Showing posts with label Finance. Show all posts
Showing posts with label Finance. Show all posts

Wednesday, 20 June 2018

Annual Percentage Rate Calculations

June 20, 2018 1
annual percentage rate calculations
Annual Percentage Rate Calculations
Thus far, we've been careful to use a discount rate that is consistent with the frequency of the cash flows--for example, 1% per month with monthly payments or 10% per year with annual payments. In practice, interest rates are typically stated in one of two ways, as an annual percentage rate calculations or as an annual percentage yield (APY), even though interest may be calculated and paid more often than annually.

Annual Percentage Rate (APR)

The annual percentage rate (APR) is the periodic rate times the number of periods in a year. The APR is a nominal rate, a rate "in name only". The true (effective) annual rate may be different from the APR because of the compounding frequency.
The Compounding frequency is how often interest is compounded. For example, the compounding frequency might be monthly (12 times per year), quarterly (4 times), or annually (once). The periodic rate is an effective rate, but recall that two periods of interest is more than double one. The second period's interest includes interest on the first period's interest.
With m compounding periods per year and a periodic rate of r, the APR is:
APR = (m)(r)

The Effect of Compounding Frequency on Future Value

How does compounding frequency affect future value? To answer this question, let's compare yearly, semiannually, quarterly, monthly and weekly compounding for saving $10,000 for a year at a 12% APR.
The future value of $10,000 in one year is shown in below table for all of these compounding frequencies. The APY equals the 12% APR for yearly compounding. But the table shows how the future value and APY increase as the compounding frequency increases.
Another way to understand an APY is to say that it's the total interest earned in a year (annual interest) divided by the principal. That is,
APY = annual interest/principal
For example, the annual interest for monthly compounding is $1268.25, which, divided by $10,000, gives the same 12.68%.

Table

Future Values and APYs for Various Compounding Frequencies
annual percentage rate calculations

Continuous Compounding

If more frequent compounding increases the future value, what if we compound daily, hourly, or even every minute? These are all examples of discrete compounding, where interest is compounded a finite number of times per year. If interest is compounded an infinite number of times per year, we have continuous compounding.

The APR and APY with Continuous Compounding

When m, the compounding frequency, becomes large enough, compounding becomes essentially continuous, Without giving the proof, it turns out that with continuous compounding:
Annual Percentage Rate
Where e is approximately 2.7182. The function  ex  is called an exponential function. It is usually found on a calculator with either an " " or "exp" on the key.
Tags # Finance Continue Reading
By at 1 comment:
Labels:

Tuesday, 19 June 2018

Present Value of Cash Flows Calculator

June 19, 2018 0
present value of cash flows calculator
Present Value of Cash Flows Calculator
Unlike an annuity, in some cases future cash flows vary in size. In this post, we demonstrate a few common-sense methods for computing the value of a set of unequal future cash flows and use the present value of cash flows calculator. We'll describe three of these methods through the use of the following example.

Example

Computing the Present Value of a Set of Unequal Future Cash Flows

Suppose you expect to receive the following cash flows at the times indicated:
Time                 0                    1                    2                    3
Cash flow      $3000          $2000            $8000          $5000         
If the required return is 10%, what is the total present value of these cash flows?
The total present value of these cash flows can be calculated by calculating the present value of each cash flow and then adding them together:
Present Value of Cash Flows

                                          PV = 3000+1818.182+6611.570+3756.574= $15,186.326
This calculation is illustrated in figure 1
An alternative method for calculating the total present value of our set of unequal future cash flows is called the "rollback" method: Start with the most distant cash flow ($5000 at time 3) and discount it back one period (at 10%). Its value at t = 2 is $4545.45 (=5000/1.10). And this amount to the time 2 cash flow of $8000 to get $12,545.45. Discount this amount back one period. Its value at t = 1 is $11,404.96(= 12,545.45/1.10). And the time 1 cash flow to this amount to get $13,404.96. Discount this amount back one period. Its value is $12,186.33 (= 13,404.96/1.10). Finally, this amount plus the $3000 time 0 cash flow equals the total present value of $15,186.33. Figure 2 illustrates the rollback method of calculating a present value.
Finally, many financial calculators provide a third method for valuing this unequal set of future cash flows. Because calculators are not identical, you'll have to use your own calculator's manual to learn how to use this method. There is an important advantage to using this calculator feature: If you already know the present value, but don't know the discount rate, the calculator can automatically compute the expected return for the set of unequal cash flows. This can eliminate the hassle of very tedious trial-and-error calculations.

