Principles of Banking and Finance
Revision Session
Section B
Jason Laws
Jason Laws
Principles of Banking and Finance
Warning
❑ Please don’t treat these slides as an extra resource to work through – that is the
role of the study guide.
❑ I share these with you so you can see examples and slides I used in the revision
sessions.
❑ They include numerous past examination questions and past prelim examination
questions. I will not go through the 2021 prelim paper as the solutions are
available.
Jason Laws
2 of 218
Sample Question
5. A firm is considering two investment projects, Gamma and Kappa. These
projects are NOT mutually exclusive. Assume the firm is not capital constrained.
The initial costs and cash flows for these projects are:
❑ The key point here that many students over looked is the words NOT mutually
exclusive. Mutually exclusive in this context means that only one project could
be chosen, for example, one use of a scarce resource or one production
technique etc. Here more than one can be chosen if they have a positive NPV.
Jason Laws
3 of 218
The NPV Rule
❑ Accept any project if its NPV > 0 or if NPV=0
❑ Reject a project if its NPV < 0
❑ Suppose a project has a positive NPV, but the NPV is small, say, only a few
hundred dollars then the firm should still undertake that project if there
are no alternative projects with higher NPV as a firms wealth is increased
every time it undertakes a positive NPV project.
❑ A small NPV, as long as it is positive, is net of all input costs and financing
costs so even if the NPV is low it still provides additional returns.
❑ A firm that rejects a positive NPV project is rejecting wealth.
❑ Observation – you are not forced to accept any of the projects. If you
accept a project with a negative NPV then you are destroying shareholder
wealth.
Jason Laws
4 of 218
NPV decision rule
❑ Activity: Please draw NPV of a standard project, i.e. cash out followed by cash in,
against the discount rate.
NPV
Discount rate
Jason Laws
5 of 218
Present Values
Example
Assume that the cash flows from the construction and sale of an office
building are as follows. Given a 7% required rate of return, create a
present value worksheet and show the net present value.
Year 0
Year 1
Year 2
− 150,000 − 100,000 + 300,000
Jason Laws
6 of 218
Present Values
Example - continued
Period
0
1
2
Discount
Factor
1.0
1
1.07 = .935
1
= .873
(1.07 )2
Cash
Present
Flow
Value
− 150,000
− 150,000
− 100,000
− 93,500
+ 300,000
+ 261,900
NPV = Total = $18,400
Jason Laws
7 of 218
NPV Example
❑ Consider three alternative projects, A, B and C.
❑ They all cost $1,000,000 to set up but project’s A and C returns $800,000
per year for two years starting one year from set up. Projects B also
returns $800,000 per year for two years, but the cash flows begin three
years after set up , i.e. in year 3.
❑ Whilst project C costs $1,000,000 to set up it requires $500,000 initially
and $500,000 at termination (a clean up cost for example).
❑ If the firm uses a discount rate of 20% which is the better project?
❑ If you can rank these projects without doing the calculations then you
really understand NPV.
Jason Laws
8 of 218
NPV Example
❑ Project A:
interest rate
Year
Cash Flow
discount factor
PV
NPV=
20%
0
1
2
-$1,000,000
$800,000
$800,000
1.000
0.833
0.694
-$1,000,000.00 $666,666.67 $555,555.56
3
$0
0.579
$0.00
4
$0
0.482
$0.00
$222,222.22
❑ Project B:
interest rate
Year
Cash Flow
discount factor
PV
NPV=
20%
0
-$1,000,000
1.000
-$1,000,000.00
1
$0
0.833
$0.00
2
$0
0.694
$0.00
3
4
$800,000
$800,000
0.579
0.482
$462,962.96 $385,802.47
-$151,234.57
Jason Laws
9 of 218
NPV Example
❑ Project C:
interest rate
Year
Cash Flow
discount factor
PV
20%
0
1
2
-$500,000
$800,000
$300,000
1.000
0.833
0.694
-$500,000.00 $666,666.67 $208,333.33
NPV=
$375,000.00
Jason Laws
3
$0
0.579
$0.00
4
$0
0.482
$0.00
10 of 218
NPV Example
❑ Project C has the highest NPV and therefore if only one project can be
undertaken it should be C.
❑ However if more than one project can be undertaken then both A and C
should be selected since they both have positive NPV’s.
❑ Project B should be rejected since it has a negative NPV and would
therefore destroy wealth.
❑ It makes sense that project C should have the highest NPV, since its cash
outflows are deferred relative to the other projects, and its cash flows are
early.
❑ In contrast project B has all the costs up front but the cash inflows are
deferred.
Jason Laws
11 of 218
Sample Question
(a) Using a discount rate of 12% calculate the net present value for each project. What
decision would you make based on your calculations? (4 marks)
Jason Laws
12 of 218
Sample Question
(b) How would your decision change if the discount rate used for calculating the net
present value is 20%? (4 marks)
Jason Laws
13 of 218
Internal Rate of Return
❑ The IRR of a project can be defined as the rate of discount which, when
applied to the projects cash flows, produces a zero NPV.
❑ The IRR decision rule is then:
❑ “invest in any project which has an IRR greater than or equal to some
predetermined cost of capital”.
❑ The comparison rate is usually the cost of capital, i.e. the discount rate
we would have used in a NPV analysis.
❑ Observation: At 12% we find the NPV of both projects to be positive, and
at 20% they are both negative. Hence the IRR must be between 10% and
20%. A 12% Gamma’s NPV is only 831 so we can expect that Gamma’s NPV
tends to zero before Kappa.
Jason Laws
14 of 218
Internal Rate of Return
Example
❑ You can purchase a turbo powered machine tool gadget for $4,000. The
investment will generate $2,000 and $4,000 in cash flows for two years,
respectively. What is the IRR on this investment?
2,000
4,000
NPV = −4,000 +
+
=0
1
2
(1 + IRR) (1 + IRR)
IRR = 28.08%
❑ Hint: You may recognize that if you multiply through by (1+IRR)2 you have
a quadratic problem that can be solved.
Jason Laws
15 of 218
Internal Rate of Return: Example
❑ If you don’t see it then here is the solution:
2,000
4,000
− 4,000 +
+
=0
1
2
(1 + IRR)
(1 + IRR)
− 4,000(1 + IRR) 2 + 2,000(1 + IRR) + 4,000 = 0
 2000 :
− 2(1 + IRR) 2 + (1 + IRR) + 2 = 0
x = 1 + IRR :
− 2x2 + x + 2 = 0
Does this look familiar?
Jason Laws
16 of 218
Internal Rate of Return: Example
− 2x2 + x + 2 = 0
− b  b 2 − 4ac
x=
, a = −2, b = 1, c = 2 :
2a
− 1  12 − 4(−2)(2) − 1  1 + 16
x=
=
2(−2)
−4
1 + 17
1 − 17
x=−
or −
−4
−4
x = −0.78 or + 1.28
x = 1 + IRR, IRR = x − 1
x = −0.78 − 1 = −1.78 or
x = 1.28 − 1 = 0.28 = 28%
Or if you are lazy:
http://www.mathsisfun.com/
quadratic-equationsolver.html
Jason Laws
17 of 218
Internal Rate of Return
2500
2000
IRR=28%
1000
500
10
0
90
80
70
60
50
40
30
-500
20
0
10
NPV (,000s)
1500
-1000
-1500
-2000
Discount rate (%)
Jason Laws
18 of 218
Internal Rate of Return
❑ The internal rate of return calculation is a trivial one to undertake assuming of course that you have a pc at your disposal.
❑ In 2023 I doubt there is anyone that undertakes an IRR calculation by hand
though it is possible:
❑ IRR = i0+ (NPV0/(NPV0+│NPV1│))*(i1 – i0)
❑ So we must take two guesses of NPV at two different interest rates and
linearly interpolate between the two.
❑ As the line is “non-linear” the closer to zero (either side the more accurate
the answer).
Jason Laws
19 of 218
Internal Rate of Return
❑ We take a guess at 25%:
ir=
0.25
Time
0
1
2
CF
$ -4,000.00 $ 2,000.00 $ 4,000.00
df
1
0.80
0.640
PV
$ -4,000.00 $ 1,600.00 $ 2,560.00
NPV=
$ 160.00
❑ The we guess again at 35%:
ir=
Time
CF
df
PV
NPV=
0.35
0
1
2
$ -4,000.00 $ 2,000.00 $ 4,000.00
1
0.74
0.549
$ -4,000.00 $ 1,481.48 $ 2,194.79
$ -323.73
Jason Laws
20 of 218
Internal Rate of Return
❑ Plugging the numbers in the formula:


NPV 0
IRR = i0 + (i1 − i0 ) 

