1. Define and distinguish between \textbf{present value} and \textbf{future value} in the context of compound interest.

2. Use the compound interest formula:

\[

FV = PV\left(1 + i\right)^{n}

\]

to calculate the future value of an investment, where

* \( FV \): future value

* \( PV \): present value (principal)

* \( i \): periodic interest rate (per compound period)

* \( n \): number of compounding periods for the investment

3. Use the rearranged formula:

\[

PV = \dfrac{FV}{\left(1 + i\right)^{n}}

\]

to determine the present value required to reach a given future value.

4. Solve problems involving compound interest with annual, semi-annual, quarterly, monthly, and daily compounding frequencies.

5. Apply compound interest formulas to real-world financial scenarios such as savings accounts, investment growth, and loan balances.

1. What is the fair market value today of 3 payments of \$1,000. The first payment is to be made 150 days from now, the second 225 days from now and the final one 300 days from now. Use 8.4\% simple interest for all calculations.

<div style="text-align: right;">[Click here for the solution](https://youtu.be/SQhAIwiVOwc)</div>

2. You deposit \$2,000 now, \$1,500 in 6 months, and \$1,000 in 1 year. If the interest rate is 4.2\% per annum simple interest, What is the total future value of all deposits at the end of 18 months?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/2wadK5_GH5E)</div>

3. A \$6,000 investment earns 3.5\% simple interest for 4 months, then 5\% for the next 8 months. Calculate the total interest earned and the final value after 1 year.

<div style="text-align: right;">[Click here for the solution](https://youtu.be/xhKMpIKqCFA)</div>

4. A payment of \$950 is due in 9 months. You want to replace it with an equivalent payment made in 2 months. What amount would be equivalent if the interest rate is 5.5\% per annum simple interest?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/zzLoWKDcHYc)</div>

Lecture material for this class come from Sections 5.1 -- 5.2 and can be found below. This material is considered review material and so it is not covered in depth.

1. **Video: [The Basics of CI](https://youtu.be/y99qpJXtx3Q)**

The total number of compounding periods is \( n = m \cdot t \). The value of \( m \) determines the compounding frequency.

* Annually: \( m = 1 \)

* Semi-annually: \( m = 2 \)

* Quarterly: \( m = 4 \)

* Monthly: \( m = 12 \)

* Daily: \( m = 365 \) <br><br>

2. **Video: [Compound Interest FV](https://youtu.be/YyV8hybwRT0)**

**Video: [Compound Interest PV](https://youtu.be/xu3JoFjM82w)**

Compound interest} is interest calculated on the initial principal \textbf{and} also on the accumulated interest from previous periods. It is calculated using the formula: \[

FV = PV(1 + i)^n

Where:

\begin{itemize}

* \( FV \): Future Value (total amount after interest)

* \( PV \): Present Value (initial investment or loan)

* \( i \): Interest rate per compounding period \( \left( i = \frac{r}{m} \right) \)

* \( n \): Total number of compounding periods \( \left( n = m \cdot t \right) \)

* \( r \): Annual nominal interest rate (as a decimal)

* \( m \): Number of compounding periods per year

* \( t \): Time in years <br><br>

1. You invest \$2,500 at an annual interest rate of 5.2\% compounded monthly for 3 years. What is the future value of the investment?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/5w9hc7thqwE)</div>

2. \$1,800 is deposited into an account earning 4.5\% annual interest compounded quarterly for 5 years. Find the amount in the account at the end of the term.

<div style="text-align: right;">[Click here for the solution](https://youtu.be/tLfjhf_mshg)</div>

3. A deposit of \$6,000 is made into a savings account with 3.9\% interest compounded annually for 10 years. What will the investment be worth at maturity?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/9CMaL7HUPEs)</div>

4. You want to have \$10,000 in 4 years. The account offers 6\% interest compounded semi-annually. How much should you invest today?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/p5kNaZKyv7g)</div>

5. How much should be invested now to grow to \$7,500 in 3 years at 5.25\% interest compounded monthly?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/d2MkmbyQmG8)</div>

6. A future amount of \$12,000 is needed in 6 years. If the account pays 4.8\% compounded quarterly, what is the present value of the investment?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/MmyATTVMOK4)</div>

Additional problems that are typically done in class (with video solutions) can be found here:

* [Present and Future Value (Compound Interest)](https://theelementsmath.github.io/M114/compound-interest.html\#present-and-future-values)

1. Determine the \textbf{number of compounding periods} (\( n \)) using the formula:

\[

n = \dfrac{\ln(FV/PV)}{\ln(1+i)}.

