages are loans taken out to buy homes. Typically you borrow a large sum of money and promise to pay it back in even amounts each month for 10, 15, or 30 years.• Car loans are similar to mortgages in that you borrow a

large sum but the loan duration is usually two to six years.

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large sum but the loan duration is usually two to six years.• How does the financing department figure out the

monthly payment at a 2.9% interest for a $32,000 car?

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large sum but the loan duration is usually two to six years.• How does the financing department figure out the

monthly payment at a 2.9% interest for a $32,000 car?• Given the interest they choose it to be the constant

payment such that the present value of my car payments at a 2.9% rate is $32,000

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MONTHLY PAYMENTS REQUIRED ON PER $1000 OF LOAN (FOR SEVERAL INTEREST RATES AND LOAN DURATIONS)

Interest rate -> Years ↓

20% 10% 5% 2% 1%

30 16.71 8.78 5.37 3.70 3.22

10 19.33 13.22 10.61 9.20 8.76

5 26.49 21.25 18.87 17.53 17.09

1 92.63 87.92 85.61 84.24 83.79


EXAMPLES FROM THIS TABLE

• If you borrow $1000 and promise to pay it back monthly over 5 years at 5% interest you will owe $18.87 per month.

• If you borrow $1000 and promise to pay it back monthly over 10 years at 20% interest you will owe $19.33 per month.

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FUTURE VALUE

• Future value: the interest-adjusted value of past payments.

• That is if I put $Y in the bank today and take it out n years from now I get

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$Y (1 + r)n

dollars when I take it out


RISK AND REWARD

• Risk: the possibility that the investor will not get those anticipated payoffs– Default Risk: the risk to the investor that

the borrower will not pay –Market Risk: the risk that the market value

of an asset will change in an unanticipated manner

• Reward– Risk Premium the reward investors receive

for taking greater risk20


THE YIELD CURVE

US Treasury Yield Curve (January 2005)

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• Yield Curve: the relationship between reward and the time until the reward is received


chapter 4 interest rates and present valuectaber/100/interestrates.pdf · present value • present...

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