Valuing Cash Flows at Other Points Along the Time Line

Thus far, we've calculated a present value (t=0) or a future value at t=n. But suppose we want to know the total value of a set of cash flows at some other point in time. Calculating such a value directly may require extra care, but it uses the same formulas. If you already know the present or future value, calculating such values is quite straightforward. Our next example illustrates this process by building on our last example.

Figure 1

Computing the present value of a set of unequal future cash flows.
present value of cash flows calculator

Figure 2

The rollback method for calculating a present value.
present value of cash flows calculator

Tags # Finance Continue Reading
By at No comments:
Labels:

Saturday, 7 April 2018

Annuity Payout Calculation

April 07, 2018 0
Annuity payments are a very common financial arrangement. An annuity payout calculation is a series of equal periodic payments. The payments occur regularly, year.


Valuing Annuities

Annuities occur in many different financial transactions. Monthly payments on a car loan, a student loan, or a mortgage are annuities. Monthly rent is an annuity. A paycheck, with a fixed salary, is an annuity. Lease, interest and dividend payments are annuities. Any series of equal, periodic payments is an annuity.
The majority of annuities have end-of-period payments. For example, car loans usually require end-of-month payments. If it's a 48-month loan, the first payment is made at the end of the first month and the 48th (and last) is made at the end of month 48. This kind of annuity, where payments occur at the end of each period, is called an ordinary annuity.
Other annuities, such as for a rental, require beginning-of-period payments. For a 12 month apartment lease, the first rent payment is due at the beginning of the first month and the 12th (and last) is due at the beginning of the 12th month. This kind of annuity, where payments occur at the beginning of each period, is called an annuity due.
We know the timing of payments affects value. Therefore, it's critical to know whether you are dealing with an ordinary annuity or an annuity due. We'll start by analyzing the future and present values of an ordinary annuity. Later, we'll show you how to handle an annuity due.

The Future Value of an Annuity

We started our discussion of the time value of money in the previous posts with an example of depositing money in a savings account. Now consider a savings plan for depositing the same amount every period for n periods. How much will you have at the end of the n periods?
 Let the periodic cash flow, PMT, be the amount deposited at the end of each time period (that is ,
CF1=CF2 = ........=CFn=PMT). Figure 1 illustrates the future value of an n-period annuity. 

Figure 1

The future value of an n-period annuity.
annuity payout calculation

The future value of an annuity is the total value that will have accumulated at the end of the annuity if the annuity payments are all invested at r per period. The future value of an annuity can be computed using the future value formula to value each payment and then adding up the individual values to get the total. If we start with the last payment and then adding up the individual values to get the total. If we start with the last payment at time t = n and proceed backward to the first payment at time t = 1, the future value of the annuity at time n, FVAn, is 
FVAn=PMT(1+r)0 + PMT(1+r)1 + …………………..+PMT(1+r)n-1
Figure 1 illustrates this calculation. Note that the first payment (at t = 1) earns interest for (n-1) periods, not n periods. Each subsequent payment earns interest for one less period than the previous one. Not that the last payment occurs exactly at the end of the annuity, so it doesn't earn any interest; (1+r)0 = 1.
The equation for FVAn has a PMT in every term on the right-hand side. If the PMT is factored out, the equation can be rewritten as
Annuity Payout Period
where ∑ is a summation. This equation can be simplified to 
Annuity Payout Period-------------(1)
The quantity in large brackets in equation (1) is called the future-value-annuity factor. The future-value-annuity factor, FVAr,n, is the total future value of $1.00 per period for n periods invested at r per period. The particular values for PMT, n, and r along with equation (1) are all that's needed to determine the future value of the annuity, regardless of the number of payments.
The Present Value of an Annuity
The present value of an annuity is the amount that, if invested today at r per period, could exactly provide equal payments of PMT every period for n periods. The present value of an annuity. PVAn, is simply the sum of the present values of the n individual payments:
Annuity Payout Period
The present value of an n-period annuity is illustrated in figure 2. Because the cash flows or payments are all identical, we can rewrite this as
Annuity Payout Period
This equation for PVAn can also be simplified; it becomes
Annuity Payout Period-------------(2)

