December 7th, 2008 · 3 Comments
When you are looking to find the value of a share of common stock, you have a very
easy solution as long as the dividends are growing at a constant rate. The
constant-growth model or Gordon Growth Model will do the trick! The equation to
find the value of a share of common stock is:
In this equation, Vcs is the value of a share of common stock.
D1 is the next dividend to be paid, kcs is the
required return, and g is the growth rate. If any of these terms seem a little
foreign to you, read our tutorials on the Cost of Equity or Three Ways To Find Growth
Rate. This equation is going to give us the value of a share of common
stock. You will see problems that ask for the intrinsic value or what the stock
is worth to us today. These all mean the same thing, so don’t get thrown off by
terms such as intrinsic value. So now that we understand all of the terms in
this simple equation, let’s put it to work in a problem!
Practice Problem 1
Marco Pollo is expected to pay a dividend next year of $2 and dividends will continue to
grow at 10% per year. If investors require a return of 15% for stocks of this level of
risk, what is the intrinsic value of a share of Marco Pollo common stock?
Solution
In the problem, we’re told that D1, the next dividend to be paid, is
$2. We’re told that the growth rate is 10% and the required return,
kcs is 15%. If we put this information into our equation, we get the
following:
"Practice Problem 1 Solution" />
So that’s it! We see that the value of Marco Pollo common stock is
$40 per share. This problem was pretty straight forward so let’s take it
a step further to make sure you have all of your bases covered.
Practice Problem 2
Peas & Carrots Produce recently paid a dividend of $3 per share and dividends are
expected to grow at a constant rate of 8% indefinitely. If the required rate for a share
of common stock is 18%, what is a share of Peas & Carrots common stock worth
today?
Solution
In this problem, we’re given a growth rate of 8% and a required return of 18%. But are we
given D1? The problem tells us that the company recently
paid a dividend of $3. That dividend has already been paid! We’re looking for the
next dividend, D1 so we must use our D1
equation.
alt="Dividend Equation" />
alt="Dividend Equation" />
Now that we know that the next dividend will be $3.24, we can put all of our
information into the constant-growth equation to get a value for this stock.
"Practice Problem 2 Solution" />
That’s it! A share of Peas & Carrots common stock is worth
$34.20. Now that you have a good understanding of the constant-growth
model, try out a few of these problems!
Practice Problem 3
TennisTime recently paid a dividend of $2.25. Their return on equity is 18% and they
retain 40% of their net income and pay the remaining 60% in dividends. If investors
require 9% for stocks of this risk level, what is the intrinsic value of a share of
TennisTime common stock?
Solution
This problem is a lot tougher but you should get $134. In this problem,
you’ll need to use the growth equation from the tutorial Three Ways To Find Growth
Rate to find a growth rate of 7.2%. Then, use the D1 equation to
get your dividend of $2.412. From there, you can plug the numbers into your
constant-growth equation for the end result!
Practice Problem 4
Sunshine State Umbrellas paid a dividend of $1.00 five years ago. Their recent dividend
was $1.61. If investors require 14% for a share of common stock, what is the intrinsic
value?
Did you get $44.17? If not, it could just be a rounding issue. But,
let’s look at what this one took. We know that D0 is $1.61 but we need
D1. You can use the time value of money to find the growth rate and
you will get 9.993%. From there you can get D1 and then use the
constant-growth equation to get the final answer.
I know these practice problems were a bit harder than usual so if you have any
questions at all, please leave a comment and we’ll get everything sorted out!
Tags: Capital Budgeting · Cost of Capital
Finding the value of a share of preferred stock could not be easier! Preferred stock
is unlike common stock in that the dividends paid to shareholders do not grow. The other
important factor to consider is that shares of preferred stock typically do not have a
maturity date. So putting these facts together, we have some payment (the dividend) that
occurs forever. Sound familiar? This is just like what we learned in our time value of
money section when we studied perpetuities.
There is a very simple formula to value preferred stock. Let’s take a look at it and
understand the notation:
"" />
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Where:
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Vps
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This is the intrinsic value of a share of preferred stock. In other words, this is
what the preferred stock is worth today. It is a fair price for a share of this preferred
stock.
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D
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This is the dollar value of the dividend. Remember, you will always use a dollar value
for D!
