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Inside Collection:

Collection by: Rupinder Sekhon. E-mail the author

# Mathematics of Finance

Module by: Rupinder Sekhon. E-mail the author

Summary: This chapter covers principles of finance. After completing this chapter students should be able to: solve financial problems that involve simple interest; solve problems involving compound interest; find the future value of an annuity; find the amount of payments to a sinking fund; find the present value of an annuity; and find an installment payment on a loan.

## Chapter Overview

In this chapter, you will learn to:

1. Solve financial problems that involve simple interest.
2. Solve problems involving compound interest.
3. Find the future value of an annuity, and the amount of payments to a sinking fund.
4. Find the present value of an annuity, and an installment payment on a loan.

## Simple Interest and Discount

### Section Overview

In this section, you will learn to:

1. Find simple interest.
2. Find present value.
3. Find discounts and proceeds.

### SIMPLE INTEREST

It costs to borrow money. The rent one pays for the use of money is called the interest. The amount of money that is being borrowed or loaned is called the principal or present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money. Although the interest rate is often specified for a year, it may be specified for a week, a month, or a quarter, etc. The credit card companies often list their charges as monthly rates, sometimes it is as high as 1.5% a month.

### SIMPLE INTEREST

If an amount PP size 12{P} {} is borrowed for a time tt size 12{t} {} at an interest rate of rr size 12{r} {} per time period, then the simple interest is given by

I = P r t I = P r t size 12{I=P cdot r cdot t} {}

The total amount AA size 12{A} {} also called the accumulated value or the future value is given by

{} A = P + I = P + Pr t A = P + I = P + Pr t size 12{A=P+I=P+"Pr"t} {}

or A=P1+rtA=P1+rt size 12{A=P left (1+ ital "rt" right )} {}

Where interest rate rr size 12{r} {} is expressed in decimals.

### DISCOUNTS AND PROCEEDS

Banks often deduct the simple interest from the loan amount at the time that the loan is made. When this happens, we say the loan has been discounted. The interest that is deducted is called the discount, and the actual amount that is given to the borrower is called the proceeds. The amount the borrower is obligated to repay is called the maturity value.

### DISCOUNT AND PROCEEDS

If an amount MM size 12{M} {} is borrowed for a time tt size 12{t} {} at a discount rate of rr size 12{r} {} per year, then the discount DD size 12{D} {} is

D = M r t D = M r t size 12{D=M cdot r cdot t} {}

The proceeds PP size 12{P} {}, the actual amount the borrower gets, is given by

P = M D P = M D size 12{P=M - D} {}

P = M Mrt P = M Mrt size 12{P=M - ital "Mrt"} {}

or P=M1rtP=M1rt size 12{P=M left (1 - ital "rt" right )} {}

Where interest rate rr size 12{r} {} is expressed in decimals.

## Compound Interest

### Section Overview

In this section you will learn to:

1. Find the future value of a lump-sum.
2. Find the present value of a lump-sum.
3. Find the effective interest rate.

In the Section 2, we did problems involving simple interest. Simple interest is charged when the lending period is short and often less than a year. When the money is loaned or borrowed for a longer time period, the interest is paid (or charged) not only on the principal, but also on the past interest, and we say the interest is compounded.

Suppose we deposit $200 in an account that pays 8% interest. At the end of one year, we will have$200+$200.08=$2001+.08=$216$200+$200.08=$2001+.08=$216 size 12{$"200"+$"200" left ( "." "08" right )=$"200" left (1+ "." "08" right )=$"216"} {}. Now suppose we put this amount,$216, in the same account. After another year, we will have $216+$216.08=$2161+.08=$233.28$216+$216.08=$2161+.08=$233.28 size 12{$"216"+$"216" left ( "." "08" right )=$"216" left (1+ "." "08" right )=$"233" "." "28"} {}.