Figure 2

    The present value of an n-period annuity.
annuity payout calculation

The quantity in large brackets in equation (2) is called the present-value-annuity factor. The present-value-annuity factor, PVAFr,n, is the total present value of an annuity of $ 1.00 per period for n periods discounted at r per period. The particular values for PMT, n, and r are all that is needed to determine the present value of the annuity.

Calculating Annuity Payments

We have shown how to compute the present and future value of an annuity, given a set of payments and a discount rate. When you borrow money, the amount is the present value , and the annuity is the loan payments. We can solve for the payments by rearranging equation (2);
Annuity Payout Period
 Now suppose you are getting ahead of the game and saving money regularly rather than paying off a loan. The accumulated amount is a future value. We can solve for the amount that must be saved regularly to accumulate a given future value, this time by rearranging equation (1):

Annuity Payout Period

Amortizing a Loan

A loan amortization schedule shows how the loan is paid off over time. That is, it shows how the principal (the original amount borrowed) and interest are paid. Because an installment loan is an annuity, an amortization schedule for such a loan shows the relationships among the payments, principal and interest rate.
To create an amortization schedule, start with the amount borrowed. To this amount add the first period's interest and then subtract the first period's payment. The result is the remaining balance, which is the starting amount for the second period. Repeat this procedure each period until the remainder becomes zero at the end of the last period.

Calculating the Discount Rate and Number of Annuity Payments

In addition to solving for the payments, future value, or present value of an annuity, we can solve for the discount rate or the number of annuity payments. However, unlike the payments, we cannot always rearrange our equation to solve for these variables. Instead, the equation must be solved using trial and error. So the calculator is especially convenient for calculating these variables because it performs the tedious trial-and-error calculations automatically.

Tables 1

    A Loan Amortization Schedule
annuity payout calculation

Valuing Annuities Not Starting Today
Sometimes, annuities start at a time other than right away (where the first payment is at t=1). The present value of such an annuity can be computed from the difference between the present values of two other annuities. The first annuity goes from now until the end of the one in question. The second annuity goes from now until the start of the one in question. The difference between the two values is the value of the annuity in question.

Perpetuities

An annuity that goes on forever is called a perpetuity. Although perpetuitites actually exist in some situations, the most important reason for studying them is that they can be used as a simple and fairly accurate approximation of a long-term annuity.
As we showed in this figure 2 of the previous post, the present-value factor becomes smaller as n becomes larger. Therefore, later payments in a long annuity add little to the present value of the annuity.

Figure 3

    Duplicating the annuity cash flows for a "postponed" annuity.
annuity payout calculation

For example, at a required return of 10% per year, the present value of getting $100 in 30 years is only $5.73. It is a mere 85 cents if payment is going to take 50 years. As it turns out, the present value of an annuity has a maximum value, no matter how many payments are expected. That maximum value is the value of a perpetuity.
To examine the present value of a perpetuity, we can start with the present value of an annuity and see what happens when the life of the annuity, n, becomes very large. Let's start by rewriting equation (2), the present-value-of-an-annuity formula:
Annuity Payout Period
Annuity Payout Period
Written this way, you can see what happens when n becomes large. The first term on the right-hand side of the bottom expression is not affected by n. But the second term gets smaller because (1+r)n  gets larger when n increases. As n gets really big, the second term goes to zero. Therefore, the present value of a perpetuity is
Annuity Payout Period