 NPV 0 + NPV1 


160
IRR = 0.25 + (0.35 − 0.25)  
=
160 + − 323.73 
IRR = 0.25 + 0.1 0.3281
IRR = 28.28%
Jason Laws
21 of 218
Sample Question
5. A firm is considering two investment projects, Gamma and Kappa. These
projects are NOT mutually exclusive. Assume the firm is not capital constrained.
The initial costs and cash flows for these projects are:
❑ The key point here that many students overlooked is the words NOT mutually
exclusive. Mutually exclusive in this context means that only one project could
be chosen, for example, one use of a scarce resource or one production
technique etc. Here more than one can be chosen if they have a positive NPV.
Jason Laws
22 of 218
Sample Question
(c) Calculate an approximate IRR for each project. Assume the hurdle rate is
12%. What decision would you make based on your calculations?
(6 marks)
❑ Applying the IRR decision rule - “invest in any project which has an IRR greater
than or equal to some predetermined cost of capital” – as the IRR is greater than
the hurdle rate of 12% we accept BOTH projects.
Jason Laws
23 of 218
Payback
❑ This investment appraisal method calculates the time period taken to
payback the initial investment.
❑ When there are mutually exclusive investments or where ranking is
required, the project that has the earliest payback will be selected.
❑ Example: You are looking at a new project and you have estimated the
following cash flows:
❑ Capital Investment =£40,000
❑ Cashflows = £16,000 per annum for three years, £12,000 per annum
for the 4th year.
❑ Project Life = 4 years
❑ Salvage Value = Nil
❑ What is the payback?
Jason Laws
24 of 218
Payback
Year
Cash Flow (£)
Cummulative Cash Flow (£)
0
Cost
-£40.00
-£40.00
1
Savings
£16.00
-£24.00
2
Savings
£16.00
-£8.00
3
Savings
£16.00
£8.00
4
Savings
£16.00
£24.00
Payback Period = 2 + 8/16 = 2.5 years
❑ Observation: Unless advised otherwise payback assumes that cash
flow occurs throughout the year. Above we note that after two
years there is still a £8,000 shortfall. In year three the cash flows
are £16,000. if the cash flow is received throughout the year then
£8,000 is received 0.5 through the year.
Jason Laws
25 of 218
Payback
❑ Disadvantages:
❑ Ignores cash flows beyond the payback period
❑ Does not consider the profile of cash flows within the payback period
❑ Ignores discounting – but can use modified packback
❑ Advantages:
❑ Assumes cash flows after payback are so risky as to be of no value
❑ Easy to understand and calculate – useful screening and
communication device
❑ Observation: Modified payback uses the PV of cash flows rather than the
raw cash flows. As PV is less than the raw cash flow modified payback
must be longer than “normal” payback.
Jason Laws
26 of 218
Sample Question
(d) Calculate the payback period for each project. The company looks to select
investment projects paying back in 2 years. What decision would you make based on
your calculations? (2 marks)
£6,000/£17,000 = 0.35, so payback = 2 + 0.35 = 2.35.
Decision: As the payback of both projects is greater than the companies payback
policy then both projects should be rejected.
Jason Laws
27 of 218
Sample Question
(e) Explain what is meant by the ‘opportunity cost of capital’ in the context of the NPV
method.
(4 marks)
Observation: Many students made the observation that this is the return on an
alternative investment and/or that it is a market determined rate of return.
However, very few students note that “this is the return on an alternative
investment with the same level of risk”
Jason Laws
28 of 218
Sample Question
(f) Explain the additivity property of the NPV method.
(5 marks)
❑ Mutually exclusive projects are a set of projects of which only one can be
chosen at a given time. Firms have to choose one project among several
that do the same job: for example, a manually controlled machine versus
a computer controlled machine.
❑ Given that present values obey the additivity principle, it follows that the NPV
also possesses the additivity property. Assume that a firm has only two projects
(X and Y); the NPV of projects X and Y is equal to the NPV of project X plus the
NPV of project Y. (Note that the additivity property holds because present values
are all measured in today’s dollars.)
❑ This can be written as:
❑ NPV (X +Y ) = NPV (X ) + NPV (Y )
Jason Laws
29 of 218
Sample Question
❑ The additivity property implies that the value of the firm is simply the sum of the
values of the separate projects. Because of the additivity property, when there are
mutually exclusive projects, the NPV method indicates that the project with the
largest positive NPV should be adopted.
❑ The reason for this is that the project with the largest NPV generates the largest
NPV of the firm’s aggregated cash flows.
❑ One example clarifies the point that the choice of project relies on the additivity
property. Assume that project X is a positive NPV project, while project Y is a
negative NPV. The joint project (X+Y) will have a lower NPV than project X on its
own. The NPV enables managers to avoid choosing bad projects just because they
are packaged with good ones.
Jason Laws
30 of 218
The time value of money
❑ When people undertake to set aside money for investment something has to be
given up now, e.g. If someone buys shares in a firm now or lends to a business
there is a sacrifice of current consumption.
❑ Hence compensation is required to induce people to sacrifice consumption.
❑ Compensation will be required for at least three things:
❑ Impatience to consume – individuals generally prefer to have £1000 today
than £1000 in say five years time. The utility of £1000 now is greater than
£1000 in five years time. The rate of exchange between certain future
consumption and certain current consumption is the pure rate of interest.
This would occur even in the absence of inflation and risk.
❑ Inflation – in addition to the above compensation for time investors will also
have to be compensated for the loss in purchasing power.
❑ Risk – the promise of a receipt of a sum of money on the future generally
carries with it and element of risk – the payout may not take place or the
amount may be less than anticipated.
Jason Laws
31 of 218
The time value of money
❑ Example
❑ Consider an investor considering a £1,000 one-year investment and requires
compensation for these three elements:
❑ 2% is required to compensate for the pure time value of money,
❑ Inflation is anticipated to be 3% over the year
❑ To compensate the investor for impatience to consume and inflation the
investment needs to generate a return of:
❑ (1 + 0.02) x (1+.0.03) – 1 = .0506 = 5.06%
❑ The 5.06% may be regarded as the risk-free return (RFR), the interest rate
which is sufficient to induce investment assuming no uncertainty about cash
flows.
❑ Lending to governments, through the purchase of bonds and bills is typically
considered risk free.
Jason Laws
32 of 218
The time value of money
❑ However different investments carry different degrees of uncertainty about the
outcome of the investment.
❑ For instance, an investment on the Russian stock market, with its high volatility
may be regarded as more risky than the purchase of a share in say Exxon Mobil.
❑ Investors require different risk premiums on top of the risk free return to reflect
the perceived level of extra risk:
❑ Required Return (or time value of money) = RFR + Risk Premium
❑ Activity: List any models that allow you do determine the risk adjusted rate of
return?
Jason Laws
33 of 218
The time value of money
❑ In order to compare like with like it is important to value cash flows at the same
point in time, that could be the current time period or some time period in the
future.
❑ The conversion process is achieved by discounting all future cash flows by the time
value of money, thereby expressing the cash flows as an equivalent amount
received at time zero:
❑ F = P(1 + i)n
❑ Where F = future value, P = present value, i = interest rate, n = number of
years
❑ Example: if a saver deposited £100 in a bank account paying interest at 8 per cent
per annum, after three years the account will contain:
❑ 100 x (1 + 0.08)3 = £125.97
Jason Laws
34 of 218
The time value of money
❑ The formula can be inverted so we can ask the question “how much money must I
deposit now to receive £125.97 in three years?”
❑ P = F/(1 + i)n
❑ P = £125.97/(1 + .08)3 = £100
❑ Here we have discounted the £125.97 back to a present value of £100.
❑ We can also say that we have discounted the future cash flow back to “year zero”.
❑ More formally, in order to find the present value of some future cash flow we
multiply by the discount factor:
DF =
1
(1+ r ) t
Jason Laws
35 of 218
The time value of money
❑ Discount Factors can be used to compute the present value of any cash flow:
Ct
PV = DF  Ct =
t
(1 + r )
❑ Where Ct is the cash flow received in time period t, and r is the required
return/time value of money.
Jason Laws
36 of 218
The time value of money
❑ Present Values can be added together to evaluate multiple cash
flows:
PV =
C1
+ (1+ r ) 2 + ....
C2
(1+ r )
1
❑ Each individual cash flow is measure in “year zero” money and so
they can be added together, this is the additivity property of
discounted cash flow.
Jason Laws
37 of 218
The time value of money
❑ Consider how the discount factor varies with r and t.
T/r
1
2
3
4
5
6
7
8
9
10
1%
0.9901
0.9803
0.9706
0.961
0.9515
0.942
0.9327
0.9235
0.9143
0.9053
2%
0.9804
0.9612
0.9423
0.9238
0.9057
0.888
0.8706
0.8535
0.8368
0.8203
5%
0.9524
0.907
0.8638
0.8227
0.7835
0.7462
0.7107
0.6768
0.6446
0.6139
Jason Laws
10%
0.9091
0.8264
0.7513
0.683
0.6209
0.5645
0.5132
0.4665
0.4241
0.3855
15%
0.8696
0.7561
0.6575
0.5718
0.4972
0.4323
0.3759
0.3269
0.2843
0.2472
20%
0.8333
0.6944
0.5787
0.4823
0.4019
0.3349
0.2791
0.2326
0.1938
0.1615
38 of 218
The time value of money
❑ Sometimes there are shortcuts that make it very easy to calculate the present
value of an asset that pays out in different periods. These tools allow us to cut
through the calculations quickly.
❑ Perpetuity - Financial concept in which a cash flow is theoretically received forever.
cash flow
C1
PV of Cash Flow =
 PV =
discount rate
r
Think of a preference
share
❑ Annuity - An asset that pays a fixed sum each year for a specified number of years.
1
1 
PV of annuity = C   −
t
r
(
)
r
1
+
r


Jason Laws
39 of 218
Other time value of money topics
❑ The time value of money, i.e. a dollar today is worth more than a dollar tomorrow,
is fundamental to many topics in finance:
a) NPV
b) Dividend Discount Model
c) Bond pricing
❑ Only a) and c) were covered on the prelim paper. Here we cover topic b) which has
been covered on many PBF exam papers in the past.
Jason Laws
40 of 218
Valuation of Equities
❑ The dividend discount model is based on the premise that the market value of
ordinary shares represents the sum of expected future dividends, to infinity,
discounted to time zero.
❑ Consider a shareholder who intends holding a share for one year. A single dividend
will be paid at the end of the holding period, d1 and the share will be sold at a
price p1 in one year.
❑ To derive the value of a share at time 0 to this investor the future cash flows d1 and
p1 need to be discounted at a rate which includes an allowance for the risk of the
share, k:
d1
p1
p0 =
+
1+ k 1+ k
Jason Laws
41 of 218
Valuation of Equities
❑ Where does p1 come from?
❑ Consider a second investor who expects to hold the share for a further year and
sell at time 2 for P2, the price, p1, will be:
d2
p2
p1 =
+
1+ k 1+ k
❑ Substituting into the equation for p0 we get:
d1
d2
p2
p0 =
+
+
2
2
1+ k (1+ k ) (1+ k )
Jason Laws
42 of 218
Valuation of Equities
❑ If a series of one year investors bought this share, and we in turn solved for p2, p3
etc. we would find:
d1
d2
d¥
p0 =
+
+....
2
¥
1+ k (1+ k )
(1+ k )
❑ The terminal stock price can effectively be ignored as its present value is zero.
However, it value feeds in to p∞-1 etc. And so by iteration it enters into p0.
Jason Laws
43 of 218
Example: Stock 1st
❑ Consider stock 1st that has just paid a dividend of $10 and is expected to pay this
dividend forever. If the cost of capital, k, is 5% what is the value of stock 1st?
❑ We could use Excel and find the PV the future dividends.
k=
Time
1
2
3
4
5
6
7
8
9
10
5%
$
$
$
$
$
$
$
$
$
$
Div
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
PV(Div)
$ 9.52
$ 9.07
$ 8.64
$ 8.23
$ 7.84
$ 7.46
$ 7.11
$ 6.77
$ 6.45
$ 6.14
Jason Laws
Sum (years 1 - 500)= $ 200.00
44 of 218
Example: Stock 1st
❑ But of course this is analogous to a perpetuity so we could have also found the
price using:
❑ P0 = Div/k = $10/0.05 = $200
❑ In this example the PV tends towards zero, in fact it is less than 10 cents after 90
(ish) years, and 1 cent after 150 (ish) years.
❑ Of course the bigger is k, the faster the PV tends towards zero.
Jason Laws
45 of 218
Example: Stock 2nd
❑ Consider stock 2nd who will not pay a dividend for the next five years but in year
six it will pay a dividend of $10 and it is expected to pay this dividend forever.
❑ If the cost of capital, k, is 5%, what is the value of stock 2nd?
❑ Again, we can map out the future dividends in Excel, find the PV’s and sum them
together.
k=
Time
1
2
3
4
5
6
7
8
9
10
5%
$
$
$
$
$
$
$
$
$
$
Div
10.00
10.00
10.00
10.00
10.00
PV(Div)
$
$
$
$
$
$ 7.46
$ 7.11
$ 6.77
$ 6.45
$ 6.14
Jason Laws
Sum (years 1 - 500)= $ 156.71
46 of 218
Example: Stock 2nd
❑ Clearly the answer is lower than that of Stock 1st. We could have found the answer
by taking the value of Stock 1st and subtracting the sum of the PV’s from years 1 to
5 (in fact this is an annuity, for which there is a formula).
❑ Or we can think laterally...
❑ Figuratively speaking, standing at the end of year 5 we know that the Stock 2nd will
pay a dividend of $10 and pay this dividend forever.
❑ In “year five money” the value of Stock 2nd is:
❑ P5 = $10/0.05 = $200
❑ In order to get the value in “year zero money” we need to discount at 5% over 5
years:
❑ P0 = $200/1.055= $156.71
Jason Laws
47 of 218
Valuation of Equities
❑ According to the discounted cash flow model, the equilibrium price of a share is
equal to the present discounted cash flow of expected future dividend payments.
 Dt +i 

Pt =  Et 
i 
i =1
 (1 + k ) 

❑ However, in order to compute this value, one has to make assumptions about
future dividends.
❑ We have considered a constant dividend but what if dividends grow at some
constant rate, g?
Jason Laws
48 of 218
Valuation of Equities
❑ So that:❑ Dt+1 = Dt(1+g)
❑ Et(Dt+2) = Dt+1(1+g)
❑ Et(Dt+3) = Et(Dt+2)(1+g) = Dt+1(1+g)2
❑ Et(Dt+i) = Et(Dt+i-1)(1+g) = Dt+1(1+g)i-1
❑ Substituting this result in the equation on the previous slide we get:-
 Dt +i 
Dt +1 (1 + g )

Pt =  Et 
=
i 
i
i =1
 (1 + k )  i =1 (1 + k )


Jason Laws
i −1
49 of 218
Valuation of Equities
❑ If we can get the summation sign from zero to infinity rather than from 1 to infinity
we can treat the result as a geometric series.
❑ Note, if we start summing from 1 we must increase the index within the
summation by 1 to give:-
Dt +1 (1 + g )
Pt = 
i
(1 + k )
i =1
i −1

