Portfolio Theory – Lecture 2
The discounted cash flow model
Note: Please read over these notes before the class. As we noted last week we are going to
systematically apply the scientific method in our study of finance. Recalling that the very first
step in the scientific approach is to start with a problem, a question, or a puzzle, that is what we
Step 1 of the scientific method.
Problem. How do we find out the fair value of a share? More generally, how can we find the
fair value of any asset, for example the fair value of other paper assets such as a corporate bond,
or physical assets such as real estate, buildings and machinery, or intangible assets such as a
company’s reputation, or its brands and trademarks?
Step 2 of the scientific method.
Build a model (i.e. a theory or a hypothesis). We have done that step already, i.e. the textbook
world or the rational expectations model is the main model we will be applying in this course,
although later on we will be improving this model somewhat to include certain behavioural
factors when we give our introduction to the new field of behavioural finance.
Step 3 of the scientific method.
Explain the puzzle in the model. When we solve the problem in the model we are giving the
theoretically correct solution to the problem. This does not necessarily mean that this
theoretically correct solution will actually work in the real world, since the theoretically correct
solution is nothing but a description of what happens inside the model world. However, a good
model will be a good approximation to reality, so we would expect that in this case the
theoretically correct solution will also be reliable in practice.
As we will see in this chapter, the theoretically correct technique to find the fair value of an
asset is to use what is called the discounted cash flow technique, or DFC technique. In other
words, the rational self-interested investors of our textbook world model use the DCF technique
to work out the fair value of assets. It turns out that this theoretically correct solution also works
very well in practice, and in fact is so useful in practice that DCF is the most important
technique in the whole of finance.
However, DCF is not a perfect technique and it is more useful in some situations than others.
This reminds us that although the textbook world is a good approximation to the real world it is,
like all theories and models still only an approximation to reality.
Step 4 of the scientific method.
Empirical research. Empirical research means looking at the world to see how well the
theoretically correct solution actually works in practice. In this chapter we will be focussing on
explaining what the DCF model is. However, we will be seeing many successful applications of
the DCF model throughout the course, and whenever we do this we are really applying step 4 of
the scientific method to the asset valuation problem.
Step 5 of the scientific method.
Applications. When we find a theoretically correct solution that works well in practice we can
apply it with confidence. In the course of your business career and in managing your personal
finances you will find many situations where DCF techniques are called for, for example finding
the cheapest mortgage, negotiating the sale of purchase of commercial property, deciding how
much to pay for an investment, etc. etc. Also of course, since DCF is the language of finance
you will need to understand the DCF technique well enough not to get ripped off in financial
2. Some examples
Before discussing the DCF model in its most general form, we first of all look at a few
examples of the model in action.
Example 1: (Bond valuation) You are employed as an analyst on the bond trading desk of a
small Austrian bank. You are considering whether to buy US$250,000 worth of corporate bonds
of Glassblower Chemicals.
Some background and jargon
Companies can borrow money in two ways. The first is simply to borrow from a bank. For
example, under an interest only loan Glassblower might borrow $10m for ten years at ten
percent interest from ABC Bank, paying $1m (= 10% of $10m = 0.1 $10m) interest at the end
of years 1 to 10 and also repaying the principle of $10m at the end of year 10. The second way
of Glassblower to borrow is to issue corporate bonds.
A corporate bond is a piece of paper entitling the owner of the bond to certain payments to be
paid by the issuer of the bond. To raise the money this way Glassblower might issue 10,000
bonds each with a face value or nominal value of $1,000, with a 10% annual coupon or
nominal interest rate and a maturity of ten years.
How does Glassblower sell these bonds? Glassblower is a chemical manufacturer, not a
financial institution with expertise and contacts in the financial markets. To issue the bonds
Glassblower approaches an investment bank, since helping corporations to raise finance is one
the main things investment banks do. Glassblower engages Sharkey, Sleight and Hand Inc.