Valuing an Annuity Due

The payments for an annuity due occur at the beginning of each period instead of at the end. Because each payment occurs one period earlier, an annuity due has a higher present value than a comparable ordinary annuity. Likewise, an annuity due has a higher future value than a comparable ordinary annuity because each payment has an additional period to compound. In fact, annuity payout calculation a simple way to value an annuity due is to multiply the value of a comparable ordinary annuity by (1 + r).
Tags # Finance Continue Reading
By at No comments:
Labels:

Present Value of Ordinary Annuity

April 07, 2018 0
We introduced the concept of present values and future values in our brief discussion of the Time-Value-of-Money Principle in last Posts. We also defined three different rates of return: expected, required and realized. The expected return is the return you expect to earn if you make the investment. 

Single Cash Flows

The required return is the minimum return you must expect to get to be willing to make the investment. The realized return is the return you actually earned on an investment during a given time period. We showed you that finding the present value or the future value of a single cash flow is a simple calculation. After a brief recap, we'll extend its logic to deal with multiple cash flows.

Finding the Future Value of an Investment

The future value (FV) is the value an investment will grow to after a given time period. Let's say you invest $1000 today. Table 1 Shows the amount of money you'll have accumulated at the end of each of the next six years if the bank is paying 10% interest. After one year.
FV1 = $1000 +$100 = $1100
In the second year, you'll earn $110 more -10% interest on your accumulated investment (=[0.10]1100), for a total of
                                                                       FV2 = $1100 + $110=$1210 
The extra $10 of interest earned in the second year is called compound interest. Compound interest is a way of computing interest earned where interest is earned on both the original investment and on the reinvested interest. As you can see in table 1, the interest earned each year grows because of compound interest.
Table 1 also shows how fast your $1000 investment grows if invested funds earn simple interest instead of compound interest. Simple interest is a way of computing interest earned where interest is earned on only the original investment. Note that in year 1 with simple interest, the interest earned is $100, the same as with compound interest. However, after that, the story changes. In year 2 with simple interest, the interest earned is again $100. No interest is earned on the first year's $100 interest. All other years also earn only $100, 10% of the original investment.
Would you rather earn compound interest or simple interest? Obviously, if the interest rates are the same, you'll have more money with compound interest than with simple interest. Because of today's technology, the use of simple interest has largely disappeared.
One way to find a future value is to calculate interest each year, adding it to the previous year's balance, and accumulating the result for the desired number of years. In table 1, we stopped at six years. Suppose you were investing for 20 years. It's repetitive and such a large number of hand calculations can cause errors. Consequently, we use shortcut methods whenever we can. One shortcut method of finding future values is to use the future-value formula:
                                                                   The Future-Value Formula
                                                                    FVn=PV(1+r)n = PV(FVFr,n)------------------------(1)
The amount (1+r)n above is called the future-value factor. The future-value factor, FVFr,n, is the value $1.00 will grow to if it's invested at r per period for n periods. Figure 1 in this post is a graph of FVFr,n as a function of n and r. As you can see there, future value is directly related to both time and the discount rate. The larger the discount rate, the larger the future value. For positive discount rates, the more time, the larger the future value.