(
1+ g )
= Dt +1 
i +1
i = 0 (1 + k )

i
(
Dt +1
1+ g )
1+ g 
= Dt +1 
=



i
1
(1 + k ) i =0  1 + k 
i = 0 (1 + k ) (1 + k )

i
Jason Laws

i
50 of 218
Valuation of Equities
❑ The last expression on the right hand side is a geometric expression.
❑ If g < k so that the entire bracketed term < 1 we can use the following rule:
1
a =
if a  1

1− a
i =0
i
❑ Then:-
Dt +1
Dt +1
Dt +1
1
1+ g 
Pt =
=

 =

(1 + k ) i =0  1 + k  (1 + k ) 1 − 1 + g k − g
1+ k

i
Jason Laws
51 of 218
Example: Stock 3rd
❑ Consider stock 3rd that has just paid a dividend of $10 and is expected to grow
dividends at 2% forever.
❑ If the cost of capital, k, is 5% what is the value of stock 3rd
❑ Again, we can map out the future dividends in Excel, find the PV’s and sum them
together
k=
g=
Time
1
2
3
4
5
6
7
8
9
10
$
$
$
$
$
$
$
$
$
$
5%
2%
Div
10.20
10.40
10.61
10.82
11.04
11.26
11.49
11.72
11.95
12.19
D0=
$ 10.00
PV(Div)
$ 9.71
$ 9.44
$ 9.17
$ 8.91
$ 8.65
$ 8.40
$ 8.16
$ 7.93
$ 7.70
$ 7.48
Jason Laws
Sum (years 1 - 500)= $ 340.00
52 of 218
Example: Stock 3rd
❑ Or we can use the formula:
❑ Pt = $10 x 1.02/(0.05 – 0.02) = $10.2/0.03 = $340.
❑ Activity:
❑ Stock A is expected to pay a dividend of £10 forever;
❑ Stock B is expected to pay a dividend of £8 next year with dividend growth
expected to be 3% per annum thereafter.
❑ Stock C just paid a dividend of £6 with dividend growth expected to be 4% p.a.
thereafter.
❑ If the required return on similar equities is 10%, calculate the value of each stock.
Jason Laws
53 of 218
Valuation of Equities
❑ Given the expected next period dividend, the anticipated dividend growth rate
and the discount rate we can calculate the theoretical equilibrium price of the
security.
❑ Alternatively, given the current price, expected end of period dividends and the
anticipated growth rate we can obtain the implicit discount rate.
❑ This is the cost of equity capital facing the company.
❑ 𝑃𝑡 =
𝐷𝑡+1
𝑘−𝑔
→𝑘=
𝐷𝑡+1
𝑃𝑡
+𝑔
Jason Laws
54 of 218
Dividend Growth Model II
❑ What if we do not anticipate a constant rate of growth of dividends ?
❑ In this case we would need to forecast all future dividend payments - impossible !
❑ One possible solution is to assume that dividends grow at a constant rate over a
certain period and then grow at a different rate over a second period etc.
❑ Clearly the simplest model to solve is the two period growth model in which
dividends grow at g1 in periods 1 to N and that at g2 in periods N+1 to infinity.
Jason Laws
55 of 218
Example: Stock 4th
❑ Consider stock 4th that has just paid a dividend of $10 and is expected to maintain
this stable dividend for the next five years and then grow dividends at 2% forever.
❑ If the cost of capital, k, is 5% what is the value of stock 4th?
❑ Again, we can map out the future dividends in Excel, find the PV’s and sum them
together.
k=
5%
D0=
$ 10.00
g=
Time
1
2
3
4
5
6
7
8
9
10
$
$
$
$
$
$
$
$
$
$
2%
Div
10.00
10.00
10.00
10.00
10.00
10.20
10.40
10.61
10.82
11.04
PV(Div)
$ 9.52
$ 9.07
$ 8.64
$ 8.23
$ 7.84
$ 7.61
$ 7.39
$ 7.18
$ 6.98
$ 6.78
Jason Laws
Sum (years 1 - 500)= $ 309.69
56 of 218
Example: Stock 4th
❑ Or again we can think laterally.
❑ In “year five money” the value of Stock 4th is:
❑ P5 = ($10x 1.02)/(0.05-0.02) = $340
❑ In order to get the value in “year zero money” we need to discount at 5% over 5
years:
❑ P0 = $340/1.055= $266.40
❑ But then of course we have the PV of the dividends received in years 1 through 5:
Time
1
2
3
4
5
$
$
$
$
$
Div
10.00
10.00
10.00
10.00
10.00
PV(Div)
$ 9.52
$ 9.07
$ 8.64
$ 8.23
$ 7.84
$ 43.29
❑ Giving a total of $309.69 ($266.40 + $43.29)
Jason Laws
57 of 218
Example: Stock 5th
❑ Consider stock 5th that has just paid a dividend of $10 and is expected to grow this
dividend at a rate of 10% p.a. for the next five years and then grow dividends at
2% forever.
❑ If the cost of capital, k, is 5% what is the value of stock 5th?
❑ Think intuitively...
Jason Laws
58 of 218
A more complex example
❑ Vornado is a company that has patent rights for a new mobile phone technology
that is expected to enable it to generate growth in earnings of 20% for the next 3
years. After that (from the start of year 4) the company expects to see earnings
growth drop to a constant rate of 5%.
❑ Assuming that the company pays out 60% of earnings as dividends and that the
last dividend payment made by the company was $2.20, calculate an estimate of
the current price of Vornado. Assume the required return on equity is 8%.
Jason Laws
59 of 218
A more complex example
❑ This is rather a difficult exercise as it has a number of complexities relative to
previous examples.
❑ Was the reference to the company paying out 60% of earnings as dividends and
growing earnings by 20% a “red-herring”.
❑ Let’s assume some Earnings and Share values.
❑
❑
❑
❑
Year 0
Earnings = £3,666,666.67
Number of shares = 1,000,000
What is the total dividend and the dividend per share?
Jason Laws
60 of 218
A more complex example
❑ Dividend = 60% x £3,666,666.67 = £2,200,000
❑ Dividend per share = £2.2
❑ Note we arrive at the same answer if the number of shares are 500,000 and the
Earnings are £1,833,333.33.
❑ If the firm grows earnings at 20% per annum find the new earnings level, total
dividend and dividend per share.
❑
❑
❑
❑
❑
Year 1
Earnings=
Dividends=
Number of shares=
Dividend per share=
£4,400,000.00
£2,640,000.00
1000000
£2.64
Jason Laws
61 of 218
A more complex example
❑ Dividends have increased from £2.20 to £2.64, an increase of:
❑ [£2.64 - £2.20]/£2.20 = 20%
❑ So if a firm is to pay a constant proportion of earnings then earnings growth and
dividend growth are the same.
❑ D0= cE/n
❑ D1 = cE(1+g)/n
❑ (D1-D0)/D0 = g
Jason Laws
62 of 218
A more complex example
❑ The example can now be expressed as:
❑ Vornado is a company that has patent rights for a new mobile phone technology
that is expected to enable it to generate growth in DIVIDENDS of 20% for the next
3 years. After that (from the start of year 4) the company expects to see
DIVIDENDS growth drop to a constant rate of 5%.
❑ The last dividend payment made by the company was £2.20, calculate an estimate
of the current price of Vornado.
❑ Assume the required return on equity is 8%.
Jason Laws
63 of 218
c  Earnings
DPS0 =
n
c  Earnings  (1 + g )
DPS1 =
n
c  Earnings  (1 + g ) c  Earnings
−
DPS1 − DPS 0
n
n
=
=
c  Earnings
DPS 0
n
c  Earnings
(1 + g − 1)
n
=g
c  Earnings
n
Jason Laws
64 of 218
A more complex example
❑ Now we need to map out the future stream of dividends:
Year
Dividend
1
£2.64
2
£2.64 + 20% = £3.17
3
£3.17 + 20% = £3.80
4
£3.80 + 5% = £3.99
❑ In year 4 dividend growth drops to 5% and continues to grow at 5% into infinity.
Jason Laws
65 of 218
A more complex example
❑ We are applying:
 Dt +i 

Pt =  Et 
i 
i =1
 (1 + k ) 

• Finding the sum over the first three periods is trivial:
• £2.64/1.08 + £3.17/1.082 + £3.80/1.083=£8.18.
• But how do we find the value, in today’s terms, of the dividends
received after year 3?
• Well if I asked you at the end of year 3, when the dividend of £3.80
had just been paid, and dividends were forecast to grow at 5%, what
the price of the share was then I hope you would say:
Jason Laws
66 of 218
A more complex example
Dt +1 £3.80 1.05
Pt =
=
= £133.00
k−g
.08 − .05
❑ However that is in “year 3 money” and so to find the value now we must find
its present value:
❑ £133.00/1.083 = £105.58
❑ Adding this to the PV of the dividends from years 1 to 3 we get a current
share price of:
❑ £8.18 + £105.58 = £113.76
Jason Laws
67 of 218
Some dividend data
❑ The dividend per share of British American Tobacco since 2002 is:
18
𝑔=
208.55
− 1 = 10.79%
33
Jason Laws
Year
Div
2002
33
2003
36.3
2004
39.7
2005
43.2
2006
48.7
2007
58.8
2008
58.8
2009
2010
69.7
89.5
2011
104.8
2012
119.1
2013
130.6
2014
144.9
2015
150
2016
155.9
2017
174.6
2018
190
2019
201.05
2020
208.55
68 of 218
Yield to Maturity
❑ A bond’s yield to maturity (ytm) is the interest rate implied by the payment
structure,
❑ i.e. the interest rate at which the PV of the income stream equals the current
bond price.
❑ Yields are always quoted on an annual basis.
❑ Let T be the maturity of the bond and C(1), C(2) … C(T) be the future cash flows,
the ytm is the rate of return which satisfies:-
C (1) C (2)
C (T )
P=
+
+ ... +
2
1 + y (1 + y )
(1 + y )T
❑ It is clear that there is an inverse relationship between the price of a security and
its ytm. If the ytm increases the market price of the bond will decreases
Jason Laws
69 of 218
Bond Pricing
❑ When the yield is below the coupon rate, the bond will be priced at a premium to
its par value.
❑ When the yield is above the coupon rate, the bond will be priced at a discount to
its par value.
❑ Hint: Remember this relationship and so you have an idea of what the price will be
before calculating a bond price.
❑ The price-yield relationship is not a straight line; rather it is convex. As yields
decline, the price increases at an increasing rate; as the yield increases, the price
declines at a declining rate.
Jason Laws
70 of 218
Price Yield Curve
400
This is a price yield chart drawn for a 30 year
bond with a face value of $100 paying a $10
coupon each year
350
300
Price
250
200
150
100
50
0
0
5
10
15
20
25
Yield to maturity
Jason Laws
71 of 218
Zero coupon Bonds
❑ A bond that pays a single cash flow at maturity is referred to as a single coupon
bond.
❑ The calculation of the yield to maturity for such bonds is very easy.
❑ For example, consider a zero-coupon bond maturing in ten years with a maturity
value of $1,000.
❑ If the current price of the bond is $311.80 then the yield to maturity is:
$311.80 = $1,000/(1+ ytm)10  ytm = 12%
Jason Laws
72 of 218
Duration
❑ Definition – how sensitive the price is to the change in yield; % change in price for
a given small change in the ytm.
P (1 + r )
D=−
r P
(
  T Ct 
1+ r)
 T
−t 
=−
 T
=−
Ct (1 + r )  