(SSH) to undertake the financing. SSH advises that in the present market the bonds can be sold
for $1050 each, and agrees to act as underwriter for the issue. Underwriting the bonds means
giving a guarantee that the bonds will be sold for at least $1050, and that if SSH is unable to sell
the whole issue of bonds for this price, SSH will buy any remaining bonds itself at the $1,050
price. For these services SSH charges a fee of $400,000.
Let’s suppose the bond issue is successful. Glassblower receives $10,100,000 (= 1,000 bonds
$1,050 $400,000 fee). SSH successfully sells the bonds to its network of institutional and
private clients for $1,050 each and pockets the fee of $400,000).
What do the buyers of the bonds receive for their $1,050? Suppose a pension fund buys 1,000
bonds for $1,050,000 (= 1,000 bonds $1,050). For each bond, at the end of years 1 to 10 the
fund receives the coupon payments of $100 (= the 10% coupon the $1,000 face value), and
also receives the face value of $1,000 per bond at the end of the maturity of 10 years. (Note
that the terms coupon, face value and maturity are just part of the legal jargon which tells us
what cash flows the holder of the bond is entitled to. Note also that the face value of the bond is
usually different from the market price of the bond, i.e. the price at which buyers and sellers
are prepared to trade the bonds. In our example the face value is $1,000 while the market price
Corporate bonds also trade in the secondary market. For example, after three years the pension
fund might decide to sell its Glassblower bonds in the bond market. How much will it get for
each bond? Well that depends on how much investors believe the bonds are worth. Like in all
markets the seller wants to receive the highest price while the buyer wants to pay the lowest
price. In trying to find the fair price both the buyer and the seller will apply the discounted cash
flow model (DCF).
An important fact about the reliability of the DCF model in practice is that it is applied
universally in the bond market, i.e. the bond market is one of the those situations where our
textbook theory works so well that it is the standard model used by all bond traders, buyers and
sellers all the time.
Now back to our example. Should you buy $250,000 of these bonds for your bank? It is now
four years since the bonds were issued, and the remaining maturity of the bond is six more
years. The fourth coupon payment has just been made, so there are six more coupon payments
due, at the end of years 1 to 6. At the end of year 6 the bondholder will receive the $1,000 face
But will we actually receive these payments? These payments are really just promises to pay, i.e.
Glassblower promises to make these payments, and is legally obliged to make the payments if it
has the money. But what if Glassblower is half way to going bust? To work out the fair value of
the bonds we need to assess Glassblower’s financial strength, i.e. we need to assess the risk of
You have just taken a call from Marco Ravioli, a bond salesman from Skalliwag Brothers Plc an
investment bank based in London active in the bond market. Ravioli tells you the bonds are
currently trading at $930, that this is cheap, and that he can get you $250,000 of these bonds at
this price. He is going to ring back in one hour. What do you tell him when he calls?
The first thing you do is check on your Bloomberg screen for the current price at which
Glassblower bonds are trading. The idea here is that people trading in the bond market are
mainly institutional buyers and sellers, and that basically they know what they are doing. If the
bonds are currently trading at the $930 price Ravioli claims, then that is what these smart
investors believe they are worth. In that case $930 is probably pretty close to fair value.
Unfortunately Glassblower is a small company and its bonds do not trade much, and in fact
there is no recent quoted price for these bonds. The problem you have is that there is not a large
market for Glassblower bonds, as there is for, say, bonds issued by the UK or US governments.
Also you do not trust Ravioli. Ravioli is trying to make money for Skalliwag, not for you. Also
you realise that the company you represent is small. You are not one of Ravioli’s big clients, so
if he rips you off on this deal he may not be too concerned that you will not be doing business
with him again.
The institutional bond market is not like the retail market. The government will not step in to
bail you out if things go wrong. The rule of the institutional bond market is caveat emptor,
Solution: Since there is no market price for Glassblower bonds we work out the fair value of
the bonds ourselves by applying the DCF model. If you own one of Glassblower’s bonds this
will entitle you to the following cash flows:
End of Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
Cash flow $100 $100 $100 $100 $100 $1,100
To find the value of these cash flows, i.e. how much the bond is worth you need to apply the
discounted cash flow model in order to find the present value. The present value is the value
of the cash flows right now, i.e. the value of the bond.
