ENSC 201 ENSC 411

Summary of Wednesday’s Class

• Purpose of the course is to evaluate business strategies based on cashflows

• Cashflows occurring at different times cannot be compared directly

• There are factors of the form “(X/Y,i,N)” which can be used to move money through time

Conversion Factors

There are formulas, found in the back of the textbook, forevaluating the conversion factors.

Warning! On no account should you rememberthese formulas!

Write out the solutions to problems leaving the conversionfactors unevaluated till the last stage. Then look them upin Appendix B.

Sometimes you will find it useful to enter the formulas onspreadsheets.

Some of the formulasfrom the back of thetextbook.

One page from Appendix B.

Cash Flow Diagrams

These are helpful in making sure we have taken all the important cash flowsinto account. They need not be exactly to scale, but it helps if they’re close.

Time

Pay out $1000 now

Receive $500 for the next 3 years

Present Value

This is an application of the notion of equivalence:

We compare a series of cash flows by bringing themall to the present and adding them up. The sum iscalled the present value of the series.

If the series represents cash flows coming to us, we want the present value to be positive and the biggerthe better.

Present Value

$1000

For example, the present value of this series of cash flows is

PV = -1000 + 500(P/F,i,1) +500(P/F,i,2) + 500(P/F,i,3)

$500

AnnuitiesA

The pattern of a regular series of annual payments comesup often enough that we give it a special name: an annuity.

By convention, an annuity starts one time period after the presentand continues for N years.

We can find its equivalent present value using another conversionfactor:

The Present

PV = A(P/A,i,N)

AnnuitiesA

The pattern of a regular series of annual payments comesup often enough that we give it a special name: an annuity.

The Present

PV = A(P/A,i,N)

As N increases, does (P/A,i,N) increase or decrease?

AnnuitiesA

The pattern of a regular series of annual payments comesup often enough that we give it a special name: an annuity.

The Present

PV = A(P/A,i,N)

As i increases, does (P/A,i,N) increase or decrease?

AnnuitiesA

The pattern of a regular series of annual payments comesup often enough that we give it a special name: an annuity.

The Present

PV = A(P/A,i,N)

As A increases, does (P/A,i,N) increase or decrease?

Present Value

$1000

$500

So a more concise expression for the present value of this series would be

PV = -1000 + 500(P/A,i,3)

Some Tips for the Assignments and Exams

Say what you're doing.

In the exams, you can get 25% credit for an answer if we can just tell what method it is you're using, and an additional 25-50% if it's the right method. You won't necessarily get exactly the numerical values we have on the model answer sheets -- in many questions there are several defensible ways of solving the problem. To make it easy for us to mark it right, say what the numbers you're writing down are supposed to be, e.g.,

``Present worth of wages = A(P/A,i,N)'’

If we're just confronted by a page of anonymous calculations, there's not much we can do except glance through it and see if any of the numbers look anything like any of the numbers in the model answer.

Use explicit conversion factors,

i.e., expressions like `(P/A,i,N)'.

Using an algebraic formula instead is more work, and there are many more opportunities to make a numerical slip.

The only time you should use the formulas is when creating a spreadsheet. Even then, it's a good idea to write out what it is you're calculating in terms of the conversion factors -- this makes it easy for us to give credit even when there's a mistake in the spreadsheet (which can easily happen).

If you don't have a copy of the text, you can find tables of conversion formulae on line, for example at:

http://www.uic.edu/classes/ie/ie201/discretecompoundinteresttables.html

Avoid excessive precision.

If you're calculating the present value of a million-dollar investment, don't bother specifying it to the nearest thousandth of a cent. Three significant figures is usually adequate, and anything after the fifth significant figure is just imaginative fiction.

Bad:

“In 10 years, your investment will be worth $121,987.12531”

Good:

“In 10 years, your investment will be worth about $122,000”

When presenting a table of numbers, they should all be given to the same level of precision, and the decimal points should align vertically. Let the table entries be in thousand-dollar or million-dollar units, so there are only a few digits on either side of the decimal point. If you do have more than three digits to one side of the decimal point, separate them into groups of 3 by commas or spaces.

Year Cashflow ($)

2013 -120000.0178

2014 -60000.09

2015 500000.111972

2016 259112.1

Year Cashflow ($ 000’s)

2013 -120.00

2014 -60.00

2015 500.00

2016 259.11

Bad

Good

Answer the question asked.