t 

Y  t =1 (1 + r )  
Y  t =1
Ct 
 

t 
(
)
1
+
r
 t =1

(1 + r )
Ct 

t 
(
)
1
+
r
 t =1

T
T
tCt

t
T
T
(
)
(
)
tC
1
+
r
1
+
r
−t −1
t =1 (1 + r )
t
= +  tCt (1 + r )  T
=

=
t +1
P


 T Ct 
(
)
1
+
r
Ct
t =1
t =1


t 
t 
(
)
(
)
1
+
r
1
+
r
 t =1

 t =1

Jason Laws
73 of 218
Duration
❑ Expanding out the summation sign:
C1
C2
CT
1
+ 2
+ ... + T 
2
T
(
1+ i)
(
(
1+ i)
1+ i)
D=
P
❑ The duration can therefore be calculated by computing the Present Value (PV) of
the cash flows, and then multiplying them by the time indices.
❑ The Duration of a zero coupon bond is simply its Maturity.
CT
T
T
(
1+ i)
D=
=T
P
Jason Laws
74 of 218
Duration
❑ The effective duration of a coupon bond is lower than its stated maturity.
❑ For example a 10 year coupon bond with an 8% ytm.
Jason Laws
75
75 of 218
Sample Question
6. (a) Explain the concept of Macaulay duration and explain why a coupon paying
bond has a duration less than its maturity.
(4 marks)
❑ Observation – Many students simply regurgitated the guide and even included
material on the relationship between duration and yield to maturity which was
not relevant.
Jason Laws
76 of 218
Sample Question
(b) Calculate the Macaulay duration of a four year 5% coupon bond where the market
interest rate is 4%. Assume the par value of the bond is $1000. And coupons are paid
annually
(4 marks)
Jason Laws
77 of 218
Modified Duration
❑ This measure of Duration can also be used to obtain the percentage change in a
bonds price as a function of a basis point change in the yield to maturity.
❑ It can be shown that:-
P0
(1 + i )
= −D
(1 + i )
P0
❑ Security firms tend to divide D by (1+i) and call the result modified duration. So
that:-
D
Dm =
1+ i
P0
(1 + i )
(1 + i )
= −D
= − Dm  (1 + i )
= − Dm  (1 + i )
(1 + i )
(1 + i )
P0
Jason Laws
78 of 218
Modified Duration
P0
= − Dm  (1 + i )
P0
❑ D (Macaulays) = 3.73 years.
❑ Current Price = $1036.30
❑ Current ytm = 4%
❑ Activity:
❑ Draw bond price against interest rate.
❑ Estimate the new price of the bond if interest rates change to 4.25%,
4.5%, 5%.
Jason Laws
79 of 218
Characteristics of Macaulays Duration
1.
2.
3.
4.
5.
The Duration of a coupon bond will always be less than its term to
maturity.
An inverse relationship exists between coupon and duration – a bond
with a larger coupon will have a shorter duration as more of the total
cash flows come earlier.
A bond with no coupon payments will have a duration equal to its term
to maturity.
A positive relationship generally holds between term to maturity and
duration, but duration increases at a decreasing rate with maturity.
There is an inverse relationship between ytm and duration.
Jason Laws
80 of 218
Jason Laws
81 of 218
Duration versus Beta
❑ If you expected a large drop in interest rates would you prefer to hold high or low
duration bonds? Why?
P0
= − Dm  (1 + i )
P0
❑ The larger is Duration the more a given change in interest rates is amplified.
❑ Analogously the larger the Beta of a portfolio then the more a movement in the mart is
amplified.
❑ Note that just like Beta’s, Durations are linearly additive.
N
D p =  wi Di
i =1
Jason Laws
82 of 218
Sample Question
(c) An investor decides to construct a bond portfolio made up of $10,000 in the four year
5% coupon bond (Par = $1000, ytm = 4%) and $30,000 in a three year zero coupon bond
(par value = $1,000). What is the Macaulay duration of this bond portfolio?
(4 marks)
Jason Laws
83 of 218
Sample Question
(c) An investor decides to construct a bond portfolio made up of $10,000 in the four year
5% coupon bond (in (b) above) and $30,000 in a three year zero coupon bond (par value =
$1,000). What is the Macaulay duration of this bond portfolio?
(4 marks)
Jason Laws
84 of 218
Sample Question
(d) For the four year 5% coupon bond described in (b) above, would the Macaulay
duration of this bond increase or decrease if the market interest rate increases from
4% to 5%? Explain your answer.
(3 marks)
Tip: Duration is approximately the slope of the price-yield curve. As interest rates
rise the price yield curve gets flatter, hence Duration falls.
C1
C2
C2
C4
1´
+2´
+ 3´
+4´
2
3
4
1+
i
1+ i)
1+ i)
1+ i)
( )
(
(
(
D=
C1
C2
C2
C4
+
+
+
2
3
4
1+
i
( ) (1+ i) (1+ i) (1+ i)
Jason Laws
85 of 218
Sample Question
❑ As i increases the denominator falls – note negative relationship between price
and interest rate.
❑ However in the numerator the weight on the 4, 3, 2 and 1 gets smaller so the
numerator gets smaller overall. Hence overall Duration decreases.
Jason Laws
86 of 218
Sample Question
(e) Estimate, using modified duration, the change in the price of the four year 5%
coupon bond if the market interest rate decreases from 4% to 3%. (4 marks)
❑ Here we use the following relationship:
P0
(1 + i )
= −D
(1 + i )
P0
❑ P/P =( -3.73 x -0.01)/1.04 = + 0.0373/1.04 = 0.03587 = 3.59%
❑ New P = (1 + 0.03587) x 1036.30 = $1,073.47.
❑ Observation: Many students simply re-estimate rather than using Duration.
Jason Laws
87 of 218
Sample Question
(f) Estimate, using modified duration, the change in the price of the three year zerocoupon bond if the market interest rate decreases from 4% to 3%. (3 marks)
❑ Here we use the following relationship:
P0
(1 + i )
= −D
(1 + i )
P0
❑
❑
❑
❑
P/P =( -3 x -0.01)/1.04 = + 0.03/1.04 = 0.0288 = 2.88%
We cannot find the new price without the original price.
Original price = $1000/1.043 = $889.00
New P = (1 + 0.0288) x 889.00 = $914.60
Jason Laws
88 of 218
Sample Question
(g) Explain why the modified duration measure only gives good estimates when the
change in the market interest rate being considered is small.
(3 marks)
❑ Note back to the equation to find the price of a bond. It includes squared terms,
cubed terms etc. This creates a curved relationship between price and yield.
❑ The greater the maturity of the bond the more curved this line will be.
❑ Not further that Duration is essentially the slope of the line at the current yield to
maturity. If we draw a tangent at the current yield we will note that the tangent
gets further away from the price-yield curve the further along the ling we go.
❑ Hence, due to the convex relationship between price and yield, if we use Duration
to estimate new bond prices, for a large change in interest rates, our estimate will
differ substantially compared to the “real price” – see diagram..
Jason Laws
89 of 218
Convexity
Price Yield Curve
Error in Estimating Price
based only on Duration
400
This is a price yield chart drawn for a 30 year
bond with a face value of $100 paying a $10
coupon each year
350
300
Price
250
200
Actual Price
150
100
50
0
0
5
10
15
20
25
Yield to maturity
Jason Laws
90 of 218
Sample Question
❑ Calculate the price and Macaulay duration of the following two bonds.
Note that both bonds pay annual coupons, have par values of £100
and the current market interest rate is 8%.
Bond
Maturity
Coupon
A
7 years
3%
B
4 years
10%
Jason Laws
91 of 218
Sample Question
❑ Bond A
Time
1
2
3
4
5
6
7
CF
£ 3.00
£ 3.00
£ 3.00
£ 3.00
£ 3.00
£ 3.00
£ 103.00
Price=
PV
£ 2.78
£ 2.57
£ 2.38
£ 2.21
£ 2.04
£ 1.89
£ 60.10
£ 73.97
D=
Mod D=
Jason Laws
t x PV
2.78
5.14
7.14
8.84
10.20
11.34
420.70
£ 466.14
6.30
5.83
92 of 218
Sample Question
❑ Bond B
Time
1
2
3
4
CF
PV
t x PV
£ 10.00 £ 9.26
9.26
£ 10.00 £ 8.57
17.14
£ 10.00 £ 7.94
23.82
£ 110.00 £ 80.85 323.40
Price= £ 106.62 £ 373.62
D=
3.500
Mod D=
3.24
Jason Laws
93 of 218
Sample Question
❑ (b) Use the modified duration formula to estimate the % change in
price of bond A and bond B if interest rates were to rise by 1%.
P0
= − Dm  (1 + i )
P0
• Modified Duration is 5.83 and 3.24 years respectively.
• The % change in price would therefore be:
• Bond A: -5.83 x 0.01 = -5.83%
• Bond B -3.24 x 0.01 - -3.24%
Jason Laws
94 of 218
Sample Question
❑ (c) For an investor looking for the lowest risk of capital loss on their
investment which bond would you recommend?
❑ Since Bond B has the lowest modified duration then an investor holding this
bond will suffer the least in the event of rise in interest rates. A risk averse
investor should therefore choose Bond B.
Jason Laws
95 of 218
Sample Question
❑ Explain, using appropriate diagrams, why the modified duration
formula only provides an estimate of the interest rate sensitivity of a
bond
❑ See price-yield relationships for Bond A – note the value of the bond when
the coupon = yield = 4%.
Jason Laws
96 of 218
Bond Convexity
140
Predicted Change using duration
120
Actual Change - less than
predicted due to convex nature of
price yield relationship.
100
80
Price A
60
40
20
Change in interest rates
0.245
0.23
0.215
0.2
0.17
0.185
Jason Laws
0.155
0.14
0.125
0.11
0.095
0.08
0.065
0.05
0.035
0.02
0.005
0
97 of 218
Alternative Exercise
❑ It is the the end of June 2021 a UK corporate bond has a coupon rate
of 3.5%, par (face) value of £1,000 and will mature in June 2024.
❑ Using the data given above and assuming semi-annual coupons and a
discount rate equal to 2.5% p.a., calculate the value of the corporate
bond.
Jason Laws
98 of 218
Alternative Exercise
❑ It is the end of June 2021 a UK corporate bond has a coupon rate of
3.5%, par (face) value of £1,000 and will mature in June 2024.
❑ Using the data given above and assuming semi-annual coupons and a
discount rate equal to 2.5% p.a., calculate the value of the corporate
bond.
Pcb, sem = PV =
17.5
17.5
17.5
17.5
17.5
1,017.5
+
+
+
+
+
1.0125 (1.0125 )2 (1.0125) 3 (1.0125) 4 (1.0125) 5 (1.0125) 6
= £1,028.73
Jason Laws
99 of 218
Alternative Exercise
❑ Calculate the duration of the UK corporate bond assuming annual coupons
and annual discount rate.
T
CF
DF
CFxDF
CFxDFxT
1
35
0.961538
33.65385
33.65385
2
35
0.924556
32.35947
64.71893
3
35
0.888996
31.11487
93.34462
4
35
0.854804
29.91815
119.6726
5
35
0.821927
28.76745
143.8372
6
1035
0.790315
817.9755
4907.853
973.7893
5363.08
❑ D= 5.51 years
Jason Laws
100 of 218
Alternative Exercise
❑ Assume annual interest rates rise by 1% from 4% to 5%. What will be the
approximate percentage change in the value of the UK bond assuming
annual coupon and annual discount rate?
%P  − D x
i
0.01
= − 5.51x
= −0.05298
1+ i
1.04
Jason Laws
101 of 218
Alternative Exercise
❑ Activity:
❑ Compare the problems of estimating future cash flows for government
bonds, corporate bonds and common stock.
❑ Write an essay plan include aspects of this essay that you would include.
Jason Laws
102 of 218
The term structure of interest rates
❑ Compare and contrast expectations theory, liquidity premium theory and market
segmentation in explaining the term structure of interest rates.
❑ A yield curve plots the yields of bonds with different maturity but the same risk.
❑ Usually the yield curve is constructed from government securities – same low
default risk,
❑ Yield = yield to maturity = IRR (discount rate) of the bond.
Jason Laws
103 of 218
The term structure of interest rates
❑ Example yield curves:
Jason Laws
104 of 218
The term structure of interest rates
❑ The current US yield curve is shown below:
Jason Laws
105 of 218
The term structure of interest rates
❑ The expectations theory of the term structure of interest rates states that in
equilibrium, the long-term rate is a geometric average of today’s short term rate
