Understand Finance - Finance Tutorials and Finance How To's

There are a number of cases where you will need to find the growth rate for a firm.
This tutorial teaches you how to find that growth rate in three easy ways.

The first way is that it will simply be given to you in the problem. Easy enough!

The second way to find growth is to use the equation:

In this equation, ROE is the return on equity which will be given to you in the problem. The lowercase r is the retention rate. Remember before when we said that companies have two options with their net income? They can either pay a dividend or retain the money for themselves. The percentage that they keep for themselves is their retention rate or r in our equation. Let’s try a quick practice problem to illustrate this point.

Practice Problem 1

Finding The Growth Rate

Busboy Corporation has a return on equity of 18% and pays out 40% of their net income in the form of a dividend. What is Busboy Corporation’s growth rate?

Solution

The problem tells us that ROE is 18%. They tell us that they pay out 40% of net income as a dividend. If they pay out 40%, they must retain the rest! So we know that r is 60%. So to find growth, our equation looks like:

In this first practice problem, we find that growth equals 10.8%.

The third and final way to find growth is by using the time value of money. If you’re unfamiliar with the time value of money, this simple tutorial on present value vs. future value will help quite a bit. To better illustrate this point, let’s try another problem.

Practice Problem 2

Finding Growth Through Time Value of Money

File Incorporated paid a dividend of $2.18 nine years ago. The firm recently paid a dividend of $4.36. What is File Inc.’s growth rate?

Solution

Take a look at this problem on a timeline.

When using the time value of money to find the growth rate, we just solve for the interest. In this example, we set the present value to -$2.18, the future value to $4.36, n=9, and we solve for I. You’ll get 8% if all went well. That’s our growth rate!

So that’s it! Growth will either be given to you explicitly in the problem, or you can calculate it yourself using the return on equity equation or the time value of money. If you’re still somewhat foggy on this tutorial or you need some additional help, please post a message in the comments section below!

In this tutorial, we will show you how to calculate a firm’s taxable income. This is really an accounting exercise more than anything, but it is worth posting on the site since I understand how much we all love financial accounting! You will need to be familiar with income statements to handle these problems, so we’ll walk you right through that process.

Income Statements can come in a variety of forms, but one thing they will all have in common is that revenue (or sales) will be on top and net income will be on the bottom. When a problem asks you to find taxable income, you’re really just looking for earnings before taxes or EBT. Let’s jump right into a problem to show how this all comes together.

Sample Problem 1

Calculating Taxable Income

Bird Seed Corporation had sales last year of $1,000,000. Their cost of goods sold amounted to 40% of sales and operating expenses amounted to 15% of sales. The firm also has packaging equipment with annual depreciation of $80,000. Bird Seed Corporation has an outstanding loan of $2,000,000 with an interest rate of 9%. Calculate the firm’s taxable income.

Solution

The problem gives us a bunch of numbers, so let’s put them into an income statement. We know sales will go up top, and then we just start subtracting out the costs. Your finished income statement should look like this:

What you should notice is that we just started with sales and subtracted out our
costs, including depreciation. You’re always going to take depreciation out in this step,
since depreciation helps us lower our tax bill. Also notice that we have a line in our
income statement called Interest Expense. This is simply the interest that we
pay each year on any debt. In this problem, Bird Seed Corporation had a loan from the
local bank. The amount of interest they pay each year is simply 9% times $2,000,000. So
once we’re all done, we have the answer to our problem! The taxable income is
$190,000! That problem was pretty straight forward. Let’s try
another.

Sample Problem 2
Black & Blue Collection Agency had revenue last year of $500,000. Their cost of goods
sold was $100,000 and operating expenses consisted of salaries of $80,000, labor of
$60,000, and maintenance on their phone system of $5,000. The phone system was has annual
depreciation of $25,000. The firm has a $250,000 loan with National Bank at 8% interest.
Finally, the firm paid dividends to its shareholders of $100,000 and received dividends
from their investments totaling $30,000. Calculate the firm’s taxable income.

Solution
So this problem does two things differently. First, it breaks the operating expenses up
to their individual items. That’s no big deal! We’ll just separate them on our income
statement. But what is that stuff about dividend income and dividends paid?