Note that in finance the only thing we care about is cash flow, not profits. In finance we only
care about profit figures if they help us to get a handle on the underlying cash flows. This is the
rational approach. Profit is only a number that appears at the end of a profit and loss account,
worked out by applying the rules of accounting. Over the long term total profits roughly equals
total cash flows, but in any particular year the profit and cash flow generated by the firm can be
very different. One of the main differences between finance and accounting is that in finance the
focus is always on cash.
The present value of a cash flow is less than the value of a cash flow in the future, i.e. a promise
of $100 in one years time is not worth $100 hard cash today. There are three reasons for this.
Firstly, a cash flow today can be spent today or invested today. Secondly, inflation will eat away
the value of a future cash flow. If inflation is running at 5% you need $105 dollars next year in
order to buy what could be bought today for only $100. Thirdly, a future cash flow is risky, in
other words you might not get the $100 you are entitled to since Glassblower may go bust over
the coming year.
In the DCF model we reduce the value of the future cash flows by applying the appropriate
discount rate. The discount rate takes into account the three reasons given above for why a
promised $100 in the future is worth less than $100 today. This is how it is done.
First of all we need to estimate the discount rate that is appropriate for Glassblower bonds. This
involves making a judgement about the financial strength of Glassblower, since the risk of the
bonds depends directly on the risk of the company that is to pay the cash flows on the bond, and
the discount rate itself depends on the risk of these cash flows. Fortunately we do not have to do
analyse the financial strength of Glassblower ourselves, because bond rating agencies like
Moody’s and Standard and Poors have already done it for us. Looking up the rating of
Glassblower bonds you see that Moody’s currently give the Glassblower bonds a Ca rating. Ca
sounds pretty good. However, according to Moody’s definition of Ca:
‘Bonds which are rated Ca represent obligations which are speculative in a high degree. Such
issues are often in default or have other marked short-comings’.
Clearly the financial strength of Glassblower has deteriorated since the bonds were first issued
four years ago. Ca rated bonds are also known in the trade as junk bonds.
You now look at the prices at which other Ca bonds are trading in the market. You find that Ca
bonds maturing in around 6 years currently have discount rates of 20%, i.e. 20% is the discount
rate appropriate to the level of risk of a Ca rated bond. So 20% is the discount rate you should
use to find the present value of Glassblowers bonds.
This is how we do DCF calculations:
The value of one Glassblower bond = the present value of the promised cash flows discounted
at 20% =
$100/1.2 + $100/1.22 + $100/1.23 + $100/1.24 + $100/1.25 + $1,100/1.26 = $667.449.
$667.449 is miles away from the $930 fair value Ravioli told you, the lowdown, lying,
*%@*&!. You probably have a pretty good idea now what to say to Ravioli when he calls back.
If you pay $667.449 for each bond rather than the $930 Ravioli asked you for, on a deal of
$250,000 worth of bonds you save your bank from a loss of $70,364. (If you buy $250,000 of
bonds at Ravioli’s price of $930 then you will get 268 bonds since $250,000/$930 = 268.8172
and you cannot buy fractions of a bond. For these 268 bonds you pay $249,240 = 268 $930.
These 268 bonds are really only worth 268 $667.449 = $178,876. So your loss is $249,240 -
$178,876 = $70,364).
Discount rates and required rates of return
There are quite are few different names used for discount rate in finance. The main ones are
discount rate = rate of return = expected rate of return = required rate of return.
In applications of DCF to the bond market we also use the terminology discount rate = interest
rate = market interest rate = yield = market yield.
In our course we will use discount rate most of the time.
Another way to understand the discount rate in this example is to ask ourselves ‘what rate of
return are we getting on our investment if we pay $667.449 for a Glassblower bond, and if
Glassblower actually pays us the promised coupon payments and principle?’