and expected short-term rates in the future.
❑ (1+R)2 = (1+r1) (1+r2)
❑ Where:
❑
❑
R = annual yield on a 2 year bond
r1 = annual return on a 1 year bond
r2 = one year forward rate beginning in 1 years time
❑ Arbitrage ensures the LHS = RHS
Jason Laws
106 of 218
The term structure of interest rates
❑ Suppose that the yield on a two-year government bond, R is 9% p.a. and the yield on an
equivalent one year bond, r1 is 8% p.a. The yield implied on a one year bond held during
year two of the two year bond’s life, r2, is given as (assuming £1,000 invested):
❑ £1,000 x (1.09) x (1.09) = £1,188.10 = £1,000 x (1.08) x (1 + r2)
❑ r2 = 10.01%
❑ So when interest rates expected to rise (i.e. r2 > r1) then the long rate, R is greater than
the short rate r1 - Hence yield curve is upward sloping.
❑ If r2 < r1 (yields expected to fall) then R < r1
❑ - Hence downward sloping yield curve
❑ So expectations of future changes in interest rates determine the yield curve
❑ However, the expectations theory does not help explain why the yield curve we observe
is normally upward sloping.
❑ (i.e. interest rates not always expected to rise)
Jason Laws
107 of 218
The term structure of interest rates
❑ Liquidity premium theory asserts that, in a world of uncertainty, investors and
lenders will want to hold assets which can be converted into cash quickly.
Therefore they will demand a liquidity premium for holding long term debt.
❑ (1+R)2 = (1+r1) (1+r2 + L)
❑ where L = liquidity premium
❑ - Hence long term rates will normally be greater than short term rates
Jason Laws
108 of 218
The term structure of interest rates
❑ The market segmentation theory, suggests that the bond market is actually made
up of a number of separate markets distinguished by time to maturity, each with
their own supply and demand conditions.
❑ Hence no relationship between yields for different maturities
❑ i.e. long rates determined in market for long maturity bonds, short rates
determined in market for short maturity bonds
❑ The theory that best describes the normal upward sloping yield curve = liquidity
preference
Jason Laws
109 of 218
Default Risk
❑ We have just discussed a number of reasons why the rates of return on bonds may
differ across maturity.
❑ It is also the case that bonds of the same maturity offer a different rate of return.
❑ Why ?
❑ Due to the risk of default.
❑ The difference between the return on “safe” bonds (i.e. government) and
“risky” bonds is called a default premium.
❑ How do you assess risk ?
❑ Moodys and S&P - evidence suggest that low ranked bonds promise higher
returns.
Jason Laws
110 of 218
Bond Ratings
❑ Since the primary function of bonds as an investment vehicle is to make fixed
payments, it's essential that the company or government issuing the debt has the
ability to make all payments on time and in full. Bond ratings evaluate the debt
issuer to determine the risk of default.
❑ The leading rating agencies, Standard & Poor's and Moody's Investors Services,
assign ratings when a bond is first issued, and that rating helps determine how
high the bond's interest rate will be. If the agencies assign a high rating, that
means there's little risk of default, so the issuer can obtain a lower interest rate.
❑ While the rating systems of Moody's and S&P differ somewhat, they're more alike
than different. Both agencies have investment-grade ratings, which connote a high
level of creditworthiness, and speculative ratings, which mean higher risk levels
and merit higher interest rates.
❑ Here are Moody's ratings, from highest to lowest. Investment grade: Aaa, Aa1,
Aa2, Aa3, A1, A2, A3, Baa1, Baa2, Baa3. Speculative grade: Ba1, Ba2, Ba3, B1, B2,
B3, Caa1, Caa2, Caa3, Ca, C1.
❑ Here are S&P's ratings. Investment grade: AAA, AA+, AA, AA-, A+, A, A-, BBB+, BBB,
BBB-. Speculative grade: BB+, BB, BB-, B+, B, B-, CCC+, CCC, CCC-, CC, D.
Jason Laws
111 of 218
Interest rate risk management
❑ Income Gap Analysis
❑ - the difference between interest sensitive assets and interest sensitive
liabilities
❑ The Fed classifies assets and liabilities into the following maturity buckets:
❑ Overnight
❑ 1 day to 3 months
❑ 3 months to 6 months
❑ 12 months to 5 years
❑ Over 5 years
Jason Laws
112 of 218
Interest rate risk management
❑ Under the income gap analysis banks report the gap in each maturity
bucket:
❑ GAP = RSA – RSL
❑ A positive gap implies sensitive assets > sensitive liabilities:
❑ So if interest rates rise the banks interest revenue will be rising faster
than interest cost and net interest margin and income will rise.
❑ Vice versa for a fall in interest rates.
❑ I = GAP x i
Jason Laws
113 of 218
Interest rate risk management
❑ Income Gap analysis is essentially a book value accounting cash flow
analysis of the gap between interest revenues and interest costs over a
period of time.
❑ It therefore ignores the time value of money.
❑ When interest rates change there is a market value effect - change in
PV of the cash flows of assets and liabilities and a income effect
(interest received or paid)
❑ Even rate insensitive assets and liabilities have a component that is rate
sensitive.
❑ A bank receives a runoff cash flow from these rate insensitive items that can
be reinvested at the current market rates.
❑ e.g. A fixed rate mortgage repaid within one year.
Jason Laws
114 of 218
Interest rate risk management
❑ Can try and estimate the size of the run off cash flows.
❑ Income Gap Analysis also ignores the effects of the changes in interest
rates on off balance sheet items.
Jason Laws
115 of 218
Interest rate risk management
❑ Consider the following balance sheet (in millions).
❑ Assume further that 20% of fixed rate mortgages are repaid within the
year and 20% of savings deposits are rate sensitive.
Assets
Liabilities and
Equity
Variable Rate
Mortgages
20
Money Market
Deposits
5
Fixed Rate
Mortgages
25
Savings deposits
20
Commercial
Loans
50
Variable Rate CD
(< 1 year)
30
Physical Capital
5
Equity
45
Total Assets
100
Total Liabilities
100
Jason Laws
116 of 218
Interest rate risk management
❑ Assume further that 20% of fixed rate mortgages are repaid within the
year and 20% of savings deposits are rate sensitive.
❑ RSA = 20 + .2 x 25 + 50 = 75m
❑ RSL = 5 + .2 x 20 + 30 = 39m
❑ GAP = RSA – RSL = 75 – 39 = 36m
❑ If interest rates increase from 8 per cent to 9.5 per cent:
❑ I = GAP x i = 36 x (.015) = 0.54m
❑ Or:
❑ Increase in income on assets = 0.015 x 75 = 1.125m
❑ Increase on payments on liabilities = 0.015 x 39 = 0.585m
❑ Increase in net income = 0.54m
Jason Laws
117 of 218
Interest rate risk management
❑ A market value -based model of measuring and managing interest rate risk
is the so called duration gap analysis.
❑ Under duration gap analysis banks are able to take into account the effects
of changes in interest rates on both income and market value.
❑ Banks are also able to immunise their balance sheets against interest rate
risk.
❑ In order to implement duration analysis bank managers need to determine
the duration of all assets and liabilities.
❑ Fortunately, like beta (to follow) and NPV duration is linearly additive.
N
D p =  wi Di
i =1
Jason Laws
118 of 218
Interest rate risk management
❑ The overall duration gap can then be calculated as:
DURgap
L
= DURassets −    DURliabilitie s
 A
❑ Where L = market value of liabilities and A = market value of assets.
❑ Note that when the gap is zero then that part of the banks balance sheet is
said to be “immunised against unexpected changes in interest rates”
❑ Immunisation therefore allows banks to lock into a fixed yield.
Jason Laws
119 of 218
Interest rate risk management
❑ Duration gap analysis can be used to calculated the change in the market
value of net worth as a percentage of total assets induced by a change in
interest rates:
NW
i
= − DURgap 
A
1+ i
❑ Note assumes a parallel “shift” in the term structure of interest rates.
❑ Term structure could be upward sloping/downward sloping or flat as long as
shift is parallel.
Jason Laws
120 of 218
❑
The following balance sheet information is
available for Bank Plus (amount in
$millions and duration is years).
DURgap
L
= DURassets −    DURliabilitie s
 A
NW
i
= − DURgap 
A
1+ i
Amount
Duration
Assets
Capital
2400
Residential Mortgages
Variable Rate
1600
9.1
Fixed Rate
1400
5.1
Commercial Loans
5600
3.5
Money market deposits
3500
1.3
Savings deposits
2800
2.3
Variable Rate CD’s (> 1year)
1200
3.1
Equity
3500
Liabilities
If interest rates changed from 4% to 3.5% what is the change in net
wealth as a proportion of equity?
Jason Laws
121 of 218
Interest rate risk management
❑ Total Assets (inc. Capital) = 11,000
❑ Total Liabilities (exc. Equity) = 7,500
❑ Duration of Assets = (1600/11000) x 9.1 + (1400/11000) x 5.1 +
(5600/11000) x 3.5 = 3.75
❑ Duration of Liabilities = (3500/7500) x 1.3 + ( 2800/7500) x 2.3 +
(1200/7500) x 3.1 = 1.96
❑ Duration Gap = 3.75 - (7500/11000) x 1.96 = 2.42 years
❑ If interest rates changed from 4% to 3.5% then the change in net wealth as
a proportion of equity is:
❑ -2.42 x (-0.005)/(1.04) = 1.16%
Jason Laws
122 of 218
Sample Question
❑ The following balance sheet is available (amounts in $m and duration in years) for
ABC bank:
Amount
Duration
Loans
3,400
4.6
T-bonds
600
2.1
Deposits
3,300
2.3
Equity
700
❑ What is the average duration of all the assets? What is the average duration of the
liabilities?
Jason Laws
123 of 218
Sample Question
❑ Total Assets
❑ Total Liabilities (exc. Equity)
=
=
4000
3300
❑ Duration of Assets
= [(3400)/4000 x 4.6]+[(600/4000)x 3.1] =
= 0.85 x 4.6 + 0.15 x 3.1 = 4.38
❑ Duration of Liabilities= 2.30
Jason Laws
124 of 218
Sample Question
❑ What is the duration gap for ABC bank? Explain what this duration gap implies for
ABC bank.
DURgap
L
= DURassets −    DURliabilitie s
 A
❑ Gap = 4.38 – (3300/4000) x 2.3 = 2.478.
❑ What does 2.478 mean?
❑ The average duration of assets is greater than the average duration of
liabilities, thus asset values change by more than liability values.
Jason Laws
125 of 218
Sample Question
❑ Positive DGAP
❑ Indicates that assets are more price sensitive than liabilities, on average.
❑ Thus, when interest rates rise (fall), assets will fall proportionately more
(less) in value than liabilities and equity will fall (rise) accordingly.
❑ Negative DGAP
❑ Indicates that weighted liabilities are more price sensitive than weighted
assets.
❑ Thus, when interest rates rise (fall), assets will fall proportionately less
(more) in value that liabilities and the equity will rise (fall).
Jason Laws
126 of 218
Sample Question
A+L=E
So :
é
ù é
Di
Di
DEquity = ê-DA ´
´ Aú - ê-DL ´
´
(1 + i)
(1 + i)
ë
û ë
Dividing through by A:
é
DEquity
Di ù é
Di
Lù
= ê-DA ´
´ ú
ú - ê-DL ´
A
(1 + i) û ë
(1 + i)
Aû
ë
é
DEquity
Di
Lù
Di
=´ êDA - DL ´ ú = -DGAP ´
A
(1 + i) ë
Aû
(1 + i)
Jason Laws
ù
Lú
û
127 of 218
Sample Question
❑ (c) What is the forecast impact on the market value of ABC bank resulting from a
0.25% decrease in interest rates from 3.5% to 3.25%?
❑
❑ The potential loss/gain to equity holders’s net worth/Equity (as a percentage of
assets is):
NW
i
= − DURgap 
A
1+ i
❑ ∆NW/A = -Durgap * ∆i / (1+i) = -2.478*(-0.0025) /(1+0.035) = +0.00598