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kps
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This is the required rate for a share of preferred stock. This will be a percent so
always put this in as decimal form (i.e. 8% = 0.08).
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Now that we understand the basic formula for preferred stock valuation, lets try a
sample problem. UnderstandFinance preferred stock pays a $3 annual dividend and investors
require 9% for a stock of this risk level. What is the intrinsic value of this preferred
stock?
This one gives us all of the puzzle pieces up front! We have $3 as our value for D and
9% as our required rate. The only trick here is to remember to put the 9% in as a
decimal. Your formula should look like this:
"" />
Before moving on to a slightly harder problem, we need to have a discussion on par
value of preferred stock. The par value is typically $25, $50, $75, or $100. This is
extremely important! Some problems will not give you a dollar value for a dividend, but
instead tell you the dividend rate. The dividend rate is simply the percent of
par value that is paid out each year in the form of dividends. The following problem will
show you what we’re talking about here:
Magic Carpet Company’s preferred stock has a par value of $50 and a dividend rate of
8%. If investors’ required rate is 10%, what is a fair price for this preferred
stock?
Here we need to turn that dividend rate into a dollar value. Just multiply the par
value times the dividend rate to get D. Your formula should look like this:
"" />
Practice Problem 1
Wooden Pencil Accounting’s preferred stock recently paid a dividend of $5.50. The par
value of their preferred stock is $75. Investors require an 11.5% rate of return on this
stock. What is the intrinsic value of the preferred stock?
Solution:
You should have found an intrinsic value of $47.83. In this case, we
didn’t even need to know the par value since the dividend was given to us in dollars.
Practice Problem 2
FinanceCorp’s preferred stock has a par value of $50 and a dividend rate of 5%. Investors
require an 7% rate of return on this stock. What is the intrinsic value of the preferred
stock?
Solution:
Did you get an answer of $35.71? If not, make sure you are using decimal
form for your required rate!
Tags: Capital Budgeting · Cost of Capital
December 4th, 2008 · 1 Comment
Finding the value of a bond is pretty easy once you understand that bonds are just
time value of money problems in disguise. Naturally, you’ll want to have a firm grasp on
time value of money–if you don’t, we have plenty of tutorials to
help you build that foundation. With that disclaimer, let’s take a look at how bonds
work.
Why would anybody invest in a bond? Bonds will pay the investor a coupon
payment every year (or twice a year), and also par value (usually $1,000) at
maturity. Those coupon payments are an annuity, with a lump-sum par value at the end.
Sounds like a time value of money problem already! To show you a little more, let’s look
at a simple example.
Sample Problem
UnderstandFinance has outstanding bonds on the market with a par value of $1,000 and
mature in 5 years. The coupon rate is 9% and coupons are paid semiannually. If investors
require an 11% rate of return on these bonds, what should the price of the bond be?
Solution
We know the bonds mature in 5 years, but since the bonds make coupon payments
semiannually, there are 10 compounding periods. They tell us that the coupon rate is 9%.
This means that 9% of $1,000, or $90, will be paid each year in coupon payments. Remember
though, these bonds make semiannual payments! If they pay $90 a year, they must pay $45
every six months. Investors require 11% on these bonds, so that will be what we plug into
our financial calculator as I. Finally, we know we’ll get par value of $1,000 back at the
maturity date. This is our future value! Let’s see what this bond looks like on a
timeline.
So what you should notice here is that our coupon payment of $45 every period is just
a simple annuity. The $1,000 is a lump sum payment at the end and can be treated as a
future value. So, here’s what we should plug into our financial calulator:
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Texas Instruments BAII Plus
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So when we solve for PV, we find out that this bond is worth $924.62.
This problem illustrates a great concept. When investors require a rate higher than the
coupon rate, the bond will sell at a discount. In this case, investors require
11% but the bond only pays 9%. So we see that the bond sold for less than the $1,000 par
value. When investors require less than the coupon rate, the bond will sell for a
premium, or more than the $1,000 par value. What happens if the investors’
required rate equals the coupon rate? You guessed it! The bond will sell at par. This is
the inverse relationship between interest rates and bond price. When interest rates rise,
bond prices fall. When interest rates fall, bond prices rise. This is an important
concept so take some time to get cozy with it.