So an initial deposit of $200 has accumulated to$233.28 in two years. Further note that had it been simple interest, this amount would have accumulated to only $232. The reason the amount is slightly higher is because the interest ($16) we earned the first year, was put back into the account. And this $16 amount itself earned for one year an interest of$16.08=$1.28$16.08=$1.28 size 12{$"16" left ( "." "08" right )=$1 "." "28"} {}, thus resulting in the increase. So we have earned interest on the principal as well as on the past interest, and that is why we call it compound interest. Now suppose we leave this amount,$233.28, in the bank for another year, the final amount will be $233.28+$233.28.08=$233.281+.08=$251.94$233.28+$233.28.08=$233.281+.08=$251.94 size 12{$"233" "." "28"+$"233" "." "28" left ( "." "08" right )=$"233" "." "28" left (1+ "." "08" right )=$"251" "." "94"} {}.

Now let us look at the mathematical part of this problem so that we can devise an easier way to solve these problems.

After one year, we had

$200 1 + . 08 =$ 216 $200 1 + . 08 =$ 216 size 12{$"200" left (1+ "." "08" right )=$"216"} {}
(16)

After two years, we had

$216 1 + . 08$ 216 1 + . 08 size 12{$"216" left (1+ "." "08" right )} {} (17) But$216=$2001+.08$216=$2001+.08 size 12{$"216"=$"200" left (1+ "." "08" right )} {}, therefore, the above expression becomes$ 200 1 + . 08 1 + . 08 = $233 . 28$ 200 1 + . 08 1 + . 08 = $233 . 28 size 12{$"200" left (1+ "." "08" right ) left (1+ "." "08" right )=$"233" "." "28"} {} After three years, we get$ 200 1 + . 08 1 + . 08 1 + . 08 $200 1 + . 08 1 + . 08 1 + . 08 size 12{$"200" left (1+ "." "08" right ) left (1+ "." "08" right ) left (1+ "." "08" right )} {}
(18)

Which can be written as

$200 1 + . 08 3 =$ 251 . 94 $200 1 + . 08 3 =$ 251 . 94 size 12{$"200" left (1+ "." "08" right ) rSup { size 8{3} } =$"251" "." "94"} {}
(19)

Suppose we are asked to find the total amount at the end of 5 years, we will get

$200 1 + . 08 5 =$ 293 . 87 $200 1 + . 08 5 =$ 293 . 87 size 12{$"200" left (1+ "." "08" right ) rSup { size 8{5} } =$"293" "." "87"} {}
(20)

We summarize as follows:

The original amount $200 =$200 The amount after one year $200 1 + . 08 =$ 216 The amount after two years $200 1 + . 08 2 =$ 233 . 28 The amount after three years $200 1 + . 08 3 =$ 251 . 94 The amount after five years $200 1 + . 08 5 =$ 293 . 87 The amount after t years $200 1 + . 08 t The original amount$200 = $200 The amount after one year$200 1 + . 08 = $216 The amount after two years$200 1 + . 08 2 = $233 . 28 The amount after three years$200 1 + . 08 3 = $251 . 94 The amount after five years$200 1 + . 08 5 = $293 . 87 The amount after t years$200 1 + . 08 t size 12{ matrix { "The original amount" {} # "$200"="$200" {} ## "The amount after one year" {} # "$200" left (1+ "." "08" right )=$"216" {} ## "The amount after thwo years" {} # "$200" left (1+ "." "08" right ) rSup { size 8{2} } =$"233" "." "28" {} ## "The amount after three years" {} # "$200" left (1+ "." "08" right ) rSup { size 8{3} } =$"251" "." "94" {} ## "The amount after five years" {} # "$200" left (1+ "." "08" right ) rSup { size 8{5} } =$"293" "." "87" {} ## "The amount after "t" years" {} # "$200" left (1+ "." "08" right ) rSup { size 8{t} } {} } } {} Banks often compound interest more than one time a year. Consider a bank that pays 8% interest but compounds it four times a year, or quarterly. This means that every quarter the bank will pay an interest equal to one-fourth of 8%, or 2%. Now if we deposit$200 in the bank, after one quarter we will have $2001+.084$2001+.084 size 12{$"200" left (1+ { { "." "08"} over {4} } right )} {} or$204.