❑ ∆NW = 0.00598*A = -0.00598*4,000 = 23.92
❑ i.e. NW rises to 700+23.92 = 723.92
Jason Laws
128 of 218
Sample Question
❑ (e) How can income gap analysis help banks manage interest rate risk? Discuss the
weaknesses of income gap analysis. What is runoff cashflow and how does this
affect the income gap analysis?
❑ Banks identify the difference between rate sensitive assets and rate sensitive
liabilities for a particular time horizon. This difference (the income gap) is then
multiplied by the forecast change in interest rate for the time horizon considered.
This will identify the net change in income (net interest) over the forecast time
horizon.
Jason Laws
129 of 218
Sample Question
❑ Problems:
❑ (1) Ignores the effect of interest rate changes on market values of assets (through
discounting)
❑ (2) Even rate insensitive assets and liabilities will have a component that is
sensitive e.g. a proportion of fixed rate mortgages will be repaid within the
forecast time horizon – these repaid mortgages can be re-invested at a (rate
sensitive) interest rate.
Jason Laws
130 of 218
Sample Question
❑ (3) Is applied to the balance sheet therefore ignores the income effects of changes
in interest rates on off-balance sheet items e.g. overdrafts – a proportion of which
may become balance sheet items over the forecast period).
❑ Runoff cashflow is the estimate of the cash flow from rate insensitive assets and
liabilities that will become rate sensitive over the forecast period. These runoff
cash flows can be reinvested at the current interest rate. Examples should be given
to illustrate.
Jason Laws
131 of 218
Sample Question
7. (a) Briefly explain each of the following terms:
(i) efficient frontier
(ii) feasible region
(iii) security market line
(iv) tangency portfolio
(8 marks)
❑ Observation: Many students did not draw diagrams. For these definitions they were
essential. Evidence of confusion between the security market line and capital market
line was also evident.
Jason Laws
132 of 218
Historical record
❑ We now examine the performance of three important US financial investments.
❑ Standard and Poor 500 – an index of the 500 largest US companies.
❑ Long term US treasury bonds.
❑ US treasury bills – a portfolio of 3 month US treasury bills.
❑ See https://www.credit-suisse.com/media/assets/corporate/docs/aboutus/research/publications/credit-suisse-global-investment-returns-yearbook-2020summary-edition.pdf
Jason Laws
133 of 218
Historical Record
Jason Laws
134 of 218
Historical Record
Jason Laws
135 of 218
Historical Record
Jason Laws
136 of 218
Historical record
❑ Given the historical record, why would any investor buy anything other than
common stocks?
❑ If you look closely at the figure you will see the answer
❑ Risk.
❑ The long-term government bond portfolio grew more slowly than did the stock
portfolio, but it also grew much more steadily.
❑ The common stocks ended up on top, but as you can see, they grew more
erratically much of the time.
Jason Laws
137 of 218
Return variability
Frequency chart of SPX monthly returns (1928-2021)
350
300
250
200
150
100
50
0
Stock Market Crash of 1987= -21.8%
Jason Laws
138 of 218
Risk and Return
❑ As you can see a variety of return outcomes are possible.
❑ Accordingly the return measure of a risk asset is considered to be a random
variable.
❑ For analytical purposes the uncertainty of the return is measured by the
expected (average) return.
❑ To provide a quantitative measure of the expected we normally use the
weighted average of all possible returns where the weights are the
probabilities of the occurrence of that return:
❑ E(R) = p1R1 + p2R2 + … + pnRn
❑ Ri = return of state of nature i. n = number of possible states of nature;
pi = probability of occurrence of return Ri.
❑ However in reality probabilities are not known so instead we use the
historical returns and the number of past observations.
❑ The variation around the expected return is a measure of the risk of a
security.
Jason Laws
139 of 218
Risk and Return
❑ To provide a quantitative measure of the degree of possible deviations we
use the variance, and its square root – standard deviation.
❑ Following the previous notation:
 = p1 (R1 − E ( R) )2 + p2 (R2 − E ( R) )2 + ... pn (Rn − E ( R))2
❑ e.g. Consider a stock second class with a return of 18% half the time and
10% the other half of the time. The Expected Return is 14% and the
standard deviation is:
 = 0.5(18 − 14)2 + 0.5(10 − 14)2 = 0.5(4)2 + 0.5(− 4)2 = 0.5 16 + 0.5 16 = 16 = 4
Jason Laws
140 of 218
The Risk and Return of a Portfolio
❑ A portfolio is a combination of different assets held by an investor.
❑ The share of each individual asset over the total value of the portfolio is
referred to as the portfolio weight.
❑ Example
❑ You have 10,000 shares in Singapore Telecom (price – 2.53) and 5,000
shares in Singapore Airlines (price – 11). What is the weight of each asset
in the portfolio?
Jason Laws
141 of 218
The Risk and Return of a Portfolio
❑ A portfolio is a combination of different assets held by an investor.
❑ The share of each individual asset over the total value of the portfolio is
referred to as the portfolio weight.
❑ Example
❑ You have 10,000 shares in Singapore Telecom (price – 2.53) and 5,000
shares in Singapore Airlines (price – 11). What is the weight of each asset
in the portfolio?
❑ Answer
❑ Portfolio Value = 10000 x 2.53 + 5000 x 11 = 25,300 + 55,000 = 80,300
❑ SAIR weight = 55,000/80,300 = 68.5%,
❑ TELC weight = 31.5%
Jason Laws
142 of 218
The Risk and Return of a Portfolio
❑ The expected return of a portfolio is simply the weighted average of the
expected returns of the individual stocks of which the portfolio is composed,
where the weights are the portfolio weights:
❑ E(R) = w1E(R1) + w2E(R2) + … + wnE(Rn)
❑ Where wi is the weight of each stock.
Jason Laws
143 of 218
Myth
Fact
In order to reduce
portfolio risk I
need the
correlation
between assets to
be negative.
If I form a portfolio
between two
assets with a
correlation of -1 I
can obtain a risk
free portfolio. Is
the return zero?
The minimum
variance portfolio
is optimal for
everybody?
Jason Laws
144 of 218
Myth
In order to reduce
portfolio risk I
need the
correlation
between assets to
be negative.
If I form a portfolio
between two
assets with a
correlation of -1 I
can obtain a risk
free portfolio. Is
the return zero?
The minimum
variance portfolio
is optimal for
everybody?
Fact
x
x
x
Jason Laws
145 of 218
The Risk and Return of a Portfolio
❑ When computing the variance we also have to concern ourselves with how
the asset returns vary together - the covariance.
❑ If returns tend to move in opposite directions then this reduces the overall
variability of the portfolio.
❑ But if returns tend to move in the same direction then the variability of the
portfolio is increased.
❑ In the analysis that follows 12 refers to the covariance between assets 1
and 2.
Jason Laws
146 of 218
The Risk and Return of a Portfolio
❑ If 12 is positive, when the return on the first asset is greater than the mean
value then the return on the second asset, is also, on average, greater than
its mean value.
❑ And vice versa when the return on the first asset is less than the mean
value.
❑ If 12 is negative, the returns on assets 1 and 2 tend to move in opposite
directions and offset each other.
❑ If the return on the asset is above the mean, the return on the second
will, on average, be below the mean.
❑ The fact that the returns on the assets move in opposite directions reduces
the variability of the portfolio.
Jason Laws
147 of 218
The Risk and Return of a Portfolio
❑ In general, the variance of a portfolio of assets is given by:N
N N
2
2 2
p
p
i
i
i k ik
i =1
i =1 k =1
k i
❑ and its standard deviation is P = P2.
❑ A measure of the association between two assets which is always in the
range +1 to -1 is the correlation coefficient. This is defined as:-
 = Var( R ) =  X  +  X X 
 12
12 =
 1 2
Jason Laws
The covariance and
correlation coefficient
always have the same sign
148 of 218
Mean, Variance and Covariance of a Portfolio
❑ When there are two assets in a portfolio the variance of a portfolio is give
by:
❑ Or:
 p2 = x12 12 + x22 22 + 2 x1 x2 12
 p2 = x12 12 + x22 22 + 2 x1 x2 12 1 2
❑ Since:
 12
12 =
 1 2
Jason Laws
149 of 218
Negative correlation
❑ When correlation between two assets is “-1” you have the ultimate in
diversification benefits and a risk-free portfolio.
❑ Perfect negative correlation gives a mean combined return for two securities
over time equal to the mean for each of them, so the returns for the portfolio
show no variability.
❑ Any returns above and below the mean for each of the assets are
completely offset by the return for the other asset, so there is no variability
in total returns, that is, no risk, for the portfolio.
Jason Laws
150 of 218
Role of correlation
❑ Consider the following market data:-
Ave=
sd=
GM_%
19.99%
39.3%
correl=
0.122
WMT_%
15.20%
20.3%
❑ graphically we consider the diversification effects of combining these two assets
under different assumptions about the correlation coefficient.
Jason Laws
151 of 218
Jason Laws
152 of 218
Correlation = - 1
Is there a combination
of assets with zero risk ?
Jason Laws
153 of 218
Jason Laws
154 of 218
A
B
C
Jason Laws
155 of 218
Efficient Portfolios
❑ Despite correlation between the two assets being positive diversification can
still reduce the risk of the portfolio.
❑ On the diagram on the previous slide we can identify a number of
“inefficient portfolios”.
❑ The portfolios which lie on the line from B to C are inefficient, since for each
one of them we can find an alternative portfolio, on the line from B to A,
which has the same standard deviation but a higher expected return.
❑ The set of efficient portfolios is now formed on the line from B to A.
Jason Laws
156 of 218
Portfolio Risk Exercise
❑ Consider the following information about two stocks:
Stock
Expected Return
Standard
Deviation
1
8%
15%
2
5%
11%
❑ Calculate the risk and returns of the following portfolios (correlation =
+0.2):
Portfolio
Proportion in 1
Proportion in 2
1
0
100
2
30
70
3
75
25
4
100
0
Jason Laws
157 of 218
Portfolio Risk Exercise
❑ E(RP) = w1*E(R1) + w2*E(R2)
❑
❑
❑
❑
E(R1) = 5%
E(R2) = 0.3*8 + 0.7*5 = 5.9%
E(R3) = 0.75*8 + 0.25*5 = 7.25%
E(R4) = 8%
Jason Laws
158 of 218
Portfolio Risk Exercise
22 = wX2 * x2 + wY2 * 2 + Y2 + 2 * WX * WY * X * Y*CORRXY
1 = 11%
22 = .32 * .152 + .72 * .112 + 2 * .3 * .7 * .15* .11* .2 = 0.00934
2 = 0.0966 = 9.66%
32 = .752 * .152 + .252 * .112 + 2 * .75 * .25 * .15* .11* .2 = 0.01465
3 = 0.1203 = 12.10%
4 = 15%
Jason Laws
159 of 218
Minimum Variance Portfolio
❑ With the exception of the perfect correlation diagram we can see that there is one
point where standard deviation is minimised.
❑ Calculus is appropriate here.
❑ Recall:-
 = X  + (1 − X 1 )  + 2 X 1 (1 − X 1 ) 12 1 2
2
P
2
1
2
2
1
2
2
❑ multiplying out the brackets:
 = X  +  − 2 X 1 + X  +
2
P
2
1
2
1
2
2
2
2
2
1
2
2
2 X 1 12 1 2 − 2 X 12 ! 2
2
1
Jason Laws
160 of 218
Minimum Variance Portfolio
❑ We should now differentiate 2p with respect to X1.
2
2
2
2
P
1 1
2
1 2
1