Here’s how I like to think about the inverse relationship. Let’s say you own a bond
with a 10% coupon rate. So, your bond pays $100 per year in coupon payments. If bonds of
similar risk are on the market with an 8% coupon rate, they’re only paying $80 a year. So
the bond you hold is more valuable, correct? If I want to buy your bond, you’re going to
charge a premium for it since it has a higher stream of coupon payments attached
to it.
Practice Problem 1
Big Apple Camera Shop has bonds on the market with a par value of $1,000 and mature in 15
years. The bonds have a 7% coupon rate and coupons are paid semiannually. If investors
require 6% for bonds of similar risk, what is the present value of these bonds?
Solution
Right away, we know that if investors require 6% but the bond pays 7%, these bonds will
sell for a premium. In fact, you should have gotten an answer of
$1,098. The coupon rate is 7% of $1,000 or $70 per year. However, since
the bond pays coupons semiannually, divide by 2 to get a PMT=$35. Also, make sure you put
N=30!
Practice Problem 2
Blue Jeans Coporation issued bonds 20 year bonds five years ago with a face value of
$1,000 and a coupon rate of 12%. If investors require 10.5% for bonds of this risk level,
what is a fair price for a Blue Jeans bond?
Solution
This one is a little tricky? If you got $1,112.08, then you’ve
definitely got the idea! If not, let’s see what could have gone wrong. These are 20 year
bonds, but there’s only 15 years left. So, we know N must be 30. Whenever you’re dealing
with bond valuation, you’re only going to be concerned with the time that is left. The
next issue is that they didn’t tell you the bonds were semiannual. A safe bet is to
assume that if the bond pays a coupon, it is a semiannual payment.
Tags: Uncategorized
December 4th, 2008 · 1 Comment
Do you have a car payment to make each month? Do you pay rent on
an apartment? Both of these are annuities! As you may have
guessed, an annuity is simply a series of equal cash flows equally
spaced out in time. There are two main types of annuities–ordinary
annuities and annuities due.
An ordinary annuity is one in which the cash flows occur at the
end of the time period. For example, if you start a new job with a
monthly salary of $3,000, you receive your check for this amount at
the end of the month. The fact that the cash flow (your salary in
this case) is received at the end of the period along with the fact
that it is the same value each month makes this an ordinary
annuity.
So what in the world is an annuity due? It is just a
series of equal cash flows that occur at the beginning of
the period. My guess is, you’re probably sufferring through an
annuity due as we speak. If you are living in an apartment, making
those rent payments, you’re paying an annuity due. When you pay
your rent on the first of the month, you’re paying for the month
ahead of time. That is, on August 1st, you pay rent for the right
to live in the apartment for the rest of the month of August. This
is an annuity due because the payments occur at the
beginning of the period.
Let’s look at a simple example of an ordinary annuity. If you
deposit $1,000 at the end of each year into an account earning 9%
compounded annually, how much will you have in the end of 5 years?
Lets look at this on a timeline:
On our timeline, we see the $1,000 payments going into our
account. The numbers of the bottom refer to the end of the year. I
think after drawing the timeline, we’re ready to put this into our
financial calculator. We have 5 years, 9% interest, no present
value (see the timeline at time period 0?), and $1,000
payments.
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Texas Instruments BAII Plus
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Hewlett-Packard 10BII
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If all went well, your calculator should have given you a future
value at time period 5 of $5,984.71. You may be
wondering why we put the $1,000 payments in as a negative number.
The reason for this is that these are cash outflows. A
good rule of thumb is that if money is leaving your wallet, we put
it in our calculator as a negative number. When money is coming
back to you, it is a positive number. This problem also taught us a
great lesson in that when we deal with ordinary annuities, the
calculator will give us our future values in the final year of the
annuity. In other words, the future value we just calculated is at
year 5 on our timeline.
Now let’s look at an annuity due (begin mode) problem. How much
would you be willing to pay for an annuity that paid you $1,000 at
the beginning of the year for 3 years if your opportunity cost is
11%?
Normally, the word “begin” makes your heart race and your hands
sweaty. It really isn’t that bad though. Again, we draw a
timeline.