If the question asks, `` which alternative is best? '', don't just calculate the value of each alternative and leave it to the reader to figure it out. Say it explicitly.

Some Minor Details

Simple Interest and Compound Interest

In the case of simple interest, we just charge intereston the principal amount.

In the case of compound interest, we add the interest to the principal at regular intervals – the compounding interval – and charge interest on the sum.

For example, suppose we borrow $100 at 10% interest.

After N years, the amount we owe is:

Years 1 2 10 100

Simple $110 $120 $200 $1 100

Compound $110 $121 $259 $1 378 061

No-one ever uses simple interest, and we will never speak of it again.

Any compound interest rate can be described as i %per time_period1, compounded every time_period2

For example, a bank may charge 12% interest per year,compounded every month.

If time_period1 is the same as time_period2, then the interest rate is an effective interest rate.

Otherwise it is a nominal interest rate.

The appendices and formulas at the back of the bookall assume that we are talking about effective interestrates. So if we are quoted a nominal interest rate, wehave to convert it to an effective interest rate beforedoing any calculations.

For example, if a bank charges 12% interest per year, compounded every month, we can convert this to 1% interest per month, compounded every month. This is now an effective interest rate, and we can do calculations with it.

An interest rate of 36% per year, compounded monthly,

is the same thing as an interest rate of 9% per quarter, compounded monthly,

which is the same as an interest rate of 3% per month, compounded monthly.

The last of these is an effective interest rate, since the two time periods

match, and we can look up its conversion factors in the back of the book.

I put $100 in the bank for one year.

Which interest rate gives me more money at the end of the year:

12% interest per year, compounded every year

1% interest per month, compounded every month?

12% interest per year, compounded every year

gives me $100(F/P,12%,1) = $112.00

1% interest per month, compounded every month

gives me $100(F/P,1%,12) = $112.68

(1+j) = (1+i)12

Suppose I have an effective interest rate – 10% per month, compounded monthly, say – and I want to transform it to an equivalent effective annual rate. How do I do this?

The two rates are equivalent if they give me the sameamount of money after the same period of time.

So if the effective monthly rate is i and the equivalenteffective yearly rate is j, we must have

(Where `biennial’ means `every two years’)

Exercise: what effective biennial interest rate isequivalent to an effective annual rate of 10%?

(1+j) = (1+0.1)2

=1.21

So j = 21%.

Answer:

Let the effective biennial interest rate be j.

Then:

Reassuring note:

Almost every interest rate you come across in reallife will be an effective annual rate.

Continuous compounding:

Suppose we keep the nominal yearly interest rate constant-- r, say -- and decrease the compounding interval towards zero, what happens to the effective interest rate, j?

Decrease to months: j = (1 + r/12)12 – 1

Decrease to weeks: j = (1 + r/52)52 – 1

Decrease to days: j = (1 + r/365)365 – 1

r j

10% 10.471%

10% 10.506%

10% 10.516%

10% 10.517%Decrease forever: j = er – 1

Further reassuring note: continuous compoundingrarely shows up in real life. When it does, there arespecial interest tables for looking it up.

Final minor detail:

If we compound monthly or annually, we cansum up weekly and daily cash flows and treatthem as occurring at the end of the month or year. But if we’re compounding continuously,we are better off treating these cash flows ascontinuous.

There are other special interest tables for this.

Example:

The SFU library charges you $1/day for an overduebook and continuously compounds the amount youowe at a nominal rate of 10% per year.

How much do you owe after two years?

Solution:

You will owe $F, where

F = A(F/A,r,2)

r is 10% per year. A is $365/year (note that we haveto express A in terms of the same time unit as r.). We look the conversion factor up in the appropriate table.

So you will owe

F = 365(2.2140) = $808.11

The mid-period convention:

An alternative way of representing continuous cashflows is to suppose that they arrive in one lump in themiddle of the period you’re considering.

On this approximate model, you consider that youowe the library two years’s fines, which is $730,from halfway through the period, that is, after oneyear. So the approximate solution would be:

F = 730(F/P,0.1,1), or slightly better, F = 730(F/P,e0.1,1)

And looking these up in Appendix B gives $803 and $806.77 respectively.

How much does it matter?

Regular Cts. Compounding Cts. Flow

N=2, i=0.01

2.0100 2.0101 2.0201

N=2, i = 0.25

2.2500 2.2800 2.5900

N=20, i=0.01

22.0200 22.0300 22.1400

N=20,i=0.25

342.9400 519.0100 589.6500

Compare the values for (F/A,i,N):

End of Digression into Minor Details

Recap of the Important Parts

Cash flows can only be usefully compared if theyare converted to equivalent cash flows at thesame time period.