There’s a simple rule here to remember on dividends. We will completely ignore the
dividends we pay out! These do not show up on our income statement when we’re finding
taxable income. However, dividends we receive have to show up! If we’re getting dividend
income from our investments, the government will definitely want their fair share.
Currently, 70% of dividend income is excluded from taxes. In other words, we will have to
include 30% of dividend income on our income statement. Always add this amount in as a
positive number because it raises our taxable income.

As you can see in this one, we ignored the dividends that the company paid. Then, we
know they received $30,000 in dividend income. But, we’re only responsible for 30% of
that, so 30% times $30,000 gave us the $9,000 that we included on our income
statement.

Feeling good about taxable income? Give these two problems a shot on your own!

Practice Problem 1
Clemens Rocket Corporation had sales last year of $5,000,000. Cost of goods sold amounted
to 50% of sales and cash operating expenses amounted to 12% of sales. Depreciation was
$100,000 for the year. The firm paid dividends of $250,000 and received dividend income
of $150,000. The firm also had an interest expense of $400,000. Calculate Clemens’
taxable income.

Solution
If all went well, you should have arrived at a taxable income of
$1,445,000. If this isn’t the answer you calculated, make sure you’ve
added only 30% of the dividend income. Remember that you will ignore the dividends the
company paid.

Practice Problem 2
Red Roses Incorporated had sales of $2,000,000 last year with cost of goods sold of
$1,200,000. Operating expenses amounted to 10% of sales. The firm also recorded $50,000
in depreciation expense for the year. The firm has $1,000,000 in outstanding bonds with
an 8% coupon rate. Finally, the firm paid $65,000 in dividends for the year. Calculate
Red Roses Incorporated’s taxable income.

Solution
Did you get $470,000 for your taxable income? If not, check to make sure
you didn’t include any dividend income. They paid dividends but remember that we don’t
include those in our income statement.

That’s a long tutorial but I hope it helped! If you have any questions or need help
with your homework, please leave me a comment below!

Yield to maturity sounds pretty scary, right? I’ve seen students cringe when they hear the term. However, once you realize that calculating the yield to maturity on a bond is just a simple time value of money problem, it will become your new best friend. Naturally, a good grasp of annuities will be necessary for this tutorial.

So what in the world is yield to maturity? Simply put, it is the rate of return that you would earn if you bought a bond today and held it until the bond matured. If you dig a little deeper, you’ll find out that it is also the internal rate of return of the bond. So now that we know what the term means, let’s take a look at how we calculate it with our financial calculator.

Whenever you see a question that asks you to find the yield to maturity of a bond, you need to think of the letter I. Since we can think of a bond as just a time value of money problem in disguise, we solve for I to get the yield to maturity. Let’s take a look at a problem to illustrate this point.

Sample Problem

Family Moving Company has recently issued 10 year bonds with a 5% coupon rate. The bonds make annual coupon payments and are currently selling for $950 in the market. What is the yield to maturity of these bonds?

Solution

We can draw a timeline to see what’s going on with this bond. Your timeline should look like this:

In year 0, or today, we pay $950 for this bond. The bond has a 5% coupon rate, so every year we receive a $50 coupon payment. Finally, in 10 years when the bond matures, we will receive par value of $1,000. To find the yield to maturity, we just need to solve for I/YR in our financial calculator.

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 10 Step 4. Set PMT = $50 Step 5. Set PV = -$950 Step 6. Set FV = $1,000 Step 7. Compute I/Y

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 10 Step 4. Set PMT = $50 Step 5. Set PV = -$950 Step 6. Set FV = $1,000 Step 7. Compute I/YR

Nice work! You just found that the yield to maturity is 5.67% for this bond! There are a few points to clear up here. At time 0, we pay $950 for the bond. Since this is a cash outflow, it must go into the calculator as a negative number. We will receive the payments and the par value at the end as cash inflows, so both of these values are positive. Calculating yield to maturity really is that easy! Try some of these practice problems on your own to make sure you’ve got it.

Practice Problem 1

CardCounter Casino has bonds on the market with 15 years left to maturity. The bonds make semiannual coupon payments and have a coupon rate of 16%. If the current market price of these bonds is $1,085, what is the yield to maturity?