After two quarters, we will have $2001+.0842$2001+.0842 size 12{$"200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{2} } } {} or$208.08.

After one year, we will have $2001+.0844$2001+.0844 size 12{$"200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{4} } } {} or$216.49.

After three years, we will have $2001+.08412$2001+.08412 size 12{$"200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{"12"} } } {} or$253.65, etc.

The original amount $200 =$200 The amount after one quarter $200 1 + . 08 4 =$ 204 The amount after two quarters $200 1 + . 08 4 2 =$ 208 . 08 The amount after one year $200 1 + . 08 4 4 = 216 . 49 The amount after two years$200 1 + . 08 4 8 = $234 . 31 The amount after three years$200 1 + . 08 4 12 = $253 . 65 The amount after five years$200 1 + . 08 4 20 = $297 . 19 The amount after t years$200 1 + . 08 4 4t The original amount $200 =$200 The amount after one quarter $200 1 + . 08 4 =$ 204 The amount after two quarters $200 1 + . 08 4 2 =$ 208 . 08 The amount after one year $200 1 + . 08 4 4 = 216 . 49 The amount after two years$200 1 + . 08 4 8 = $234 . 31 The amount after three years$200 1 + . 08 4 12 = $253 . 65 The amount after five years$200 1 + . 08 4 20 = $297 . 19 The amount after t years$200 1 + . 08 4 4t size 12{ matrix { "The original amount" {} # "$200"="$200" {} ## "The amount after one quarter" {} # "$200" left (1+ { { "." "08"} over {4} } right )=$"204" {} ## "The amount after two quarters" {} # "$200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{2} } =$"208" "." "08" {} ## "The amount after one year" {} # "$200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{4} } ="216" "." "49" {} ## "The amount after two years" {} # "$200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{8} } =$"234" "." "31" {} ## "The amount after three years" {} # "$200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{"12"} } =$"253" "." "65" {} ## "The amount after five years" {} # "$200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{"20"} } =$"297" "." "19" {} ## "The amount after "t" years" {} # "$200" left (1+ { { "." "08"} over {4} } right ) rSup { size 8{4t} } {} } } {}
(21)

Therefore, if we invest a lump-sum amount of PP size 12{P} {} dollars at an interest rate rr size 12{r} {}, compounded nn size 12{n} {} times a year, then after tt size 12{t} {} years the final amount is given by

A = P 1 + r n nt A = P 1 + r n nt size 12{A=P left (1+ { {r} over {n} } right ) rSup { size 8{ ital "nt"} } } {}

### Example 10

#### Problem 1

If a bank pays 7.2% interest compounded monthly, what is the effective interest rate?

Interest can be compounded yearly, semiannually, quarterly, monthly, daily, hourly, minutely, and even every second. But what do we mean when we say the interest is compounded continuously, and how do we compute such amounts. When interest is compounded "infinitely many times", we say that the interest is compounded continuously. Our next objective is to derive a formula to solve such problems, and at the same time put things in proper perspective.