\]

2. Identify and calculate the \textbf{term} (\( t \)) of an investment or loan in years, months, or other units,

\[

t = \frac{n}{m},

\]

where \( m \) is the number of compounding periods per year.

3. Calculate the \textbf{periodic interest rate}:

\[

i = \left(\dfrac{FV}{PV} \right)^{1/n} - 1.

\]

where \( r \) is the annual nominal interest rate.

4. Calculate the \textbf{nominal interest rate}:

\[

r = i \times m,

\]

where \( r \) is the annual nominal interest rate.

1. You invest \$4,000 at an annual interest rate of 5\% compounded monthly for 3 years. What is the future value of the investment?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/Csdbw7AIGSA)</div>

2. You invest \$7,200 in an account that earns 4.25\% compounded annually for 8 years. How much will the investment be worth at the end of the term?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/4vo1QE6fmXE)</div>

3. You want to have \$10,000 in 4 years. If the interest rate is 5.5\% compounded semi-annually, how much should you invest today?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/AZg71adyIfo)</div>

4. A savings goal of \$8,000 must be met in 3 years. If the bank offers 4.2\% compounded monthly, what is the required initial deposit?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/hCXRx8LGUe4)</div>

Lecture material for this class come from Section 5.3 and can be found below. This material is considered review material and so it is not covered in depth.

1. **Video: [Periodic and Nominal Interest Rate](https://youtu.be/n2Z1aLFtaiE)**

The **annual nominal interest rate** is the stated annual interest rate. The **periodic interest rate** is the interest rate applied each compounding period. Examples include:

* Annual compounding: \( i = r \)

* Semi-annual: \( i = \frac{r}{2} \)

* Quarterly: \( i = \frac{r}{4} \)

* Monthly: \( i = \frac{r}{12} \)

* Daily: \( i = \frac{r}{365} \)

\[

i = \left(\dfrac{FV}{PV} \right)^{1/n} - 1.

\] <br><br>

2. **Video: [Term of Investment](https://youtu.be/rfa1WBzjMaY)**

The **term of the investment** is how long the money is invested or borrowed. The **number of compounding periods** is the number of times interest is applied over the entire term.

\[

n = \dfrac{\ln(FV/PV)}{\ln(1+i)}.

\] <br><br>

1. \$3,000 grows to \$3,828.81 in a bank account earning 6\% interest compounded annually. How long was the money invested?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/ala74SWNy-Y)</div>

2. A deposit of \$2,500 grows to \$3,262.78 at an interest rate of 4.5\% compounded quarterly. How many years did it take to reach that value?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/FTwVwzi5Nf0)</div>

3. \$4,000 is invested for 4 years and grows to \$5,041.60. The interest is compounded annually. What is the annual nominal interest rate?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/3plLnxnEtHU)</div>

4. An investment of \$6,500 grows to \$8,312.34 over 6 years, compounded monthly. Find the annual interest rate.

<div style="text-align: right;">[Click here for the solution](https://youtu.be/fnOVWqeE4gI)</div>

Additional problems that are typically done in class (with video solutions) can be found here:

* [Calculating the Periodic Interest Rate and Number of Compounding Periods](https://theelementsmath.github.io/M114/compound-interest.html\#calculating-the-periodic-interest-rate-and-number-of-compounding-periods)

1. Define a payment stream and describe how it differs from a single lump sum investment.

2. Calculate the future value and present value of a stream of compound interest payments.

3. Define the concept of equivalent payments and when it applies in financial contexts.

4. Determine the time value equivalence of payments made at different times and under different interest rates.

5. Convert a series of unequal payments into a single equivalent payment at a specified point in time.

6. Solve problems involving multiple interest rates applied over different time intervals.

7. Use piecewise calculations to find the total future or present value when rates change partway through an investment or loan.

8. Define continuously compounded interest and compare it to standard compound interest.

9. Use the formula:

\[

FV = PV \cdot e^{rt}

\]

to calculate future value with continuous compounding.