Solution

If you got a yield to maturity of 14.59%, you’re doing a great job! If not, there were a few tricks in this one that might have got you. Since the bonds make semiannual coupon payments, you’ll need to set P/Y=2 and N=30. Also, you’ll find out that the bonds make $160 in coupon payments annually, so they will make $80 payments semiannually.

Practice Problem 2

Taco Town issued 30 year bonds 22 years ago. The bonds now sell in the market for $915, make semiannual coupon payments, and have a coupon rate of 11.5%. What is the yield to maturity?

Solution

Tough question, eh? 13.26% is the correct answer. The bonds have 8 years left until maturity but with semiannual coupon payments, N=16. Whenever we deal with bonds, we’re only going to worry about the time that is left. Also, the bonds pay $115 per year in coupon payments, but since the bonds are semiannual, divide by 2 to get PMT=57.50.

That’s all for this one, folks. If you have any questions, leave a comment or email me and I’ll be happy to help!

Calculating the cost of equity can be tricky since there are two different types of equity. Firms have retained earnings (the money left over after dividends are paid) as the first type of equity. The second form of equity comes from issuing new shares of stock. Don’t fear, we’ll discuss both of these types of equity in this tutorial!

Let’s talk about retained earnings first. So in the course of business, a
firm will generate some sales or revenue. They pay cost of goods sold, taxes, interest expense, and so on. The final amount then is their net income. So what do they do with net income? The firm either pays a dividend to their shareholders, or they keep that money for themselves. The portion that the firm keeps is called their retained earnings!

Retained earnings have a cost, so to find that cost, we use the following equation:

In this equation, D₁ is the next dividend the firm will pay, P₀ is the current price of a share of stock, and g is the growth rate. In most problems, P₀ will be given to you. However, we will usually have to find the growth rate,g, on our own.

There are three different ways to get the growth rate,g. The first way is that it will simply be given to you in the problem. Easy enough! The second way to find growth is to use the equation:

In this equation, ROE is the return on equity which will be given to you in the problem. The lowercase r is the retention rate. Remember before when we said that companies have two options with their net income? They can either pay a dividend or retain the money for themselves. The percentage that they keep for themselves is their retention rate or r in our equation. Let’s try a quick practice problem to illustrate this point.

Practice Problem 1

Finding The Growth Rate

Busboy Corporation has a return on equity of 18% and pays out 40% of their net income in the form of a dividend. What is Busboy Corporation’s growth rate?

Solution

The problem tells us that ROE is 18%. They tell us that they pay out 40% of net income as a dividend. If they pay out 40%, they must retain the rest! So we know that r is 60%. So to find growth, our equation looks like:

In our practice problem, we find that growth equals 10.8%.

The third and final way to find growth is by using the time value of money. To better illustrate this point, let’s try another problem.

Practice Problem 2

Finding Growth Through Time Value of Money

File Incorporated paid a dividend of $2.18 nine years ago. The firm recently paid a dividend of $4.36. What is File Inc.’s growth rate?

Solution

Take a look at this problem on a timeline.

When using the time value of money to find the growth rate, we just solve for the interest. In this example, we set the present value to -$2.18, the future value to $4.36, n=9, and we solve for I. You’ll get 8% if all went well. That’s our growth rate!

So now that we understand how to find the growth rate in a number of ways, let’s go ahead and try to find the cost of retained earnings with this practice problem.

Practice Problem 3

Finding the Cost of Retained Earnings

Baseball Bats & Bags Corporation’s common stock currently sells for $50 per share. The firm will pay a dividend of $2.00 per share next year, and their earnings are expected to grow at 10% annually. What is their cost of retained earnings?

Solution

Sounds tough, but it really isn’t. We are given the stock price, P₀ and the growth rate of 10%. Now, remember that D₁ is the next dividend that the firm will pay. Clearly in this problem, they tell us that the next dividend will be $2.00. Putting it all together, your equation looks like this:

That’s it! You should have found that their cost of retained earnings is 14%. Now let’s take a look at the cost of new common stock.