Suppose we put $1 in an account that pays 100% interest. If the interest is compounded once a year, the total amount after one year will be$11+1=$2$11+1=$2 size 12{$1 left (1+1 right )=$2} {}. If the interest is compounded semiannually, in one year we will have$11+1/22=$2.25$11+1/22=$2.25 size 12{$1 left (1+1/2 right ) rSup { size 8{2} } =$2 "." "25"} {} If the interest is compounded quarterly, in one year we will have$11+1/44=$2.44$11+1/44=$2.44 size 12{$1 left (1+1/4 right ) rSup { size 8{4} } =$2 "." "44"} {}, etc. We show the results as follows:  Frequency of compounding Formula Total amount Annually$ 1 1 + 1 $1 1 + 1 size 12{$1 left (1+1 right )} {} $2 Semiannually$ 1 1 + 1 / 2 2 $1 1 + 1 / 2 2 size 12{$1 left (1+1/2 right ) rSup { size 8{2} } } {} $2.25 Quarterly$ 1 1 + 1 / 4 4 $1 1 + 1 / 4 4 size 12{$1 left (1+1/4 right ) rSup { size 8{4} } } {} $2.44140625 Monthly$ 1 1 + 1 / 12 12 $1 1 + 1 / 12 12 size 12{$1 left (1+1/"12" right ) rSup { size 8{"12"} } } {} $2.61303529 Daily$ 1 1 + 1 / 365 365 $1 1 + 1 / 365 365 size 12{$1 left (1+1/"365" right ) rSup { size 8{"365"} } } {} $2.71456748 Hourly$ 1 1 + 1 / 8760 8760 $1 1 + 1 / 8760 8760 size 12{$1 left (1+1/"8760" right ) rSup { size 8{"8760"} } } {} $2.71812699 Every second$ 1 1 + 1 / 525600 525600 $1 1 + 1 / 525600 525600 size 12{$1 left (1+1/"525600" right ) rSup { size 8{"525600"} } } {} $2.71827922 Continuously$ 1 2 . 718281828 . . . $1 2 . 718281828 . . . size 12{$1 left (2 "." "718281828" "." "." "." right )} {} $2.718281828... We have noticed that the$1 we invested does not grow without bound. It starts to stabilize to an irrational number 2.718281828... given the name "ee size 12{e} {}" after the great mathematician Euler.

In mathematics, we say that as nn size 12{n} {} becomes infinitely large the expression 1+1nn1+1nn size 12{ left (1+ { {1} over {n} } right ) rSup { size 8{n} } } {} equals ee size 12{e} {}.

Therefore, it is natural that the number ee size 12{e} {} play a part in continuous compounding. It can be shown that as nn size 12{n} {} becomes infinitely large the expression 1+rnnt=ert1+rnnt=ert size 12{ left (1+ { {r} over {n} } right ) rSup { size 8{ ital "nt"} } =e rSup { size 8{ ital "rt"} } } {}.

Therefore, it follows that if we invest $P$P size 12{$P} {} at an interest rate rr size 12{r} {} per year, compounded continuously, after tt size 12{t} {} years the final amount will be given by A=PertA=Pert size 12{A=P cdot e rSup { size 8{ ital "rt"} } } {}. ### Example 11 #### Problem 1 If$3500 is invested at 9% compounded continuously, what will the future value be in four years?

Next we learn a common-sense rule to be able to readily estimate answers to some finance as well as real-life problems. We consider the following problem.

### Example 12

#### Problem 1

If an amount is invested at 7% compounded continuously, what is the effective interest rate?

### Example 13

#### Problem 1

If an amount is invested at 7%, estimate how long will it take to double.

By doing a few similar calculations we can construct a table like the one below.

 Annual interest rate 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Number of years to double money 70 35 23 18 14 12 10 9 8 7

The pattern in the table introduces us to the law of 70.

Definition 1: The Law of 70:
The number of years required to double money = 70 ÷ interest rate

It is a good idea to familiarize yourself with the law of 70, as it can help you to estimate many problems mentally.

### Example 14

#### Problem 1

If the world population doubles every 35 years, what is the growth rate?

We summarize the concepts learned in this chapter in the following table:

### THE COMPOUND INTEREST

1. If an amount PP size 12{P} {} is invested for tt size 12{t} {} years at an interest rate rr size 12{r} {} per year, compounded nn size 12{n} {} times a year, then the future value is given by

A = P 1 + r n nt A = P 1 + r n nt size 12{A=P left (1+ { {r} over {n} } right ) rSup { size 8{ ital "nt"} } } {}
(30)
2. If a bank pays an interest rate rr size 12{r} {} per year, compounded nn size 12{n} {} times a year, then the effective interest rate is given by

r e = 1 1 + r n n 1 r e = 1 1 + r n n 1 size 12{r rSub { size 8{e} } =1 left (1+ { {r} over {n} } right ) rSup { size 8{n} } - 1} {}
(31)
3. If an amount PP size 12{P} {} is invested for tt size 12{t} {} years at an interest rate rr size 12{r} {} per year, compounded continuously, then the future value is given by

