Principles of Finance C708 V4 - UG: Unit 4 Module 8

You plan to invest $100,000 in a 3 year Certificate of Deposit that has a 5% compound interest rate. What is its future value? A) $115,927 B) $115,763 C) $115,000 D) $105,000

B) $115,763 *FV = $100,000 × 1.053 = $115,763 Using Calculator: N = 3, I/Y = 5, PV = 100,000. [CPT] FV = $115,763.

Discounting*

1. Opportunity Cost: The cost of not having the cash on hand at a certain point of time. If the investor/creditor had the cash s/he could spend it, but since it has been invested/loaned out, s/he incurs the cost of not being able to spend it. 2. Inflation: The real value of a single dollar decreases over time with inflation. That means that even if everything else is constant, a $100 item will retail for more than $100 in the future. Inflation is generally positive in most countries at most times (if it's not, it's called deflation, but it's rare). 3. Risk: There is a chance that you will not get your money back because it is a bad investment; the debtor defaults. You require compensation for taking on that risk. 4. Liquidity: Investing or loaning out cash necessarily reduces your liquidity.

Time period assumption

Business profit or losses are measured on a timely basis.

You plan to invest $100,000 in a 3 year Certificate of Deposit that has a simple interest rate of 5%. What is its future value? A) $115,763 B) $115,927 C) $115,000 D) $105,000

C) $115,000 *FV = $100,000 + (3 × 5% × $100,000) = $115,000.

What is the future value in 30 years of $100,000 invested today in a savings account earning a 1% simple interest rate every year (rounded up to the nearest dollar)? A) $30,000 B) More than $134,785 C) $130,000 D) $134,785

C) $130,000 *FV = $100,000 + (30 × 1% × $100,000) = $130,000.

Calculating FV

Calculating FV is a matter of identifying PV, i (or r), and t (or n), and then plugging them into the compound or simple interest formula. Key Points *The "present" can be moved based on whatever makes the problem easiest. Just remember that moving the date of the present also changes the number of periods until the future for the FV. *To find FV, you must first identify PV, the interest rate, and the number of periods from the present to the future. *The interest rate and the number of periods must have consistent units. If one period is one year, the interest rate must be X% per year, and vice versa. there are four things that you need to know in order to find the FV: How does the interest accrue? Is it simple or compounding interest? Present Value Interest Rate Number of periods Equation: FV = PV ∙(1 + i)t

Calculating Present Value (PV)

Calculating the present value (PV) is a matter of plugging FV, the interest rate, and the number of periods into an equation. Key Points *The first step is to identify if the interest is simple or compound. Most of the time, it is compound. *The interest rate and number of periods must have consistent units. *The PV is what a future sum is worth today given a specific interest rate (often called a "discount rate").

What is the future value in 30 years of $100,000 invested today in a savings account earning a 1% compound interest rate every year (rounded up to the nearest dollar)? A) More than $134785 B) $130,000 C) $30,000 D) $134,785

D) $134,785 *FV = $100,000 × 1.0130 = $134,785 Using Calculator: N = 30, I/Y = 1, PV = 100,000. [CPT] FV = $134,785.

You have $300,000 that you want to invest in a one year Certificate of Deposit (CD) with a 4% annual interest rate. What will be the value of that CD in a year? A) $301,200 B) $315,000 C) $420,000 D) $312,000

D) $312,000 *FV = $300,000 × 1.04 = $312,000 Using Calculator: N = 1, I/Y = 4, PV = 300,000. [CPT] FV = $312,000.

Discount Rate

Discounting is the procedure of finding what a future sum of money is worth today. Key Point *The discount rate represents some cost (or group of costs) to the investor or creditor. *Some costs to the investor or creditor are opportunity cost, liquidity cost, risk, and inflation. *The discount rate is used by both the creditor and debtor to find the present value of an amount of money.

Present Value (PV)

Example: FV = PV ∙(1+rt) Equation: PV = FV/(1+i)^n

Effective Annual Rate (EAR)

Finding the Effective Annual Rate (EAR) accounts for compounding during the year, and is easily adjusted to different period durations. Key Points *The units of the period (e.g., one year) must be the same as the units in the interest rate (e.g., 7% per year). *When interest compounds more than once a year, the effective interest rate (EAR) is different from the nominal interest rate. *The equation in skips the step of solving for EAR, and is directly usable to find the present or future value of a sum.