1. An investment of \$3,000 grows to \$3,915.79 in 4 years with quarterly compounding. What annual interest rate was earned?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/vQc-7wXZqlk)</div>

2. \$5,000 grows to \$6,734.29 at 6.5\% annual interest compounded monthly. How long was the money invested?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/M5UPZ7Sv0Hg)</div>

3. A deposit of \$1,200 increases to \$1,579.35 over 5 years with monthly compounding. Find the annual nominal interest rate

<div style="text-align: right;">[Click here for the solution](https://youtu.be/SNO9GQOu4Bs)</div>

4. A \$1,500 investment becomes \$2,100 at 5\% compounded quarterly. Find the number of years required.

<div style="text-align: right;">[Click here for the solution](https://youtu.be/sesqd-K7qTM)</div>

Lecture material for this class come from Sections 5.4 -- 5.7 and can be found below. This material is considered review material and so it is not covered in depth.

1. **Video: [Equivalent Payments and Payment Streams](https://youtu.be/HLUDkVUHNwM)**

A **payment stream** is a series of cash flows (payments or deposits) made at regular or irregular intervals over time. These are often used in loan payments, investments, or savings plans.

* Each payment may earn interest depending on when it is made and when it is evaluated.

* Time diagrams are useful for visualizing and organizing payment streams.

* Payment streams can be evaluated using the concept of future value or present value by summing each individual amount's value at the chosen focal date. <br><br>

2. **Equivalent payments** are payments made at different times that have the same value when brought to a common point in time using interest calculations.

* Payments are equivalent if their present or future values are equal at a given focal date.

* Used to simplify or compare payment schedules.

Steps:

* Choose a focal date.

* Convert each payment to its value at the focal date using simple or compound interest.

* Equate values and solve for unknowns (e.g., payment amount, date, interest rate).

<br><br>

3. In some financial problems, the interest rate changes at specific intervals during the term of the investment or loan.

Approach:

* Split the time period into segments where each rate applies.

* Apply the interest formula separately for each segment.

* Use the output of one period as the input (principal) for the next.

Formula:

FV = PV \cdot (1 + i_1)^{n_1} \cdot (1 + i_2)^{n_2} \cdots (1 + i_k)^{n_k}

Where:

* \( i_k \) is the periodic interest rate during segment \( k \)

* \( n_k \) is the number of periods for rate \( i_k \)

4. The \textcolor{red}{Continuous compounding} assumes interest is being added at every possible instant. It represents the mathematical limit of compounding frequency.

Formula:

FV = PV \cdot e^{rt}

Where:

* \( e \approx 2.71828 \) is Euler's number

* \( r \) is the annual nominal interest rate (in decimal)

* \( t \) is time in years

<br><br>

1. Three payments of \$800 are made at the end of years 1, 2, and 3 into an account earning 5\% compound interest annually. What is the total value of the payment stream at the end of year 3?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/Tqoxyk1YPeo)</div>

2. What single payment today is equivalent to two future payments of \$600 in 1 year and \$700 in 2 years, assuming 6\% annual compound interest?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/i35Q8YoohtI)</div>

3. A business wants to replace two payments of \$2,000 due at the end of years 2 and 4 with a single equivalent payment at the end of year 3. If the interest rate is 5\% compounded annually, what should the equivalent payment be?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/Q_GTSXLL-9c)</div>

4. An investment of \$5,000 earns 4\% interest compounded annually for the first 2 years and then 6\% compounded annually for the next 3 years. What is the value of the investment at the end of 5 years?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/0_4bwUvlFeg)</div>

5. A loan of \$10,000 is repaid over 4 years. For the first 2 years, the interest rate is 5\% annually, and then it changes to 7\% annually. What is the total amount owed at the end of 4 years?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/DjsulaiwpTc)</div>

6. An investment of \$2,500 grows to \$3,200 under continuous compounding over 5 years. What is the annual interest rate?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/evQHobJj7GY)</div>

7. What is the future value of \$1,200 invested for 6 years at an interest rate of 4.2\% compounded continuously?

<div style="text-align: right;">[Click here for the solution](https://youtu.be/KuNy6RN3H3k)</div>

Additional problems that are typically done in class (with video solutions) can be found here:

* [Equivalent Payments and Payment Streams](https://theelementsmath.github.io/M114/compound-interest.html\#equivalent-payments-and-payment-streams)

* [Changing Interest Rates](https://theelementsmath.github.io/M114/compound-interest.html\#changing-interest-rates-1)

* [Continuously Compounded Interest](https://theelementsmath.github.io/M114/compound-interest.html\#continuously-compounded-interest)