When a firm decides they’re going to issue new common stock, they will have to pay something called floatation costs to the bank that handles their stock issue. To account for these floatation costs, we simply deduct them from the current stock price. Here’s the equation for the cost of new common stock:

Notice that this equation is almost identical to the cost of retained earnings. The only difference is that the denominator is NP₀ instead of simply P₀. Not a big deal… NP₀ is just price - floatation costs.

Practice Problem 4

Finding The Cost of New Common Stock

Cookie Company has decided to raise money by issuing new shares of common stock. The firm recently paid a dividend of $3.25 and their common stock currently sells for $30 per share. Dividends are expected to grow at 12% per year for the foreseeable future and the company will have to pay $5 per share in floatation costs. What is Cookie Company’s cost of new common stock?

Solution

Right away, you should recognize that the dividend we are given is D₀ but we need D₁. Any time you see terms such as recently paid, you know you’re dealing with D₀. To find D₁, use this simple equation:

Finally, we have to find NP₀. We know that the price of
the shares are $30 and the floatation costs are $5. So, NP₀
is $25. Putting this all together with the cost of new common stock equation, we get:

So the cost of new common stock for Cookie Company is 26.56%. Try
some practice problems on your own to make sure you’ve got it!

Practice Problem 5

New Common Stock Cost

Bill’s Blinds is building a new curtain factory and needs to issue some new common stock. Their stock sells for $45 per share and they will pay floatation costs of $2.50 on each new share they sell. The company’s return on equity is 12% and they retain 75% of their net income. The most recent dividend paid was $1.25 per share. What will be their cost of new common stock?

Solution

Did you get 12.21%? Make sure you calculate D₁ for this one
since you’re given D₀ in the problem!

Practice Problem 6

Finding The Cost of Internal Equity

Sunglass World is trying to determine their cost of internal equity. Five years ago, the firm paid a dividend of $3.40 and their most recent dividend was $4.55. Current shares of stock sell for $65, but if they issue new shares, they will have to pay floatation costs of $2.75 per share. What is their cost of retained earnings?

Solution

If you got 13.42%, you have mastered cost of equity! If not, there are a couple of tricks to this one. Since you’re looking for the cost of retained earnings, ignore flotation costs! Those only come into play when you’re issuing new stock! Also, you know that D₀ is $4.55 so make sure you find D₁.

That was one long tutorial! If any part is still unclear, please leave a comment below so I can help you fully understand!

Finding the cost of preferred stock is a really easy task. We’re always going to use the equation below, and there aren’t too many curve balls you can be thrown with this one.

In this equation, D is the dollar value of the dividend, and NP₀ is the net price or in other words, the price minus any floatation costs. That’s it! Let’s check out a sample problem to see just how easy this is.

Sample Problem 1

Cost of Preferred Stock Example Problem

JumboHost has recently issued preferred stock with a $5 dividend. The preferred stock has a $75 par value and sold for $72.25. Floatation costs on this issue were 3% of par. What is JumboHost’s cost of preferred stock?

Solution

This problem gives us $5 for the dollar value of the dividend. They tell us that the price is $72.25. Furthermore, floatation costs are 3% of par value, or $2.25. Your equation should now look like this:

Great! You got the correct answer of 7.14% with only a few minutes of studying! Try another one!

Sample Problem 2

Another Cost of Preferred Stock Problem

GuitarCorp preferred stock has a par value of $50 and currently sells on the market for $56.00. The dividend rate is 7% and floatation costs on this issue amount to $4.00 per share. What is GuitarCorp’s cost of preferred stock?

Solution

This one looks a little different, but we’ll go through the same process. Remember that our equation needs a dollar value for a dividend but this problem is giving us something new called dividend rate. To get the dollar value, simply multiply the dividend rate times the par value. That gives you a dividend of $3.50 per share. Then in our equation, we have price minus floatation costs. These two numbers are both given to you in the problem!

You should have 6.73% as your answer. Easy! Just remember that when you’re given a dividend rate, always multiply by the par value and not the market price! Try the practice problem below to make sure you’ve got the hang of this.

Practice Problem 1

Cost of Preferred Stock Practice Problem

Bats N’ Balls Sporting Goods is looking to calculate their cost of preferred stock. The par value of their preferred stock is $25 and is currently selling for $23.80. The dividend rate is 8% and floatation costs amount to 3% of par. What is the cost of this preferred stock?