The answer is we get a 20% annual return each year for six years.
The cash flows we receive are lumpy, since we do not get exactly the same amount every year.
There is a big payoff at the end of year 6. In the first year we get only a 14.98% income return,
since the $100 coupon payment after year 1 is just 14.98% of the $667.449 (0.1498 $667.449 =
$100) we paid for the bond. However, at the end of year 1 the present value of the remaining
cash flows has gone up, so in addition to the income gain we also receive a capital gain, i.e. the
value of the bond goes up as we get closer and closer to the end of the bond’s life.
Let’s check this. At the end of year 1 there are 5 more payments to come.
End of Year 2 Year 3 Year 4 Year 5 Year 6
Cash flow $100 $100 $100 $100 $1,100
The value of the bond at the end of year 1 is now the present value of the remaining future cash
flows, i.e. the value of the bond at the end of year 1 = $100/1.2 + $100/1.22 + $100/1.23 +
$100/1.24 + $1,100/1.25 = $700.94. This is an increase of $33.49 (= $700.94 $667.449).
The total return after 1 year therefore = income gain + capital gain = $100 coupon + $33.49
increase in bond value = $133.49. Now $133.49 is indeed a 20% return on our investment of
$667.449 (0.20 $667.449 = $133.49).
So, the discount rate of 20% is the exactly the same thing as the rate of return we require
for taking on the risk of owning Glassblower bonds.
Discounted cash flow calculations can be done either using a calculator or by using tables. Note
that in calculations we always use decimal numbers rather than percentage figures, so that we
use 0.20 not 20% in the calculation (0.20 = 20/100). However, when we have a series of cash
flows that are all the same we can do the calculation more quickly using tables. We do this in
Using tables to do the present value calculation
Example 2 (share valuation): Sears, Roebuck and Company is a major retailer. Use the DCF
model and the information given below to work out the fair value of one share of Sears.
If you buy and hold a share the only cash flows you receive will be the series of dividend
payments D1, D2, D3, ... paid out by the firm. The value of the share is just the present value of
this series of dividend payments. So to find the value of the share you need to discount the
dividend payments to find their present value using the appropriate discount rate. There are two
problems, estimating the dividend payments and estimating the discount rate. Both of these
problems are difficult in practice.
ESTIMATING THE DIVIDENDS
The following simple approach is used by many professional analysts. To use this approach you
make the following assumption: The dividend payments will grow at a constant rate g. This
assumption is more reasonable for some companies than others. It is a reasonable assumption
for stable companies such as Sears, Roebuck and Company, but not so good for a high-growth
company such as a high-tech computer software manufacturer.
Suppose you work out that Sears dividends will grow at a rate of around 6% per year into the
foreseeable future. Suppose also that you estimate that Sears next dividend will be $2.00 per
share to be paid in one years time. Then the dividend payments you will receive will be $2.00 in
one years time, $2.12 (= $1.00 1.06) in two years time, $2.2472 (= $2.00 1.062) in three years
time, and so on forever. To find the present value you now need to discount these payments at
the right discount rate. Assume that the discount rate is 12%. (Later in the course we will see
how to estimate discount rates for shares using the Capital Asset Pricing Model. Until then
discount rates will just be given to you as part of the information in the examples we discuss).
Then the present value of the dividend stream is:
D1/1.12 + D2/1.122 + D3/1.123 + ...
= $2.00/1.12 + $2.12/1.122 + $2.2472/1.123 + ...
Now how do we discount an infinite stream of cash flows? The answer is to use the Gordon
Growth Model, also called the Gordon Growth Formula, which gives us a formula for
calculating present values of a constantly growing series of cash payments. The Gordon Growth
P0 = Present Value = D1/(k - g), where D1 is the next dividend which we receive in one
periods time, k is the discount rate and g is the growth rate.
Applying the formula to Sears stock gives us:
Value of one share = present value of dividends = $2.00/(0.12 - 0.06) = $2.00/0.06 = $33.33.