= 2 X  − 2 + 2 X  +
X
2 12 1 2 − 4 X 1 12 1 2
❑ Setting this equal to zero and dividing through by 2.
¶s P2
2
2
2
= X1 s1 + X1 s 2 - 2X1 r12s 1s 2 = s 2 - r12s 1s 2
¶X!
 P2
= X1 (12 +  22 − 2 12 1 2 ) =  22 − 12 1 2
X 1
Jason Laws
161 of 218
Minimum Variance Portfolio
❑ Hence:-
 22 − 12 1 2
X1 = 2
2
1 +  2 − 2 12 1 2
❑ and X2 = 1 – X1.
❑ A special case of this result is when we have perfect negative correlation between
assets (12 = -1). Then:2
2
1 2
2
2
1
2
1
2
2
2
1
2
1 2
1
2
1
2
 + 
 ( +  )

X =
=
=
 +  + 2 
( +  ) ( + 
)
❑ Activity: Find the minimum variance portfolio for the previous exercise.
Jason Laws
162 of 218
Indifference Curves
❑ Every individual will exhibit unique preferences for risk and return and so everyone
has a unique set of indifference curves.
❑ Consider the following hypothetical set of indifference curves.
❑ Consider an individual holding
portfolio W.
❑ How much extra return would
they require in order to increase
the risk to 20% (i.e. Portfolio Z)?
❑ All the risk-return combinations
along the indifference curve
offer the same level of
desirability.
Jason Laws
163 of 218
Indifference Curves
❑ Now let us consider a map of indifference curves.
❑ An individual would be
indifferent between points
W and Z but not between:
❑ W and S – as S gives a
higher return for the
same level of risk.
❑ Z and T – as T gives
the same return but
for a lower level of
risk.
❑ The further “north west”
the indifference curve, the
higher the desirability.
Jason Laws
164 of 218
Indifference Curves
❑ Other investors may be less risk averse than the individual we have just considered
and therefore the increase in return required to compensate for risk may be less.
The indifference curves will therefore have a lower slope.
❑ Alternatively they may less tolerant of risk and required large increases in return
for small increases in risk. The indifference curves for these individuals will have a
steep slope.
Jason Laws
165 of 218
Indifference Curves
❑ In the absence of a risk free asset investors will identify
the efficient frontier which is that part of the meanstandard deviation frontier that contains portfolios that
give the highest expected return for a given risk.
❑ That is all portfolios to the right of the minimum variance
portfolio on the mean standard deviation frontier.
❑ Investors will locate somewhere on this efficient frontier.
❑ An investor will locate on the efficiency frontier according
where his/her indifference curve is just tangential to the
efficient frontier.
Jason Laws
166 of 218
Expected
Return, kp
IB I
2 B
1
Efficient Frontier
Optimal
Portfolio
Investor B
IA
2
IA
1
Optimal Portfolio
Investor A
Risk p
Jason Laws
167 of 218
Expected
Return, kp
IB I
2 B
1
Efficient Frontier
Optimal
Portfolio
Investor B
IA
2
IA
1
Optimal Portfolio
Investor A
Risk p
Jason Laws
168 of 218
Feasible Region/Set
❑ Investors first need to identify the mean standard deviation frontier for
N risky assets using every possible combination of assets. This defines
the feasible region. The outer points of this feasible region (left and up)
define the mean standard deviation frontier.
Expected
Portfolio
Return, p
Efficient Set
Feasible Set
Jason Laws
Risk, p
169 of 218
Diversification and Risk
❑ What happens to the variance of a portfolio as the number of assets
increases ?
❑ Recall again the formula for calculating the variance of a portfolio:N
N
N
 2p = Var ( R p ) =  X i2 i2 +  X i X k ik
i =1
i =1 k =1
k i
❑ assume further that all assets are held in equal proportions such that Xi =
1/N.
N
1 2 N N 1 1
1 1
 =  2  i +    ik =
N
N N
i =1 N
i =1 k =1 N
2
p
k i
Jason Laws
1 1 ( N − 1) N N
 +
 ik


N N ( N − 1) i =1 k =1
i =1
N
2
i
k i
170 of 218
Diversification and Risk
❑ We could write expressions for the average covariance and variances as:-
−2
1
i =
N
N
N N
1
2