Solution

If you think the answer is 8.68%, then you’ve definitely got the hang of this!

Practice Problem 2

Scissor Snips Hair Salon, home of the 1-minute haircut, issued $100 par value preferred stock with a 16% dividend rate. The stock is currently selling at par, and the company paid $8 per share in floatation costs. What is the cost of their preferred stock?

Solution

A 1-minute haircut sounds a little risky. That really explains why their cost of preferred stock that you just got, 17.39% is higher than most.

That’s really enough fun for now, folks. If you have any questions about this tutorial, or you’d just like to meet other finance students, drop a note in the comment section below.

Take a seat, grab your calculator, and get ready for an easy tutorial. Generally,
we’re either going to find the cost of debt for a firm by looking at their cost of bank
loans or their cost of bonds. The most important concept to remember here is that either
way, we’re looking for the after-tax cost of debt! Remember that interest is
tax-deductible for a company, so it is very important to always find that after-tax cost
of debt.

Let’s take a look at how this all fits together when we find the cost of bank loans.
Assume our local grocery store has a line of credit with a bank at a pretax cost of 9%.
Furthermore, the grocery store is in the 34% tax bracket. Believe it or not, we’re almost
done already! We have a very simple equation to find the after-tax cost of debt.

In this equation, the lower-case k_d is our pretax cost of debt, t is the
tax rate, and the equation outputs upper-case K_d which is our after tax cost
of debt.

Going back to our grocery store example, we are given the pretax cost of debt as 9%
and the tax rate at 34%. Easy enough! Let’s run that through our equation to find the
after-tax cost of the bank loans.

That simple equation tells us that the grocery store has an after-tax cost of
bank loans of 5.94%. Believe it or not, that’s all there is to it!

To understand how we find the cost of bonds, we are simply finding the
yield-to-maturity first to find our pretax cost of bond financing. The next step is to
run that through our cost of debt equation above to get our after-tax cost of
bonds. If you are unfamiliar with finding the yield-to-maturity, please take a look at
our tutorial on the subject.

When we are dealing with the cost of bonds, it is important to note that the firm will
have to pay floatation costs. The floatation costs are simply the fee that the
company pays the investment banker to issue the bonds. The following table shows you
exactly what you will need for each problem.

Sample Problem 1.
Understand Corporation has recently issued 15 year bonds. The bonds sold at par but
floatation costs amounted to 5% of par. The bonds have a 12% coupon rate and make
semiannual coupon payments. The firm is in the 35% tax bracket. What is the cost to the
firm for these bonds?

Solution.
Let’s put all of this into a table to see what we’re working with.

Our calculator spits out a value of 12.76%. Remember that this is our
pretax cost. We want the after-tax cost so we just run it through our
K_d equation. When all is said and done, you should get a final answer for this
problem of 8.29% You’re getting the hang of it! Test your skills with some of these
practice problems.

Practice Problem 1.
Studious Inc. has decided to borrow $25,000 from their local bank. The bank will charge
them 7.5% for this loan and the firm is in the 34% tax bracket. What is the cost for the
firm after taxes?

Solution.
Hopefully you just ran the numbers through the K_d equation given above and
found their after-tax cost of 4.95%

Practice Problem 2.
Just Sweatpants, a fancy uptown clothing company, has recently borrowed some money from
the local bank at an after-tax cost of 8.7%. They are in the 34% tax bracket. What is the
firm’s cost for this loan?

Solution.
Your answer should be 8.7% believe it or not! Remember that when we find the cost of
debt, we always want the after-tax value. This problem gave you the after-tax
number up front! I’m sorry. I know we’re here to build trust, but I have to prove that
point somehow.

Practice Problem 3.
Collegiate Corporation has recently issued 10 year bonds with an 8% coupon rate. The
bonds will make semiannual coupon payments and the company is in the 30% tax bracket. If
the bonds sell at par and floatation costs amount to 3%, what is Collegiate Corporation’s
cost for this bond issue?

Solution.
Did you get 5.92%? Good! If you got an answer of 8.45%, run that through
your K_d equation to get the after-tax cost.