To apply the Gordon Growth Formula we need estimates for D1, k and g.
ESTIMATING D1 and g.
Estimating D1 is not too difficult. Sears is a stable company which usually tries to pay dividends
in a predictable manner. Many stable companies have a fairly predictable payout ratio. The
payout ratio is the proportion of accounting earnings paid out as dividends. Suppose in the past
Sears payout ratio is 0.6. So nearly every year Sears pays out 60% of its accounting profits as
dividends. All we need to do now is to estimate next years profits. This is not too difficult for
large, well-known and stable companies. Financial analysts spend a lot of time guessing next
years profits of blue chip stocks. Blue chip companies are also helpful in providing quarterly
earnings reports and other information through their public relations departments. They also
issue profits warnings if they believe they will not be able to meet the markets expectations for
the reported end of year earnings. So if we estimate next years dividend to be $2.00 per share
this estimate will probably be quite reasonable.
Now we need to estimate g. Again this is not impossible for stable companies. Here is one
common way of doing it. Looking back over the last ten years we see that Sears return of equity
has been 15%. That is most years Accounting Profits/Book Value of Equity has been 15%.
Since Sears is a stable company we think that the future will be the same as the past. Since Sears
pays out 60% of profits that means 40% of profits will be ploughed back into the company for
investment. So the ploughback ratio is 40%. Since Sears will get a 15% return on profits
ploughed back, this means that profits will grow at 15% x 40% every year. Therefore dividends
will also grow at about 15% x 40% every year. The formula is:
g = growth rate of dividends = ploughback ratio x return on equity.
Return on equity is given by the formula:
Return on equity = Accounting profits/Book value of equity.
We now have D1 and g. To use the Gordon Growth Formula, P0 = D1/(k - g), we need also to
have an estimate for the discount rate k. We know that k depends on the level of risk of the
dividends. But what we want is a technique that can actually allow us to put a precise numerical
value on k. At the moment we do not have such a technique. Later in the course we will discuss
the CAPM, the Capital Asset Pricing Model. CAPM will give us a simple formula for
estimating discount rates. To understand how difficult the problem of estimating k really is, you
need to know that CAPM is one of the most important discoveries in the whole of finance. The
discoverer of CAPM, William Sharpe was awarded the Nobel Prize for his work in finance.
In this example we have assumed that k is 12%. The Gordon Growth Model therefore gives us
an estimate for the value of the share as P0 = $2.00/(0.12 - 0.06) = $33.33. Should we trust this
answer of $33.33? For example, suppose Sear's shares are actually trading in the market right
now at $25.00 per share. Does this mean we should buy the shares at $25 because we have
worked out they are really worth $33.33? No. If the calculated price of $33.33 is different from
the price at which Sears shares are trading in the market it is more likely that our estimates of k
and g are wrong than that all the buyers and sellers in the market are wrong.
Probably the best way to estimate the fair value of Sear's shares is to use the current market
price. For some shares however, there is no market price. For example, when a new company is
floated on the stock market, or when a government owned enterprise is privatised. In that case
we need a model to tell us at what price the stock should be offered for sale in the market. The
discounted cash flow model is the best model available. Unfortunately it very often gives the
wrong answer. There is nothing wrong with the model; the problem is that it is very difficult to
estimate future dividends and to estimate the discount rate that appropriately reflects the level of
Note: This is why the DCF model is not used so much in valuing stocks in practice.
Example 3 (capital budgeting) We can use the same present value approach to decide which
capital investments a company should undertake.
Note: Corporate finance is the other main area where the DCF model is used all the time. DCF
is the standard project evaluation technique used by nearly all large UK listed companies.
You will see more applications of the DCF model to the corporate investment decision later in
the course when we focus on corporate finance in the last part of the unit.
You are the financial executive of Mears, Mutjack and Co. Mears is a shirt manufacturer. The
company wishes to replace its cutting machine with a new machine. The new machine should
be more efficient than the old machine and is expected to result in after tax cash savings of
$10,000 per year for the next five years, five years being the expected life of the new machine.