 ik


i ;  ik =
N ( N − 1) i =1 k =1
i =1
_
k i
❑ note there are N(N-1) elements inside the double summation signs since
each of the N assets can be combined with any of the remaining N-1
assets.
❑ Thus:-
_
1
N
−
1
2
p = i +
 ik
N
N
_ 2
Jason Laws
171 of 218
Diversification and Risk
(N − 1) 
N
1
1
= 2
N ( N − 1) N
❑ Note that:❑ this is the “trick” used to write the last expression.
❑ Look at the last expression on the previous slide:❑ What happens to contribution to portfolio variance of the individual
variances as N gets large ?
❑ It goes to zero !
❑ Therefore, in the limit portfolio variance equals average covariance.
❑ Individual risks of securities can be diversified away but contribution
to total risk of the covariance cannot.
Jason Laws
172 of 218
Jason Laws
173 of 218
Even a little diversification
can substantially reduce
variability
Unique risk
Market risk
Jason Laws
174 of 218
How Diversification reduces risk
❑ Market risk stems from the fact that there are many other economy-wide
perils which threaten businesses. That is why stocks have a tendency to
move together.
❑ That is why investors are exposed to market uncertainties, no matter how
many stocks they hold.
❑ Unique risk may also be called unsystematic risk, residual risk, specific risk
or diversifiable risk.
❑ Market risk may be called systematic risk or undiversifiable risk.
Jason Laws
175 of 218
Market risk
❑ If you want to know the contribution of an individual security to the risk of a
well diversified portfolio, it is no good thinking about how risky that security
is in isolation – you need to measure its market risk.
❑ i.e. how sensitive it is to market movements.
❑ This sensitivity is called beta ().
❑ Stocks with betas greater than 1 tend to amplify the overall movements of
the market.
❑ Stocks with betas between 0 and 1 tend to move in the same direction as
the market, but not as far.
❑ The market is the portfolio of all stocks and so has a beta of 1.
Jason Laws
176 of 218
Market Risk - Beta
❑ Some sample Beta’s for S&P 500 companies (source finance.yahoo.com – March
22nd 2021):
❑ KELLOGG
0.64
❑ WAL-MART STORES
0.47
❑ WALT DISNEY
1.21
❑ Goodyear Tire Company
2.26
❑ Verizon
0.47
❑ What would be predicted to happen to the returns of each of these stocks if the
market went up 10% or down 10%?
Jason Laws
177 of 218
Methods for Estimating Betas
❑ Collect data on historical returns of all the securities which could enter the
portfolio;
❑ Compute the average return and variance of each security;
❑ Collect data on the market index;
❑ Compute the average return and variance of the market index;
❑ Compute the covariance of each security with the market index;
❑ estimate the alpha and beta using:❑ Beta = im/2m ;
❑ alpha = ave(Ri) - i x ave(Rm)
❑ This is identical to fitting a trend line in Excel with the return on the stock on the
vertical axis and the return on the market on the horizontal axis and hypothesising
the relationship:
❑ Ri=a + Rm
Jason Laws
178 of 218
Jason Laws
179 of 218
CAPM Assumptions
❑
The CAPM assumes that:❑ Investors rely on two factors in making their decisions: expected return
and variance.
❑ Investors are rational and risk averse and subscribe to Markovitz
methods of portfolio diversification.
❑ Investors all invest for the same period of time.
❑ They share all expectations about assets.
❑ There is a risk free investment, and investors can borrow and lend any
amount at the risk-free rate.
❑ Capital markets are completely competitive and frictionless.
Jason Laws
180 of 218
Capital Market Theory
❑ The major factor that allowed Portfolio theory to develop into capital market
theory is the concept of the risk free asset.
❑ Risk free asset – asset with zero variance
❑ Risky Asset – one from which future returns are uncertain.
❑ Recall that covariance between two sets of returns is:
1 n
 ij =  (Ri − E ( Ri ) )(R j − E ( R j ) )
n i =1
❑ But because the returns for the risk free asset are certain, Ri – E(Ri) = 0.
Hence the covariance (and correlation) will also be zero.
Jason Laws
181 of 218
Capital Market Theory
❑ What happens to expected return and standard deviation of returns when
you combine risky assets and risk free assets?
❑ Expected Return = wRFRF + (1-wRF) E(Ri)
❑ Variance = (1-wRF)2i2
❑ standard deviation = (1-wRF)i
❑ Standard deviation is therefore a linear proportion of the standard deviation
of the risky asset portfolio.
❑ Because both expected return and standard deviation are linear
combinations a graph of possible risks and returns looks like a straight line
between the two assets.
Jason Laws
182 of 218
Capital Market Theory
❑ The introduction of a risk free security to the universe of risky securities allows
investors to move to higher levels of utility as they can allocate their wealth to the
risk free security and a portfolio of risky assets on the efficient frontier. This
creates a new (linear) efficient frontier – the CML.
❑ The line from the risk free rate to the efficient frontier that is the steepest (i.e.
where the line is just tangential to the (old) efficient frontier is the optimal new
efficient frontier – as this will yield the set of combinations of risk free asset and
efficient risky portfolio that delivers the highest expected return for a given risk.
❑ Every combination of the risk free asset and the Markowitz efficient portfolio
(M) is shown on the Capital Market Line (CML).
Jason Laws
183 of 218
CML
P2
P1
Jason Laws
184 of 218
Capital Market Theory
❑ All the portfolios on the capital market line are feasible for the investor to
construct.
❑ Portfolios to the left of M represent combinations of risky assets and the risk
free asset.
❑ Portfolios to the right of M included purchases of risky assets made with
funds borrowed at the risk free rate.
❑ Such a portfolio is called a leveraged portfolio.
❑ Compare portfolio P1, on the Markowitz efficient frontier, with portfolio P2
which is on the CML.
❑ Note that for the same risk the expected return is greater for P2 than P1.
Jason Laws
185 of 218
Capital Market Theory
❑ A risk averse investor will prefer P2 to P1. This is true for all but one portfolio
on the line: portfolio M.
❑ With the introduction of the risk free asset we must now modify the
conclusion from portfolio theory such that we now say that an investor will
select a portfolio on the line representing a combination of borrowing or
lending at the risk free rate and purchases of the Markowitz efficient
portfolio, M.
❑ The portfolio that includes all risky assets is referred as the market portfolio.
It includes all risky assets – stocks, bonds, real estate, options etc.
❑ This result is known as the two-fund separation theorem.
❑ The tangent portfolio will therefore be the portfolio of risky assets that all
investors will choose to invest in.
❑ Investors will locate somewhere on the new efficient frontier (CML)
according to their preference for risk.
Jason Laws
186 of 218
Decomposing Total Risk Using the Market
Model
❑ Recall:❑ Ri = ai + iRm + ei
❑ Taking the variance of this:❑ Var(Ri) = i2 Var(Rm) + Var(ei)
❑ Such that the total risk is measured by:❑ Systematic/market risk = i2 Var(Rm)
❑ Unsystematic/unique risk = Var(ei)
❑ Another product of the statistical technique used to estimate beta is the
percentage of systematic risk to total risk.
❑ In statistical terms this is measure by the coefficient of determination or the
R-squared value.
Jason Laws
187 of 218
The Security Market Line
❑ Noting that:-
 iM
i = 2
M
❑ If this is substituted into our CML them we have the beta version of the SML
or CAPM:-
E ( Ri ) = RF +  i E ( RM ) − RF 
❑ This equation states that given the assumptions of the CAPM, the expected
(or required) return on an individual asset is a linear function of its index of
systematic risk as measured by beta.
❑ The higher the beta, the higher the expected return.
Jason Laws
188 of 218
The Security Market Line
❑ Notice also that only an assets Beta determines its expected return.
❑ When Beta = 0:-
E ( Ri ) = RF + 0  E ( RM ) − RF  = RF
❑ When Beta = 1:-
E ( Ri ) = RF + 1 E ( RM ) − RF  = E ( RM )
Jason Laws
189 of 218
The Security Market Line
Expected
Return
E(Rm)
Rf
1
Jason Laws
Beta
190 of 218
The Security Market Line
❑ Even firms within one industry have different levels of debt, and increasing
debt increases leverage. Increasing leverage increases beta.
❑ We can "synthesize" a security with a beta of 1.3 by borrowing 30% of our
wealth, and investing the total in an asset with a beta of one.
❑ Suppose, for instance, that investor A hold a portfolio of $100 invested in an
S&P 500 index trust.
❑ In order to increase his expected return, investor B, who also has $100,
borrows an additional $30 for one year at 0% interest, and invests $130 in
the S&P 500 index trust.
❑ What will happen if the S&P goes up by next year?
❑ A will have $110, for a gain of 10%,
❑ while B will have $143 - $30, leaving a gain of 13%!
Jason Laws
191 of 218
The Security Market Line
❑ What will happen if the market drops by 10% next year?
❑ A will have $90, a loss of -10%,
❑ while B will have a net loss of $87, a 13% loss.
❑ B's leverage increased his exposure to market risk.
❑ Leverage can be used by corporations as well as individuals to increase
their expected returns, and in fact, this is exactly what some firms do.
❑ Even if they are in a low-beta business, such as a utility, they can increase
expected return through leverage
Jason Laws
192 of 218
Portfolio Risk Exercise
Introducing a riskless asset
❑ e.g. a government bill
❑ The risk free rate is the certain return on the riskless
asset:
❑Note σF = zero
❑ Assume investors can lend/borrow unlimited funds at risk
free rate.
❑ An investor will therefore invest some funds in a portfolio
of risky assets and lend/borrow at RFR.
❑ What portfolio of risky assets?
Jason Laws
193 of 218
E(R port )
D
M
C
RFR
B
A
 port
B superior to A (in risk-return trade-off)
Optimal point = M
Jason Laws
194 of 218
Portfolio Risk Exercise
❑ To attain a higher expected return than is available at
point M (in exchange for accepting higher risk)
❑ Either invest along the efficient frontier beyond point M,
such as point D
❑ Or, add leverage to the portfolio by borrowing money at
the risk-free rate and investing in the risky portfolio at
point M
Jason Laws
195 of 218
E(R port )
ing
d
n
Le
ing
w
o
rr
Bo
L
CM
M
RFR
 port
Everybody will want to invest in Portfolio M and borrow or lend to be
somewhere on the CML
n
Risk averse investors will locate to the left of M
n
Risk tolerant investors will locate to the right of M
Therefore this portfolio M must include ALL RISKY ASSETS
Jason Laws
196 of 218
Portfolio Risk Exercise
Distinguish between the capital market line and the
security market line
❑ The CML will contain all efficient portfolios and is defined
in E(R), standard deviation space.
❑ The SML will contain all securities and portfolios and is
defined in E(R), Beta space.
❑ Both are anchored at the riskless portfolio.
❑ Both contain the market portfolio.
Jason Laws
197 of 218
Jason Laws
198 of 218
Beta of a portfolio
❑ For a portfolio of N assets the historical beta is simply:❑ a weighted average of the observed historical beta is simply a weighted
average of the observed historical betas for the individual assets in the
portfolio.
N
b P = åwi bi
i=1
❑ So for example, the historical beta for a portfolio consisting of 30% of
Microsoft (beta = 0.81) and 70% of Sun Microsystems (beta =0.47) is:❑ .3 x 0.81 + .7 x .47 = 0.57.
Jason Laws
199 of 218
Beta of a portfolio
❑ Example
❑ You own 200,000 shares of Sembcorp Industries (Beta = 0.887, Price =
1.78 ), 30,000 of DBS Group (Beta = 1.172, Price = 28.38), 100,000 of
Wilmar Int’l (Beta = 0.866, Price = 5.39) and 200,000 of Comfort Del
Gro (Beta = 1.114, Price = 1.7).
❑ What is the Beta of your portfolio?
Jason Laws
200 of 218
Beta of a portfolio
❑ Example
❑ You own 200,000 shares of Sembcorp Industries (Beta = 0.887, Price =
1.78 ), 30,000 of DBS Group (Beta = 1.172, Price = 28.38), 100,000 of
Wilmar Int’l (Beta = 0.866, Price = 5.39) and 200,000 of Comfort Del
Gro (Beta = 1.114, Price = 1.7).
❑ What is the Beta of your portfolio?
❑ Solution
Stock
Sembcorp
DBS Group
Wilmar International
Comfort Del Gro
SP
1.78
28.38
5.39
1.7
N
200000
30000
100000
200000
Beta
0.887
1.172
0.866
1.114
Jason Laws
N x SP
SGD 356,000.00
SGD 851,400.00
SGD 539,000.00
SGD 340,000.00
SGD 2,086,400.00
wi
17.06%
40.81%
25.83%
16.30%
wi x Beta
0.151348
0.47826
0.223722
0.181538
1.03
201 of 218
Sample Question
❑ Consider the following portfolio composed of three stocks (X, Y, Z):
Stock
Quantity
Price
Beta
X
100
1.5
0.7
Y
120
1.7
0.95
X
210
1.1
1.05
❑ What is the beta of this portfolio? [5 marks]
Jason Laws
202 of 218
Sample Question
Stock Quantity Price Beta
X
100
£ 1.50 0.7
Y
120
£ 1.70 0.95
X
210
£ 1.10 1.05
Value of Holding
£
150.00
£
204.00
£
231.00
£
585.00
Weight
25.64%
34.87%
39.49%
100.00%
Weight x Beta
0.179487179
0.331282051
0.414615385
0.925
❑ Hence the Beta of the portfolio is 0.925.
Jason Laws
203 of 218
Sample Question
if Stock market rises by 10%
X rises by:
7.00%
Y rises by:
9.5%
Z rises by:
10.50%
New Price=
New Price=
New Price=
£ 1.61 New Value=
£ 1.86 New Value=
£ 1.22 New Value=
New Total=
£ 160.50
£ 223.38
£ 255.26
£ 639.14
or directly;
Portfilio beta x 10%=
9.25%
New Total=
£ 639.14
Jason Laws
204 of 218
Sample Question
❑ You are given the following information:
(a) A stock with a beta of 0 has an expected return of 6%
(b) A portfolio made up of 50% invested at the risk free rate and 50% invested
in the market portfolio has an expected return of 9%.
❑ What is the expected return of the market portfolio?
❑ Beta of zero implies that the risk free rate is 6%. From (b):
❑ 9% = 0.5 x 6% + 0.5 x E(Rm)
❑ 6% = 0.5 x E(Rm)
❑ E(Rm) = 12%
Jason Laws
205 of 218
Alternative Exercise
Securities I and J lie on the security market line:
I
J
Expected return
14%
18%
Beta
1
1.5
Assume the CAPM holds.
I. What is the risk free rate of return and the risk premium
on the market portfolio?
II. Security K has an expected return of 24% and a Beta of
1.8. What is likely to happen to the return and price of
security K? Explain your answer.
Jason Laws
206 of 218
Alternative Exercise
(i) CAPM: E(R*) = Rf + β[E(RM) – Rf]
For I:
14 = Rf + 1 [E(RM) – Rf]
(1)
For J:
18 = Rf + 1.5 [E(RM) – Rf]
(2)
Subtracting equation (1) from (2)
4=
0.5 [E(RM) – Rf]
[E(RM) – Rf] = 4/0.5 = 8%
Jason Laws
207 of 218
Alternative Exercise
Substituting this result into equation (1) gives:
14 = Rf + 1 * 8
Rf = 14 – 8 = 6%
The risk free rate = 6%
Jason Laws
208 of 218
Alternative Exercise
(ii) For a Beta of 1.8 the CAPM predicts the
expected return on security K will be:
E(R*) = 6 + 1.8 [8] = 20.4
i.e. security K has an expected return that is
higher than that predicted by the CAPM
Investors will buy security K which will push up
the price and drive down the expected return
until it is line with the CAPM
Jason Laws
209 of 218
Security K
Jason Laws
210 of 218
Sample Question
(d) Discuss the problems with empirically testing the Capital Asset
Pricing Model.
❑
❑
❑
❑
Some of the assumptions of the CAPM are unrealistic:
e.g. Unlimited amounts can be borrowed/lent at the risk free rate
However, the best way of judging a model is how well it predicts.
However, Roll (1977) argues that the CAPM is not testable because the
market portfolio is not observable and using proxies is measures it with error.
❑ Also we proxy expected returns with actual (historical) returns but actual
returns may be poor proxies for expected returns.
Jason Laws
211 of 218
Arbitrage
❑ In its simplest form arbitrage is the simultaneous buying and selling of
securities at two different prices in two different markets.
❑ The arbitrageur profits without risk by buying cheap in one market and
simultaneously selling at the higher price in the other market.
❑ Investors don’t hold their breath waiting for such situations to occur because
they are rare.
❑ Less obvious situations occur where a package of securities can produce a
payoff identical to another security that is priced differently.
Jason Laws
212 of 218
Arbitrage
❑ This arbitrage relies on a fundamental principle of finance called the law of
one price.
❑ i.e. a security must have the same price regardless of the means of
creating that security.
❑ The law of one price implies:❑ that if the payoff of a security can be synthetically created by a package
of other securities, the price of the package and the price of the security
whose payoff it replicates must be equal.
❑ When this situation is discovered not to hold rational investors will trade
these securities in such a way to restore equilibrium.
Jason Laws
213 of 218
Assumptions of the APT
❑ The arbitrage pricing theory (APT) postulates that a security’s expected
return is influenced by a variety of factors, as opposed to just the single
market index of the CAPM.
❑ The APT in contrast states that the return on a security is linearly related to
H “factors”.
❑ The APT does not specify what these factors are, but it is assume that the
relationship between security returns and the factors is linear.
Jason Laws
214 of 218
Derivation of the APT
❑ For now and to illustrate the APT model let us assume a simple world
consisting of three securities and with two factors.
❑ I will use the following notation:❑ Ri – random rate of return on security i (i = 1,2,3)
❑ E(Ri) – the expected return on security i (i = 1,2,3)
❑ Fh – the h’th factor that is common to the returns of all three assets (h
=1,2).
❑ i,h - the sensitivity of the i’th security to the h’th factor.
❑ ei – the unsystematic return for security i.
Jason Laws
215 of 218
The APT equation
❑ Ross (1976) has shown that the following risk return relationship will result
for security i.
❑ E(Ri) = RF + i,F1[E(RF1) – RF] + i,F2[E(RF2) – RF]
❑ Or, E(Ri) = RF + i,F1lF1+ i,F2lF2
❑ Where:❑ i,Fj – the sensitivity of security i to the j’th factor.
❑ [E(RFj) – RF] – the excess return of the j’th systematic factor over the
risk free rate – can be thought of as the price for the j’th systematic risk.
❑ This can of course be generalised to the case where there are H factors as
follows:-
Jason Laws
216 of 218
The APT equation
❑ E(Ri) = RF + i,F1[E(RF1) – RF] + i,F2[E(RF2) – RF]
+ ….. + i,FH[E(RFH) – RF]
❑ This is the APT model.
❑ It states that investors want to be compensated for all the factors that
systematically affect the return of a security.
❑ The compensation is the sum of the products of each factors systematic risk
(i,FH) and the risk premium assigned to it by the financial market [E(RFH) –
RF].
❑ As usual investors are not compensated for accepting unsystematic risk.
Jason Laws
217 of 218
Alternative Exercise
❑ How does the APT differ from the CAPM?
❑ The advantage of the APT is that it does not require us to identify and
measure the market portfolio
❑ This solves most of the theoretical limitations of the CAPM.
❑ The disadvantage is that it does not tell us what the underlying factors are
(unlike the CAPM, which collapses all the macroeconomic factors into the
market portfolio).
Jason Laws
218 of 218