FV Periodic Compounding

Finding the FV (At) given the PV (Ao), nominal interest rate (r), number of compounding periods per year (n), and number of years (t). Equation: A(t) = A0(1+r/n)^[nt]

Real Interest Rates

Fisher Equation: The nominal interest rate is approximately the sum of the real interest rate and inflation. Equation: i ≈ r + π

Compound Interest

Interest is paid at the total amount in the account, which may include interest earned in previous periods. interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest. FV=PV(1+i)t

Number of Periods

Key Points * A period is just a general term for a length of time. It can be anything—one month, one year, one decade—but it must be clearly defined and fixed. * For both simple and compound interest, the number of periods varies jointly with FV and inversely with PV. * The number of periods is also part of the units of the discount rate: if one period is one year, the discount rate must be defined as X% per year. If one period is one month, the discount rate must be X% per month. Equation: FV = PV (1 + i)n

Multi-period investments

Multi-period investments require an understanding of compound interest, incorporating the time value of money over time. Key Points *A dollar today is worth more than a dollar tomorrow, and the time value of money must take into account foregone opportunities. Single period investments are relatively simple to calculate in terms of future value, applying the interest rate to a present value a single time. *Multi-period investments require a slightly more complex equation, where interest gets compounded based on the number of periods the investment spans. *As a result of multiple periods, it is usually a good idea to calculate the average rate of return (cumulatively) over the lifetime of the investment.

Multi-period Investment

Multi-period investments take place over more than one period (usually multiple years). They can either accrue simple or compound interest. Key Points *Investments that accrue simple interest have interest paid based on the amount of the principal, not the balance in the account. *Investments that accrue compound interest have interest paid on the balance of the account. This means that interest is paid on interest earned in previous periods. *Simple interest increases the balance linearly, while compound interest increases it exponentially. Equation: FV = PV ∙(1 + rt)

Present Value and Future Value

Present value (PV) and future value (FV) measure how much the value of money has changed over time. Key Points *The future value (FV) measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return. The FV is calculated by multiplying the present value by the accumulation function. *PV and FV vary jointly: when one increases, the other increases, assuming that the interest rate and number of periods remain constant. *As the interest rate (discount rate) and number of periods increase, FV increases or PV decreases.

Simple Interest Formula

Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously earned. FV = PV ∙(1 + rt)

Single-Period Investment

Since the number of periods (n or t) is one, FV = PV (1 + i), where i is the interest rate. Key Points *Single-period investments use a specified way of calculating future and present value. *Single-period investments take place over one period (usually one year). *In a single-period investment, you only need to know two of the three variables PV, FV, and i. The number of periods is implied as one since it is a single-period.

FV of a single payment

The FV is related to the PV by being i% more each period. Equation: FV = PV ∙(1 + i)^n

FV of a single payment

The FV of multiple cash flows is the sum of the future values of each cash flow. FV= PV(1+i)n

Future Value

The Future Value can be calculated by knowing the present value, interest rate, and number of periods, and plugging them into an equation. Key Point *The future value is the value of a given amount of money at a certain point in the future if it earns a rate of interest. *The future value of a present value is calculated by plugging the present value, interest rate, and number of periods into one of two equations. *Unless otherwise noted, it is safe to assume that interest compounds and is not simple interest.

Present Value, multiple flows

The PV of multiple cash flows is simply the sum of the present values of each individual cash flow. Key Points *To find the PV of multiple cash flows, each cash flow must be discounted to a specific point in time and then added to the others. *To discount annuities to a time prior to their start date, they must be discounted to the start date, and then discounted to the present as a single cash flow. *Multiple cash flow investments that are not annuities unfortunately cannot be discounted by any other method but by discounting each cash flow and summing them together.

The Time Value of Money

The Time Value of Money is the concept that money is worth more today that it is in the future. Key Points *Being given $100 today is better than being given $100 in the future because you don't have to wait for your money. *Money today has a value (present value, or PV) and money in the future has a value (future value, or FV). *The amount that the value of the money changes after one year is called the interest rate (i). For example, if money today is worth 10% more in one year, the interest rate is 10%. Equation: FV = PV ∙ (1 + rt)

Interest Rate (i or r)

The cost of not having money for one period, or the amount paid on an investment per year.

Calculating the effective annual rate

The effective annual rate for interest that compounds more than once per year. Equation: r =(1 +r/n )^n − 1

EAR with Continuous Compounding

The effective rate when interest compounds continuously. Equation: r =e^i − 1

Compounding period

The length of time between the points at which interest is paid.

Principal

The money originally invested or loaned, on which basis interest and returns are calculated.

Perpetuities

The present value of a perpetuity is simply the payment size divided by the interest rate and there is no future value. Key Points *Perpetuities are a special type of annuity; a perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. *To find the future value of a perpetuity requires having a future date, which effectively converts the perpetuity to an ordinary annuity until that point. *Perpetuities with growing payments are called Growing Perpetuities; the growth rate is subtracted from the interest rate in the present value equation.