The new machine will cost $40,000. Should you buy the new machine?
Buying the new machine is really just the same as buying a bond or a share. To get the cash
flows of $10,000 per year for the next five years you have to pay $40,000 now. If the present
value of the series of $10,000 cash flows is more than the cost of $40,000 you should buy the
machine. If it is less you should not.
So all you need to do is to discount these cash flows back at the appropriate discount rate.
What is the right discount rate? Later in the course we will show you a standard technique
called the Capital Asset Pricing Model for estimating the appropriate discount rate. Until then
discount rates will just be given to you as part of the example.
Suppose the discount rate on Mears shares is 11%. Then the value of the cash flows is:
$10,000/1.11 + $10,000/1.112 + ... + $10,000/1.115 = $36,959.
This is less than the cost of the machine, i.e. the present value of the cash savings you get from
the machine is less than what you have to pay out to acquire the machine. So you should not go
ahead with the purchase.
Summary of the discounted cash flow model
The general rule for applying the DCF model is this. To find the value of any asset go through
the following steps:
STEP 1: Estimate the CASH FLOWS the asset will generate.
STEP 2: Estimate the DISCOUNT RATE.
The discount rate will depend on the risk level of the cash flows. The greater the risk, the
greater the discount rate. Later on we will develop the CAPM which gives us the theoretically
correct technique for calculating the discount rate. Until then just remember that the discount
rate is used as a measure of the level of risk of the asset, and that the higher the level of risk the
greater is the discount rate. This makes sense, the higher the discount rate the lower is the
present value of the cash flows, i.e. a risky asset is worth less than a safe one. Also, the discount
rate is the same thing as the required rate of return on an asset, i.e. if we buy a risky asset we
want a greater rate of return to compensate for taking a greater risk.
STEP 3: Find the PRESENT VALUE of the cash flows. The present value is the value of
The model can be used to find the value of any asset. It is used to value bonds, shares and
Value of a bond = present value of cash flows generated by the bond = PV = C/(1 + k) + C/(1 +
k)2 + ... + (C + F)/(1+ k)n
C = the coupon payments
F = the face value
n = the number of periods. So if coupon payments are made annually the maturity of the bond
is n years.
k = the discount rate. The discount rate reflects the riskiness of the cash flows. The higher the
risk level of the bond the higher k will be. In bond terminology k is also called the interest rate
and the yield on the bond.
PV = the present value of the cash flows. So PV is the fair market value of the bond. In a
rational market like in the textbook world the bond should sell for its present value.
This method of valuing bonds is used literally all the time in the real world. It is the standard
technique that everyone in the bond market uses all the time.
Value of a share =
D1/(1 + k) + D2/(1 + k)2 + D3/(1 + k)3 + D4/(1 + k)4 + ...
D1 = the expected dividend at the end of period 1. Di = the expected dividend at the end of
period i (we assume here that the next dividend to be paid will be paid at the end of year 1).
k = the discount rate. The discount rate reflects the level of risk of the expected dividend
payments. So k will be high for a risky stock such as a construction company. k will be low for a
safe company such as a utility.
If we estimate that the dividends will grow at a constant rate of g (as a decimal number) then the
Gordon Growth Model gives us the present value of the share as D1/(k g).
The DCF method is not used so much for valuing stocks in the real world, and when it is used it
tends to be used in conjunction with other methods. This is because we have less confidence in
our estimates for the future dividends and for the discount rate for stocks than for the cash flows
and discount rates for bonds. The present value is very sensitive to small changes in the discount
rate and the estimated growth rate of dividends so DCF applied to stocks tends to give us only a
Net Present value = NPV = CF0 + CF1/(1 + k) + CF2/(1 + k)2+ ... + CFn/(1 + k)n
CF0 = the expected cash flow at the start of the project. Usually this will be negative. CF1, …,
CFn are the expected cash flows over the life of the project.
k = the discount rate which reflects the risk level of the project.