Now that you understand the cost of debt, check out some other tutorials from this
category. If you have a question, please leave it in the comments section below.

Before reading this tutorial, be sure you have a good grasp on moving cash flows
around on a timeline. You will really need this foundation to be able to undersand
uneven cash flows. In the world of finance, we will often deal with different
cash flows occurring at very different times. To illustrate this point, take a look at
the following timeline:

You should notice right away that we aren’t dealing with an annuity. These are cash
flows of different amounts and they do not occur at the end of every year. If our
appropriate discount rate to use is 12%, what is this cash flow stream worth to us
today?

To find the answer to this problem, we’re going to need to discount each of these cash
flows, one by one, back to time zero. This is extremely simple. First, set your
future value equal to $200. N is 3, Interest is 12, and solve for present value. On your
calculator, you will press (for the first cash flow in year 3):

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 3 Step 4. Set I/Y = 12% Step 5. Set FV = 200 Step 6. Compute PV

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 3 Step 4. Set I/YR = 12% Step 5. Set FV = 200 Step 6. Compute PV

We found out that the $200 cash flow in year 3 is worth $142.36 to us
today. We’re not done yet! Now find what the other two cash flows are worth at time zero.
Remember, you just set them as the future values and change your N. Everything else
should stay the same. Here is the finished timeline:

When you’ve found the present values of each cash flow, you can add them up to find
out the total value today of $659.14. Try the next problem on your own
to make sure you understand this concept.

Practice Problem
ABC Company has the opportunity to receive $50,000 at the end of year 6, $65,000 at the
end of year 7, and $25,000 at the end of year 10. If their opportunity cost of funds is
8%, how much is this worth to them today?

Solution:
Not too bad, right? Just discout these three cash flows back in the same way we did
above, add the time 0 values up, and you will get an answer of
$81,015.20.

I think you’ve got it! If you have a question on this tutorial, please leave us a note in the comment section. I love hearing from the readers!

So we’ve learned in our annuity tutorial that an annuity is just a series of equal cash flows, equally spaced out in
time. This tutorial will talk about a special kind of annuity called a
perpetuity. Simply put, a perpetuity is an annuity that goes on forever. This is
a really interesting type of investment, with a very easy to use formula.

This formula is just telling us that the value of our perpetuity at time 0 is the
annual payment divided by our required return. Let’s take a look at a problem and see how
this fits together.

Practice Problem 1.
How much would you be willing to pay today for an annuity that promises to pay $9,000 per
year forever? Assume your required rate of return is 8%.

Solution:
This couldn’t get easier, folks. They tell us the annual payment is $9,000 and that our
required rate is 8%. We just need to plug those two numbers in our trusty perpetuity
equation to get the value today! Just remember to use the decimal form for your interest
rate. Your formula should now look like this:

After doing that simple math, you found out that we would be willing to spend
$112,500 for that perpetuity today. These problems really are just that
easy!

Practice Problem 2.
While strolling down a dark alley, you’re approached by a man offering to give you $1,000
every year forever if you give him just $10,000 today. If your opportunity cost of funds
is 11%, is this a good deal?

Solution:
First of all, you probably don’t want to give money to anybody on the street. That having
been said, let’s find out what he’s offering. He will give you $1,000 payments and your
opportunity cost (or your required rate) is 11%. Plugging this into your equation, you
are left with the following:

Push a couple buttons and you’ll find out that this perpetuity is only worth
$9,090.90 to you today. He wanted to sell it to you for $10,000 so you’d
better keep on walking… quickly.

If you’re still a little hazy on perpetuities, post your questions below in the comments section and we’ll get you some answers!

Hopefully you’ve read the other time value of money tutorials and feel pretty good
about things. There is however one type of problem in which we won’t use the calculator’s
function. It is the problem of continuous compounding. Whenever you see these
two words, you’re going to use the following formula:

This formula is very simple. FV is the future value that we will solve for. PV is the
present value (usually the money we deposit today). The e that you see there is simply
the “natural number.” Don’t worry about that part. It is just a key on the calculator
that we’ll use. N is just the number of years. I is the interest rate. The trick to the
interest rate is to make sure you use the decimal form of the number. This will all come
together when we check out the following sample problem.