A = Pe rt A = Pe rt size 12{A= ital "Pe" rSup { size 8{ ital "rt"} } } {}
(32)
4. The law of 70 states that

The number of years to double money = 70 ÷ interest rate

## Annuities and Sinking Funds

### Section Overview

In this section, you will learn to:

1. Find the future value of an annuity.
2. Find the amount of payments to a sinking fund.

In Section 2 and Section 4, we did problems where an amount of money was deposited lump sum in an account and was left there for the entire time period. Now we will do problems where timely payments are made in an account. When a sequence of payments of some fixed amount are made in an account at equal intervals of time, we call that an annuity. And this is the subject of this section.

To develop a formula to find the value of an annuity, we will need to recall the formula for the sum of a geometric series.

A geometric series is of the form: a+ar+ar2+ar3+...+arna+ar+ar2+ar3+...+arn size 12{a+ ital "ar"+ ital "ar" rSup { size 8{2} } + ital "ar" rSup { size 8{3} } + "." "." "." + ital "ar" rSup { size 8{n} } } {}.

The following are some examples of geometric series.

3 + 6 + 12 + 24 + 48 3 + 6 + 12 + 24 + 48 size 12{3+6+"12"+"24"+"48"} {}
(33)
2 + 6 + 18 + 54 + 162 2 + 6 + 18 + 54 + 162 size 12{2+6+"18"+"54"+"162"} {}
(34)
37 + 3 . 7 + . 37 + . 037 + . 0037 37 + 3 . 7 + . 37 + . 037 + . 0037 size 12{"37"+3 "." 7+ "." "37"+ "." "037"+ "." "0037"} {}
(35)

In a geometric series, each subsequent term is obtained by multiplying the preceding term by a number, called the common ratio. And a geometric series is completely determined by knowing its first term, the common ratio, and the number of terms.

In the example, a+ar+ar2+ar3+...+arn1a+ar+ar2+ar3+...+arn1 size 12{a+ ital "ar"+ ital "ar" rSup { size 8{2} } + ital "ar" rSup { size 8{3} } + "." "." "." + ital "ar" rSup { size 8{n - 1} } } {} the first term of the series is aa size 12{a} {}, the common ratio is rr size 12{r} {}, and the number of terms are nn size 12{n} {}.

In your algebra class, you developed a formula for finding the sum of a geometric series. The formula states that the sum of a geometric series is

a r n 1 r 1 a r n 1 r 1 size 12{ { {a left [r rSup { size 8{n} } - 1 right ]} over {r - 1} } } {}
(36)

We will use this formula to find the value of an annuity.

Consider the following example.

### Example 17

Robert needs $5000 in three years. How much should he deposit each month in an account that pays 8% in order to achieve his goal? ### Example 18 #### Problem 1 A business needs$450,000 in five years. How much should be deposited each quarter in a sinking fund that earns 9% to have this amount in five years?

If the payment is made at the beginning of each period, rather than at the end, we call it an annuity due. The formula for the annuity due can be derived in a similar manner. Reconsider Example 15, with the change that the deposits be made at the beginning of each month.

### Example 19

If at the beginning of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years? So, in the case of an annuity due, to find the future value, we increase the number of periods nn size 12{n} {} by 1, and subtract one payment. The Future Value of an Annuity due = m 1 + r / n nt + 1 1 r / n m The Future Value of an Annuity due = m 1 + r / n nt + 1 1 r / n m size 12{"The Future Value of an Annuity due"= { {m left [ left (1+r/n right ) rSup { size 8{ ital "nt"+1} } - 1 right ]} over {r/n} } - m} {} Most of the problems we are going to do in this chapter involve ordinary annuity, therefore, we will down play the significance of the last formula. We mentioned the last formula only for completeness. Finally, it is the author's wish that the student learn the concepts in a way that he or she will not have to memorize every formula. It is for this reason formulas are kept at a minimum. But before we conclude this section we will once again mention one single equation that will help us find the future value, as well as the sinking fund payment. ### The Equation to Find the Future Value of an Ordinary Annuity, Or the Amount of Periodic Payment to a Sinking Fund If a payment of mm size 12{m} {} dollars is made in an account nn size 12{n} {} times a year at an interest rr size 12{r} {}, then the future value AA size 12{A} {} after tt size 12{t} {} years is A = m 1 + r / n nt 1 r / n A = m 1 + r / n nt 1 r / n size 12{A= { {m left [ left (1+r/n right ) rSup { size 8{ ital "nt"} } - 1 right ]} over {r/n} } } {} ## Present Value of an Annuity and Installment Payment ### Section Overview In this section, you will learn to: 1. Find the present value of an annuity. 2. Find the amount of installment payment on a loan. In Section 4, we learned to find the future value of a lump sum, and in Section 6, we learned to find the future value of an annuity. With these two concepts in hand, we will now learn to amortize a loan, and to find the present value of an annuity. Let us consider the following problem. ### Example 20 #### Problem 1 Suppose you have won a lottery that pays$1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?