NPV = the net present value of the project. NPV is the total value of the project, net of all
costs. If NPV is positive we are getting something for nothing. So a positive NPV means that
we should proceed with the project, (assuming we believe the estimates for the expected cash
flows and the discount rate). If NPV is negative we should not proceed with the project.
The DCF model is used all the time in corporate finance in evaluating the firm’s capital
The discounted cash flow asset valuation model is the most important technique in modern
finance. To repeat the general rule:
To find the value of ANY asset
(1) Estimate the future cash flows generated by the asset
(2) Find the appropriate discount rate
(3) Find the present value of the expect cash flows
THE PRESENT VALUE OF THE EXPECTED CASH FLOWS IS THE
VALUE OF THE ASSET
THE DISCOUNT RATE IS DETERMINED BY THE RISK
Required: Use present and future value tables or a calculator to answer the following
Note: Some of these questions are quite tricky, even for those who are good at this sort of
thing. Generally the questions get harder as you go on.
Don’t panic! To understand the applications of discounted cash flow to the rest of the
course, it is only the very basic DCF calculations you need to be able to do.
However, try to do some of the tougher questions as well, and come along to the seminars
with any questions or problems you had.
See Arief if you get really stuck on this material.
(1) You are considering depositing £300 in an account for eight years with an interest rate of
10%. The present value in this case is £__________.
(2) The value of £300 in eight years if it earns 10% compounded annually will be
(3) You borrow £2,000 from a loan shark for six years at an interest rate of 75% per year with
annual compounding. The future value of the loan after six years will be £__________.
(4) Suppose you convince the shark to charge simple interest instead of compound interest. In
this case the future value of the loan will be £__________.
(5) A customer deposits £1,000 in a savings account earning 14%. The value of the account in
one year if the bank uses semi-annual compounding will be £__________.
(6) Suppose the bank uses quarterly compounding instead. In this case the value of the account
in one year will be £__________.
(7) The future value of £2,000 invested for eight years at 12% with monthly compounding is
(8) The annual yield of an investment, or the __________ rate of interest is the simple interest
equivalent to a stated rate that is compounded.
(9) The present value of £200,000 to be paid in year 10 if the discount rate is 8% will be
(10) The present value of £10,000 to be paid in four years if interest rates are 10% compounded
quarterly is £__________.
(11) Find the present value factor for an interest rate of 15% and 10 periods.
(12) If one speaks of an annuity without any qualification, a (an) __________ is being
A. simple annuity
B. first annuity
C. time annuity
D. annuity due
E. real annuity
(13) You plan to save £1,700 per year for 24 years as a retirement fund and expect to earn 14%
per year on all invested funds. The first payment is made at the end of this year, year 1, and the
last payment is made at the end of year 24. You will have accumulated £__________ at the end
of 24 years.
(14) You want to create an account from which you can withdraw £550 per month for 20
months. Assuming that the withdrawals are to start one month from now and that interest rates
are 1% per month, you have to put £__________ into the account today.
(15) A five year annuity of £1,000 costs £4,000. The implied rate of interest is between
__________ and __________ %.
A. 1, 2
B, 4, 5
C. 7, 8
D. 10, 11
E. 13, 14
(16) Consider a lump sum of £900,000 paid into an account that is to be used to generate an
eight year annuity. With interest rates of 10%, the annuity payment can be as large as
(17) Assume a retiree has £90,000 and needs £12,000 per year to live. If interest rates are 8%
per year, how long will the funds last?
A. 11 years (with a little left over)
B. 14 years
C. 16 years (with a little left over)
D. 23 years
E. 25 years
(18) The future value of a four year annuity of £1,200 with interest rates of 10% per year is
(19) The future value of an annuity of £85 for 17 periods, with an interest rate of 12% is
(20) The present value of an annuity of £40 for 25 periods, with an interest rate of 14% is
Use the discounted cash flow model to find the value of a bond with the following
The bond has an annual coupon of 12%, a face value of £100 and a maturity of 10 years. The
market interest rate for bonds of this level of risk is 8%.