Practice Problem 1: Let’s Deposit Some Money
If we deposit $500 in an account paying 4% compounded continuously, how much money will
be in our account at the end of 25 years?

Solution:
Pretty easy, huh? Let’s put these numbers into our formula and solve for FV. You should
get the following:

First, multiply 25 by 0.04. Then, hit the e^x key on your calculator.
Finally, multiply by 500. This should give you a value in this account of
$1,359.14.

Practice Problem 2: Now it’s your turn!
You place $15,000 into an account paying 6.5% compounded continuously. How much money
will you have in the account after 10 years?

Solution:
You should have $28,733.11 as your answer. If you didn’t get this value,
make sure you’re putting in your interest as a decimal, and that you’re working from
right-to-left.

We’ve learned in previous tutorials that an annuity is simply a series of equal cash
flows that occur over some period of time. But what is a deferred annuity?
Simply put, it is just an annuity that doesn’t start until some point out in the future.
I’m sure everyone has seen an ad on television that promises, “If you buy now, you won’t
make payments until 2008.” The company is then dealing with a deferred annuity since your
equal payments won’t start until some time out in the future!

You’ve probably seen students studying deferred annuities with tears pouring down
their faces, wondering how their life will go on. Well, maybe not. The point is, deferred
annuities are very simple to value. It just takes one extra step. The first step is the
same–we’ll just find the value of the annuity at some point in time. Once we have the
value of the annuity somewhere on our timeline, we’ll just move that lump-sum anywhere on
the timeline that we want it. Let’s jump right into this one with a sample problem.

Suppose your construction company has been offered a job that will pay you $50,000 per
year from years 5 through 9. If your opportunity cost of funds is 9%, what is this job
worth to you today? Let’s take a look at this offer on a timeline.

On our timeline, we see the $50,000 payments we’ll receive, but they’re out in the
future. This is why it is called a deferred annuity. To find out what this is
worth to us today, as in time 0 on the timeline, we just have to do two easy steps.
First, we will find the value of the annuity. Then we can move that value back to time
zero as a lump-sum cash flow.

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 5 Step 4. Set I/Y = 9% Step 5. Set PMT = 50,000 Step 6. Compute PV

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 5 Step 4. Set I/YR = 9% Step 5. Set PMT = 50,000 Step 6. Compute PV

If your calculator gave you a value of $194,482.56 you’re halfway
there. Before we move on, lets clear up some of the parts that are somewhat confusing.
When dealing with annuities, just count the number of payments to find N. Here, we will
receive 5 payments so N=5. Also, notice we don’t have any future value. Now on with the
show…

The key to understanding deferred annuities is to realize where this value we computed
is on the timeline. It is in period 4. All you need to remember is that when we’re
dealing with an ordinary annuity, your calculator will give you the present value of the
annuity one period before the annuity starts. In this case, you can clearly see that the
annuity starts at year 5 on the timeline. Thus, the calculator gives us a present value
one period before, which is year 4.

Since we know what the annuity is worth in year 4, we can ignore the payments now. The
problem just got really easy! We now have a timeline like this:

All we need to do is set the $194,482 as our future value and solve for the present
value at time 0 to answer this problem!

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Set N = 4 Step 3. Set I/Y = 9% Step 4. Set FV = 194,482 Step 5. Compute PV

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Set N = 4 Step 3. Set I/YR = 9% Step 4. Set FV = 194,482 Step 5. Compute PV

Did you get $137,776? That is what our deferred annuity is worth at
time 0. Let’s quickly recap deferred annuities.

1. First find the value of the annuity anywhere on the timeline. Remember that if you
   solve for PV, it will be in the period before the annuity starts.
2. Move this value back to zero by setting it as your future value, setting payments to
   0, and solving for present value.
3. If you are asked to solve for the value of the annuity at some point out in the
   future, then just move the value at time 0 to that future point. For example, if we were
   asked what the annuity is worth in year 25, we could just move the value at time 0
   ($137,775) out to time 25 by setting it as our present value and solving for the future
   value.

That wasn’t too hard now was it? These deferred annuities take some practice though,
so keep trying. If you have any questions or comments, please leave them below.