Time value of money

The value of an asset accounting for a given amount of interest earned or inflation accrued over a given period.

Fractional time periods

The value of money and the balance of the account may be different when considering fractional time periods. Key Points *The balance of an account only changes when interest is paid. To find the balance, round the fractional time period down to the period when interest was last accrued. *To find the PV or FV, ignore when interest was last paid and use the fractional time period as the time period in the equation. *The discount rate is really the cost of not having the money over time, so for PV/FV calculations, it doesn't matter if the interest hasn't been added to the account yet.

Solving for n

This formula allows you to figure out how many periods are needed to achieve a certain future value, given a present value and an interest rate. Equation: n = log(FV)−log(PV)/log(1+i)

Time Value of Money

Time Value of Money Time value of money is integral in making the best use of a financial player's limited funds. Key Points *Money today is worth more than the same quantity of money in the future. You can invest a dollar today and receive a return on your investment. *Loans, investments, and any other deal must be compared at a single point in time to determine if it's a good deal or not. *The process of determining how much a future cash flow is worth today is called discounting. It is done for most major business transactions during investing decisions in capital budgeting. Equation: FV = PV ∙ (1 + i)t

Accrue

To add, or grow.

Future Value, Multiple Flows

To find the FV of multiple cash flows, sum the FV of each cash flow. Key Points *The FV of multiple cash flows is the sum of the FV of each cash flow. *To sum the FV of each cash flow, each must be calculated to the same point in the future. *If the multiple cash flows are a fixed size, occur at regular intervals, and earn a constant interest rate, it is an annuity. There are formulas for calculating the FV of an annuity.

Comparing Interest Rates

Variables, such as compounding, inflation, and the cost of capital must be considered before comparing interest rates. Key Points *A nominal interest rate that compounds has a different effective rate (EAR), because interest is accrued on interest. *The Fisher Equation approximates the amount of interest accrued after accounting for inflation. *A company will theoretically only invest if the expected return is higher than their cost of capital, even if the return has a high nominal value.

Amortization

When borrowing money to be paid back via a number of installments over time, it is important to understand the time value of money and how to build an amortization schedule. Key Points *Amortization of a loan is the process of identifying a payment amount for each period of repayment on a given outstanding debt. *Repaying capital over time at an interest rate requires an amortization schedule, which both parties agree to prior to the exchange of capital. This schedule determines the repayment period, as well as the amount of repayment per period. *Time value of money is a central concept to amortization. A dollar today, for example, is worth more than a dollar tomorrow due to the opportunity cost of other investments. *When purchasing a home for $100,000 over 30 years at 8% interest (consistent payments each month), for example, the total amount of repayment is more than 2.5 times the original principal of $100,000. This is the process of scheduling intervals of payment over time to pay back an existing debt, taking into account the time value of money. Equation: P=A∙1 − (1/1+r)^n/r

Compound interest

a n interest rate applied to multiple applications of interest during the lifetime of the investment.

Quarter

a period of three consecutive months (1/4 of a year).

Annuity

a specified income payable at stated intervals for a fixed or a contingent period, often for the recipient's life, in consideration of a stipulated premium paid either in prior installment payments or in a single payment. For example, a retirement annuity paid to a public officer following his or her retirement.

Inflation

an increase in the general level of prices or in the cost of living.

Multi-period investment

an investment that takes place over more than one periods

Single-period investment

an investment that takes place over one period, usually one year.

Interest Rate (Discount Rate)

represented as either i or r. This is the percentage of interest paid each period

Incremental cash flows

the additional money flowing in or out of a business due to a project.

Growth rate

the percentage by which the payments grow each period. PVGP = A1/(i − g)

Interest rate

the percentage of an amount of money charged for its use per some period of time. It can also be thought of as the cost of not having money for one period, or the amount paid on an investment per year.

Net present value

the present value of a project or an investment decision determined by summing the discounted incoming and outgoing future cash flows resulting from the decision. Equation: PV=∑^nt=0 FV>t/(1+i)^t Sum FV The PV of an investment is the sum of the present values of all its payments. the cash flows need to have three traits: 1. Constant payment size 2. Payments occur at fixed intervals 3. A constant interest rate

Discounting

the process of determining how much money paid/received in the future is worth today. You discount future values of cash back to the present using the discount rate.

Future Value (FV)

the value of the money in the future.

Present Value (PV)

the value of the money today.

Periods (t or n)

units of time. Usually one year.