A share in ABC Company is expected to pay a dividend of 20 pence in one years time. The
company is expected to maintain this level of dividend payment for the foreseeable future. The
discount rate for the firm's equity is 18%. Estimate the value of one share of ABC.
As for question 2 except that now the dividends are expected to grow at a rate of 5% per year
As for question 3, except that the dividends are expected to grow at a rate of 20% over years 2
and 3 and at 5% from year 4 onwards.
Bull plc has a dividend payout ratio of 70%. The next dividend to be paid in one year’s time is
expected to be $1.50. Over the past ten years Bull's return on book value of equity has averaged
18%. Assume that Bull has a required return on equity of 10%; i.e. assume that investors require
a 10% rate of return as compensation for taking on the risk of investing in Bull's shares. Use this
information to estimate Bull's share price.
Present value of £1 receivable in n years time
Years Discount rate as a percentage
n 1 2 3 4 5 6 7 8 9 10
1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091
2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264
3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513
4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830
5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209
6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645
7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132
8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665
9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241
10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855
11 12 13 14 15 16 17 18 19 20
1 0.9009 0.8929 0.8850 0.8772 0.8696 0.8621 0.8547 0.8475 0.8403 0.8333
2 0.8116 0.7972 0.7831 0.7695 0.7561 0.7432 0.7305 0.7182 0.7062 0.6944
3 0.7312 0.7118 0.6931 0.6750 0.6575 0.6407 0.6244 0.6086 0.5934 0.5787
4 0.6587 0.6355 0.6133 0.5921 0.5718 0.5523 0.5337 0.5158 0.4987 0.4823
5 0.5935 0.5674 0.5428 0.5194 0.4972 0.4761 0.4561 0.4371 0.4190 0.4019
6 0.5346 0.5066 0.4803 0.4556 0.4323 0.4104 0.3898 0.3704 0.3521 0.3349
7 0.4817 0.4523 0.4251 0.3996 0.3759 0.3538 0.3332 0.3139 0.2959 0.2791
8 0.4339 0.4039 0.3762 0.3506 0.3269 0.3050 0.2848 0.2660 0.2487 0.2326
9 0.3909 0.3606 0.3329 0.3075 0.2843 0.2630 0.2434 0.2255 0.2090 0.1938
10 0.3522 0.3220 0.2946 0.2697 0.2472 0.2267 0.2080 0.1911 0.1756 0.1615
Present value of an annuity of £1 payable at the end of each of n years
Years Discount rate as a percentage
n 1 2 3 4 5 6 7 8 9 10
1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091
2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355
3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869
4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699
5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908
6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553
7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684
8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349
9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590
10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446
11 12 13 14 15 16 17 18 19 20
1 0.9009 0.8929 0.8850 0.8772 0.8696 0.8621 0.8547 0.8475 0.8403 0.8333
2 1.7125 1.6901 1.6681 1.6467 1.6257 1.6052 1.5852 1.5656 1.5465 1.5278
3 2.4437 2.4018 2.3612 2.3216 2.2832 2.2459 2.2096 2.1743 2.1399 2.1065
4 3.1024 3.0373 2.9745 2.9137 2.8550 2.7982 2.7432 2.6901 2.6386 2.5887
5 3.6959 3.6048 3.5172 3.4331 3.3522 3.2743 3.1993 3.1272 3.0576 2.9906
6 4.2305 4.1114 3.9975 3.8887 3.7845 3.6847 3.5892 3.4976 3.4098 3.3255
7 4.7122 4.5638 4.4226 4.2883 4.1604 4.0386 3.9224 3.8115 3.7057 3.6046
8 5.1461 4.9676 4.7988 4.6389 4.4873 4.3436 4.2072 4.0776 3.9544 3.8372
9 5.5370 5.3282 5.1317 4.9464 4.7716 4.6065 4.4506 4.3030 4.1633 4.0310
10 5.8892 5.6502 5.4262 5.2161 5.0188 4.8332 4.6586 4.4941 4.3389 4.1925