We now consider another problem that involves the same logic.

### Example 23

#### Problem 1

The Orange computer company needs to raise money to expand. It issues a 10-year $1,000 bond that pays$30 every six months. If the current market interest rate is 7%, what is the fair market value of the bond?

An amount of $500 is borrowed for 6 months at a rate of 12%. Make an amortization schedule showing the monthly payment, the monthly interest on the outstanding balance, the portion of the payment contributing toward reducing the debt, and the outstanding balance. Most of the other applications in this section's problem set are reasonably straight forward, and can be solved by taking a little extra care in interpreting them. And remember, there is often more than one way to solve a problem. ## Classification of Finance Problems We'd like to remind the reader that the hardest part of solving a finance problem is determining the category it falls into. So in this section, we will emphasize the classification of problems rather than finding the actual solution. We suggest that the student read each problem carefully and look for the word or words that may give clues to the kind of problem that is presented. For instance, students often fail to distinguish a lump-sum problem from an annuity. Since the payments are made each period, an annuity problem contains words such as each, every, per etc.. One should also be aware that in the case of a lump-sum, only a single deposit is made, while in an annuity numerous deposits are made at equal spaced time intervals. Students often confuse the present value with the future value. For example, if a car costs$15,000, then this is its present value. Surely, you cannot convince the dealer to accept $15,000 in some future time, say, in five years. Recall how we found the installment payment for that car. We assumed that two people, Mr. Cash and Mr. Credit, were buying two identical cars both costing$15, 000 each. To settle the argument that both people should pay exactly the same amount, we put Mr. Cash's cash of $15,000 in the bank as a lump-sum and Mr. Credit's monthly payments of xx size 12{x} {} dollars each as an annuity. Then we make sure that the future values of these two accounts are equal. As you remember, at an interest rate of 9% the future value of Mr. Cash's lump-sum was$15,0001+.09/1260$15,0001+.09/1260 size 12{$"15","000" left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } } {}, and

the future value of Mr. Credit's annuity was x1+.09/12601.09/12x1+.09/12601.09/12 size 12{ { {x left [ left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } - 1 right ]} over { "." "09"/"12"} } } {}.

To solve the problem, we set the two expressions equal and solve for xx size 12{x} {}.

The present value of an annuity is found in exactly the same way. For example, suppose Mr. Credit is told that he can buy a particular car for $311.38 a month for five years, and Mr. Cash wants to know how much he needs to pay. We are finding the present value of the annuity of$311.38 per month, which is the same as finding the price of the car. This time our unknown quantity is the price of the car. Now suppose the price of the car is yy size 12{y} {}, then

the future value of Mr. Cash's lump-sum is y1+.09/1260y1+.09/1260 size 12{y left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } } {}, and

#### Problem 6

Mr. A has just donated \$25,000 to his alma mater. Mr. B would like to donate an equivalent amount, but would like to pay by monthly payments over a five year period. If the interest rate is 8.2%, determine the size of the monthly payment?

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#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

### Reuse / Edit:

Reuse or edit collection (?)

#### Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

#### Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.

| Reuse or edit module (?)

#### Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

#### Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.