The first thing that you should notice about the timeline above
is that we start at year 1 instead of year 0. When we’re dealing
with an annuity due, the numbers on our timeline represent the
beginning of the year. So the 1 tells you that the $1,000 payment
occurs at the beginning of year 1. We’re going to handle
this annuity due the same way we would handle an ordinary annuity.
The only difference is that we will set our calculator to begin
mode.
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Texas Instruments BAII Plus
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Hewlett-Packard 10BII
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Piece of cake! Your calculator shows you that you’d be willing
to pay $2,712.52 today for this annuity due.
You’ve gotten the hang of annuities now, so try these practice
problems on your own.
Practice Problem 1
If you deposit $100 at the end of each month into an account
earning 11.5% annually, how much will you have saved at the end of
25 years?
Solution
Wow! You’ve saved up $172,011.55. If you’re not
getting the same answer that I do, make sure your calculator isn’t
in begin mode.
Practice Problem 2
How much would you have
to deposit at the beginning of each month to have $10,000 saved at
the end of the year? Assume your account earns 4% compounded
annually.
Solution
Only $815.45 a
month!
Tags: Annuities · Time Value of Money
December 4th, 2008 · 1 Comment
This tutorial will give you a great foundation to continue learning about Time Value
of Money. You know that a dollar in your hand today is worth more than a dollar sometime
out in the future. But how much is that dollar worth today? This tutorial will give you
the tools to answer that question.
We’re dealing with present values and future values here. A present value is what
something is worth to us today, whereas a future value is… well, its out in the future!
If we invest $100 today in a savings account earning 3% annually, how much will our $100
be worth in 25 years? Let’s look at this on a timeline and see what is going on here:
"http://www.understandfinance.com/img/1/timeline1.gif" />
We see that at time period 0, our present value, we have our $100. It is a negative
number because it is a cash outflow. That is to say, our wallet is now lighter by
$100 because we have put that money into a savings account. On the right hand side of the
timeline, you see that 25 years later, we want to know the value of our deposit. We’re
solving for our future value! This problem now becomes a simple calculator exercise.
Here’s the steps you take…
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Texas Instruments BAII Plus
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If all went well, you saw that your $100 grew to $209.38 after 25
years! Now, try some of the following practice problems on your own. If you are unable to
get the same answers we do, try getting some help in our
"http://www.understandfinance.com/forum/">forum.
Practice Problem 1: Calculating Present Values.
Your beautiful newborn baby girl deserves the best. On her 16th birthday, you want to buy
her a brand new car which will cost $35,000 at that time. If you can earn 11.5% annually
on your investments, how much will you need to invest today in order to buy her this
gift?
Solution:
Wow, kids are expensive (see our tutorial on opportunity cost)! We know that N is 16
since we’re buying this car in 16 years. We also know the FV will be $35,000 and our I is
11.5%. Plug this in and solve for PV to get $6,132.96. Maybe this gift
isn’t too expensive afterall!
Practice Problem 2: Calculating the number of compounding
periods.
You just won the lottery and received a check for $450,000! Being someone that
understands the time value of money, you decide to invest it in a mutual fund earning 8%
annually. How many years will it take for your winnings to become $1,000,000?
Solution:
In this one, we are solving for N. Try setting up a timeline and working this one out on
your own. Remember that today we have $450,000 and sometime in the future we want to have
$1,000,000. If you set this up correctly, you should see that it will only take
10.38 years. If you see no solution on your HP10BII or Error
5 on your TI BAII Plus calculator, make sure you put your PV in as a negative number.
Remember, that money is a cash outflow when it goes into the mutual fund!
Practice Problem 3: Finding the Interest Rate
Fifty years ago, your Uncle Tyrell invested $1,000 in a savings account that pays
interest quarterly. During your weekly visit, you spot his bank statement and see that it
is now worth $59,500. What rate of interest has Uncle Tyrell earned over the years?
Solution:
This one is a little tricky. First of all, “quarterly” tells us that we are dealing with
4 payments per year. Make sure your calculator is set up correctly! Now, if we’re dealing
with quarterly payments, what should N be? That’s right–N = 50 years * 4 payments per
year = 200. The rest should be easy enough if you’ve drawn your timeline. You will find
that Uncle Tyrell earned 8.26% on his deposit.
Tags: Time Value of Money