This is accomplished through the use ofconversion factors.

Among the conversion factors we have met sofar are:

Some Important Conversion Factors

Present worth of a future cashflow: (P/F,i,N)

Future worth of a present cashflow: (F/P,i,N)

Present worth of an annuity: (P/A,i,N)

Future worth of an annuity: (F/A,i,N)

General Approach:

I want to know X

I only know Y

So I write down an equation of the general form

X = Y(X/Y,i,N)

Then I look up (X/Y,i,N) in the back of the book.

More Important Conversion Factors

`Capital Recovery Factor’: (A/P,i,N)

`Sinking Fund Factor’: (A/F,i,N)

(This is how you calculate mortgage payments,for example.)

(`I want to have a million dollars by the time I retire’.)

`Capitalised Cost’: P = A/i

(Present worth of an infinite annuity.)

The purpose of all the conversion factors is to help usmake choices.

For example:

A drafting company employs 10 drafters at $800/week each. The CEO considers three alternatives:

1.Buy 8 low-end workstations at $2 000 each. Give two drafters 1 year’s notice. At the end of the year they each get $5 000 severance pay. Train the remaining eight in AutoCAD. The first training course is available in a year, and costs $2 000 per participant. After completing the course, each drafter gets a $100/week raise.

2.Buy 5 high-end workstations at $5 000 each. All the drafters get a year’s notice and $5 000 severance pay at the end of the year. Five new graduates are hired at $1 200 per week. They are trained in Pro-Engineer; to keep current, they will each need a $5 000 retraining session every six months.

3.Do nothing.

Any of these options would allow the company to service its current customers. Any money saved can be invested at 10%, which is also the cost of borrowing money. What should they do?

A drafting company employs 10 drafters at $800/week each. The CEO considers three alternatives:

1.Buy 8 low-end workstations at $2 000 each. Give two drafters 1 year’s notice. At the end of the year they each get $5 000 severance pay. Train the remaining eight in AutoCAD. The first training course is available in a year, and costs $2 000 per participant. After completing the course, each drafter gets a $100/week raise.

2. Buy 5 high-end workstations at $5 000 each. All the drafters get a year’s notice and $5 000 severance pay at the end of the year. Five new graduates are hired at $1 200 per week. They are trained in Pro-Engineer; to keep current, they will need a $5 000 retraining session every six months.

3. Do nothing.

Any of these options would allow the company to service its current customers. Any money saved can be invested at 10%, which is also the cost of borrowing money. What should they do?

What time frame should we use?

How do we represent `Do nothing’?

Sketch the cash-flow diagrams.

What non-monetary factors would matter?

Option 1

$16000 $16 000

$10 000

PW = -16,000 – (16,000+10,000)(P/F,0.1,1) + 40,000(P/A,0.1,N)(P/F,0.1,1)

$40,000

$16 000

$10 000

PW = -16,000 - ….

$40,000

P=F(P/F,i,1)

PW = -16,000 – 26,000(P/F,i,1)…

$40,000

26,000(P/F,i,1)

PW = -16,000 – 26,000(P/F,i,1) + …

$40,000

P’=40,000(P/A,i,4)

P’

PW = -16,000 – 26,000(P/F,i,1) + …

P’=40,000(P/A,i,4)

P=F(P/F,i,1)

PW = -16,000 – 26,000(P/F,i,1) + 40,000(P/A,i,4)(P/F,i,1)

P=40,000(P/A,i,4)(P/F,i,1)

Option 2

$25 000

$50 000

$100 000

PW ($000) = - 25 – …

$50,000

$50 000

$100 000

PW ($000) = - 25 – …

$50,000

P=F(P/F,i,1)

$100 000

PW ($000) = - 25 – 50(P/F,i,1)

$50,000

50,000(P/F,i,1)

$50 000

PW ($000) = - 25 – 50(P/F,i,1) + …

$50 000

PW ($000) = - 25 – 50(P/F,i,1) + …

P’=50,000(P/A,i,4)

PW ($000) = - 25 – 50(P/F,i,1) + …

P=50,000(P/A,i,4)(P/F,i,1)

PW ($000) = - 25 – 50(P/F,i,1) + 50 (P/A,i,4)(P/F,i,1)

P=50,000(P/A,i,4)(P/F,i,1)