Table 1

    Future Value of an Investment of $1000
Present Value of Ordinary Annuity

Figure 1

The future-value factor, FVFr,n as a function of time and various discount rates.
Present Value of Ordinary Annuity

An easier way to make our future-value calculation is to use a financial calculator: Put in PV=1000, n=6, r=10%, and PMT = 0, then compute FV = $1771.56. Note that, for most financial calculators, you enter the discount rate as a whole percent, 10, not as a decimal number, 0.10. Throughout the rest of the book, we'll show you such calculator calculations in a standardized format. The amount the calculator solves for is in bold type. The other amount are inputs.
N = 6 r = 10 PV = 1000  PMT = 0 FV = 1771.56

Present Value of a Future Cash Flow

Now, let's find the present value of an expected future cash flow. The present value (PV) is the amount that if invested today at r per period would provide a given future value at time n. We can compute a PV using the present-value formula:
                                                                  The Present-Value Formula
                                                               PV= FVn[1/(1+r)n] = FVn(FVFr,n)--------------------------(2)
The present-value formula is simply a rearrangement of the future-value formula. We are solving for PV instead of FV. In the present-value formula, the amount [1/(1+r)n] is called the present-value factor. The present-value factor, PVFr,n is the amount that, if invested today at r per period will grow to exactly $1.00 n years from today.
Figure 2 is a graph of PVFr,n as a function of time and various discount rates. It shows that present value is inversely related to both time and the discount rate. That is, the larger the discount rate, the smaller the present value. For positive discount rates, the more time until you get the cash flow, the smaller the present value will be. Like two kinds on a seesaw, when one goes up the other goes down.

Solving for a Return

If you look back at the basic calculator formula, you can see how the present-value formula is part of it. You can also see that if you know any four of the five input variables, the formula can be solved for the fifth.

Figure 2

The present-value factor, PVFr,n as a function of time and as a function of time and various discount rates.
Present Value of Ordinary Annuity

 For example, to find a PV, we put in FV (the expected future cash flow), n (the time the cash flow will occur), r (the required return), and PMT = 0, However, suppose you already know PV from a market price, but you don't know the discount rate. You can rearrange the formula to solve for the expected return. Solving for r, with PMT = 0, we get
                                                                     r = (FV/PV)1/n - 1

Solving for the Number of Time Periods

We also rearrange the basic calculator formula to solve for n, using natural logarithms. However, it's much easier to let the calculator do the work.

Tags # Finance Continue Reading
By at No comments:
Labels:

Friday, 6 April 2018

Concept of Time Value of Money in Financial Management

April 06, 2018 0
Concept of Time Value of Money in Financial Management
Concept of Time Value of Money in Financial Management

Have you ever paid for something with monthly payments? Suppose you wanted to buy a $10,000 car and were told the payments would be $273.11 per month for 48 months. How would you know whether you were being offered a great deal, a fair deal, or a bad deal?
The Time Value of Money
Now suppose you have $10,000 to invest for a long time and someone tells you about an investment that will double your money, without any risk: Invest your $10,000 now, and you'll get back $20,000 in 15 years. How does this compare with other no-risk investments?
This and next coming posts will teach you how to answer such question; it's devoted entirely to the Time-Value-of-Money Principle. You'll learn how to value at one point in time cash flows that actually occur at other points in time. We develop the logic underlying these calculations and show you procedures for solving problems using a financial calculator. We urge you, however, not to use these calculator procedures like cookbook recipes. Understanding the logic will prepare you to apply the Time-Value-of-Money Principle in the business world to new types of problems, ones that don't fit neatly into classroom examples.
Like you, companies also have to choose among investments and borrowing alternatives. In fact, their success depends on those choices. Financial decisions are measured by their net present value (NPV). Recall that NPV is the present value of the expected future cash flows minus the cost. The NPV is the value created or lost by a decision. Therefore, to be successful, companies must find positive-NPV opportunities and avoid negative-NPV choices.

The Time Value of Money and The Principles of Finance

  • Time-Value-of-Money: Note that the value of a cash flow depends on when it will occur.
  • Two-Sided Transactions: Be specific about the timing of cash flows to be fair to both sides of a transaction.
  • Risk-Return Trade-Off: Recognize that a higher-risk investment has a higher required return. Therefore, the time value of money is especially important to the profitability of long-term investments.
  • Capital Market Efficiency: Use efficient capital markets to estimate an investment's expected and required returns.
Tags # Finance Continue Reading
By at No comments:
Labels: