代写 Capital Budgeting STAT2032/6046 Financial Mathematics
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代写 Capital Budgeting STAT2032/6046 Financial Mathematics
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 1
CAPITAL BUDGETING – COMPARING INVESTMENT PROJECTS
In this section, we are attempting to choose between various investment projects with
different cash flow streams. There are a number of useful measures that help an investor
select between potential investment projects. The criteria that we will consider are:
• Accumulated profit.
• Net present value.
• Internal rates of return.
• Discounted payback period.
CASH FLOWS
Comparison of investment projects involves comparing the cash flow payments for the
projects. The net cash flow
t
c at time t is:
t
c = cash inflow at time t - cash outflow at time t
If any payments may be regarded as continuous then ) (t ρ , the net rate of cash flow per unit
time t , is defined as:
( ) ( ) ( )
I O
t t t ρ ρ ρ = −
where ( )
I
t ρ and ( )
O
t ρ denote the rates of inflow and outflow at time t respectively.
By the time the project ends (at time T ) the accumulated value (the balance in the account)
will be:
dt i t i c T AV
T
t T t T
t
t ∫ ∑
− −
+ + + =
0
) 1 )( ( ) 1 ( ) ( ρ
This is one criterion that can be used to assess an investment project. This measure suffers
from the disadvantage that it can only be used in situations where there is a definite fixed
time horizon for the project.
Another problem is that the accumulated profits for two different projects cannot be
compared directly if they have different time horizons, since the calculated values will relate
to different dates.
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 2
EXAMPLE
Consider the following cash flows:
Immediate outflow of 100
Inflow of 200 after 3 months
Inflows of 1,000 per annum paid continuously
What is the accumulated value of the net cash flows at time t , where > t 3 months, using an
annual effective interest rate of i?
Solution
0.25
0
( ) 100(1 ) 200(1 ) 1000(1 )
t
t t t s
AV t i i i ds
− −
= − + + + + +
∫
Recall from previous notes that: dt i s
n
t n
n ∫
−
+ =
0
) 1 (
0.25
( ) 100(1 ) 200(1 ) 1000
t t
t
AV t i i s
−
⇒ = − + + + +
The problems mentioned above can be avoided by calculating the "net present value" instead.
NET PRESENT VALUE
The net present value at the rate of interest i of the net cash flows is usually denoted by
NPV( i ).
dt i t i c i NPV
T
t t
t
t ∫ ∑
− −
+ + + =
0
) 1 )( ( ) 1 ( ) ( ρ
This is equivalent to the accumulated profit, except now we are looking at the value at the
outset, rather than the value at the end of the project. A higher net present value indicates a
more "profitable" project.
The rate of interest i used to calculate the net present value is often referred to as the risk
discount rate. As the risk discount rate increases, the equivalent NPV decreases.
We will discuss how net present values are used in comparing projects after introducing the
internal rate of return.
EXAMPLE
What is the present value at time 0, of the cash flows in the previous example, assuming the
continuous inflow payment is paid indefinitely.
Solution
0.25
0
100 200(1 ) 1000(1 )
s
PV i i ds
∞
− −
= − + + + +
∫
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 3
Recall from previous notes that:
0
(1 )
n
t
n
a i dt
−
= +
∫
0.25
0.25
100 200(1 ) 1000
1000
100 200(1 )
ln(1 )
PV i a
i
i
−
∞
−
⇒ = − + + +
= − + + +
+
YIELD RATES (INTERNAL RATE OF RETURN)
Previously we introduced problems where we had to solve for an unknown rate of interest.
For each problem, an equation of value was developed, and an interest rate was found that
solved the equation.
The yield rate or internal rate of return (IRR) is the effective rate of interest that
equates the present value of income and outgo, ie makes the net present value of the
cash flows equal to zero.
Methods of solving for unknown interest rates were covered in week 5 and include:
• Solving quadratic equations (when an analytical solution exists)
• Linear interpolation
EXAMPLE
$100,000 is used to purchase an annuity-immediate
Institution A offers annual payments in arrears of $17,000 for 10 years
Institution B offers annual payments in arrears of $19,000 for 10 years
Solution
Obviously an investor would prefer Institution B since it offers higher payments.
The investment from Institution A has a yield that can be found by solving:
10
17,000 100,000 0 11.0%
i
a i − = ⇒ ≅
The investment from Institution B has a yield that can be found by solving:
10
19,000 100,000 0 13.8%
i
a i − = ⇒ ≅
EXAMPLE
Institution A offers annual payments of $17,000 for 10 years in exchange for a purchase price
of $100,000
Institution B offers annual payments of $19,000 for 12 years in exchange for a purchase price
of $130,000
Solution
In this example it is not immediately obvious which investment offers the higher yield.
The investment from Institution A has a yield that can be found by solving:
10
17,000 100,000 0 11.0%
i
a i − = ⇒ ≅
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 4
The investment from Institution B has a yield that can be found by solving:
12
19,000 130,000 0 9.9%
i
a i − = ⇒ ≅
Although Institution A offers the higher yield, this does not necessarily mean that the
investment offered by Institution A is superior.
A more important criterion than internal rate of return for comparing different investment
projects is to consider the rate of interest at which the investor may lend or borrow money.
Rather than comparing yields, choosing between investment projects is better achieved by
comparing net present values at specific interest preference rate (or hurdle rate).
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 5
LINK BETWEEN THE NPV AND IRR
Since NPV is the present value of the net cash flows associated with a project, if an investor
lends or borrows money at an interest rate
1
i , then the project will be profitable if:
0 ) (
1
> i NPV .
If the internal rate of return on a project is
0
i , then NPV(
0
i )=0. In other words, a project turns
from profitable to unprofitable when
0
i i = . Assuming that our inflows are relatively in the
future compared to our outflows, the NPV decreases as the risk discount rate increases, the
NPV will fall below zero when
0
i i >
Therefore, a project is profitable if
0 1
i i < . This makes sense intuitively: if the rate of interest
at which an investor can lend or borrow funds is less than the yield on the investment, then
the project will be profitable.
Many projects will need to provide a return to shareholders and so there will not be a specific
fixed rate of interest that has to be exceeded. Instead a target, hurdle, or interest preference,
rate of return may be set for assessing whether a project is likely to be sufficiently profitable.
In this context, a project may be considered profitable if
0 ) (
1
> i NPV
where
1
i is the hurdle rate.
When comparing two investment projects A and B, project A is more profitable than project
B, if:
) ( ) (
1 1
i NPV i NPV
B A
>
Note: If the IRR for project A is greater than that for project B:
B A
i i > , this DOES NOT
imply that ) ( ) (
1 1
i NPV i NPV
B A
> . This is because the NPV for each project depends on
1
i .
EXAMPLE
Institution A offers annual payments of $17,000 for 10 years in exchange for a purchase price
of $100,000
Institution B offers annual payments of $19,000 for 12 years in exchange for a purchase price
of $130,000
Find the net present value of these two projects for an investor at:
(i) a risk discount rate of 9% per annum
(ii) a risk discount rate of 6% per annum
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 6
Solution
At a risk discount rate of 9% per annum, the net present values of both of these projects are:
100 , 9 $ 000 , 100 000 , 17 ) 09 . 0 (
09 . 0 10
= − = a NPV A
054 , 6 $ 000 , 130 000 , 19 ) 09 . 0 (
09 . 0 12
= − = a NPV B
At a risk discount rate of 6% per annum, the net present values of both of these projects are:
121 , 25 $ 000 , 100 000 , 17 ) 06 . 0 (
06 . 0 10
= − = a NPV A
293 , 29 $ 000 , 130 000 , 19 ) 06 . 0 (
06 . 0 12
= − = a NPV B
Both projects return a positive NPV. This means that both projects will be profitable to an
investor if the investor borrows the purchase price and reinvests payments at the risk discount
rates quoted.
Using net present values as a criterion, with a risk discount rate of 9%, Project A appears
more favourable (higher NPV). However, with a risk discount rate of 6%, Project B appears
more favourable. This is despite the fact that the yield on project A is 11% and the yield on
project B is 9.9%.
The diagram below illustrates how the NPV changes for different interest rates. Note: the
yields for the two projects are where the NPV=0.
In the above example there is one cross-over point. In more complicated examples there may
be more than one cross-over point.
-20,000
-
20,000
40,000
60,000
80,000
100,000
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
interest rate
NPV
NPV for A
NPV for B
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 7
In summary, if an investor is trying to choose between different investment projects, when
advising the investor of whether to invest in any of the projects, and if so, in order to choose
the most profitable project:
1) find the yield
0
i (IRR) for each project by solving 0 ) (
0
= i NPV
2) If the interest preference rate is
1
i , then the projects with
0 1
i i < are profitable and need to
be compared. Those projects where
0 1
i i > can be rejected.
3) For projects where
0 1
i i < , find ) (
1
i NPV . The project with the highest ) (
1
i NPV will give
the higher profit.
REINVESTMENT RATES
The calculations so far have assumed that the lender can reinvest payments received from the
borrower at a reinvestment rate equal to the original investment rate.
In practice, a borrower may pay a different rate of interest i on the borrowings than the rate
2
i they would receive on investment of income.
EXAMPLE
Institution A offers annual payments at the end of each year of $17,000 for 10 years in
exchange for a purchase price of $100,000.
What is the yield for a purchaser of the annuity if the rate of reinvestment of annual payments
is 8%?
Solution
Before considering the impact of specific reinvestment rates, we consider the case where the
yield is equal to the reinvestment rate.
From the borrower's point of view (Institution A), the yield on the investment is the solution
to the equation:
0 ) ( ) ( 0 ) (
0 0 0
= − ⇒ = i PV i PV i NPV
O I
0
0
10
100,000 17,000 0 11%
i
a i − = ⇒ ≅
where ) (
0
i PV I and ) (
0
i PV O are the present values calculated at the interest rate
0
i of the
income and outgo respectively.
For the investor purchasing the annuity, the yield depends on the rate of reinvestment applied
to the payments received.
If the payments of $17,000 are reinvested by the investor at the same rate of interest of 11%,
then the yield is also 11%. This is shown below.
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 8
If the payments of 17,000 are invested at 11% then the accumulated value of the payments at
the date of the last payment is:
11 . 0 10
000 , 17 s
The present value of these payments is:
10
11 . 0 10 0
000 , 17
i
v s
So the yield in the case where payments are reinvested at 11% is the solution to the equation
below:
0 ) ( ) (
0 0
= − i PV i PV
O I
0 000 , 100 000 , 17
10
11 . 0 10 0
= −
i
v s .
If we solve this we find that the yield (the internal rate of return) is 11%.
However, if the payments of $17,000 are reinvested at a rate of % 8
2
= i , then the yield on the
investment will differ from % 11 .
If the payments of 17,000 are invested at 8% then the accumulated value of the payments at
the date of the last payment is:
08 . 0 10
000 , 17 s = $246,272
In order to find the yield we need to solve the equation below:
0 000 , 100 000 , 17
10
08 . 0 10 0
= −
i
v s
0
10 10
0 0
100,000
(1 ) 9.43%
246,272
i
v i i
−
= + = ⇒ ≅
In general, if the lender is not able to reinvest the repayments at the initial rate of investment,
but reinvests them instead at rate
2
i per period, where i i <
2
, then the lender achieves an IRR
less than i .
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 9
DISCOUNTED PAYBACK PERIOD
The discounted payback period is the number of years before the project starts making money
(ie. positive balance).
If an investor can borrow and reinvest funds at the effective rate of interest i , then the time
1
t
until the project is making money is the smallest value of t such that the accumulated value
) (t AV of the net cash flows is greater than or equal to zero, where:
h t
t h
h
i c t AV
−
≤
+ = ∑ ) 1 ( ) (
and where
h
c is the net cash flow at time h (ie. income minus outgo).
In other words, the discounted payback period
1
t is the smallest value of t such that:
0 ) 1 ( ) ( ≥ + =
−
≤
∑
h t
t h
h
i c t AV
Here we are only assuming that the project consists of discrete cash flows. If continuous cash
flows are also present, the discounted payback period
1
t is the smallest value of t such that:
⎟
代写 Capital Budgeting STAT2032/6046 Financial Mathematics
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ + + =
∫ ∑
− −
≤
ds i s i c t AV
t
s t h t
t h
h
0
) 1 )( ( ) 1 ( ) ( ρ 0 ≥
We have assumed so far that the investor may borrow or lend money at the same rate of
interest. In practice, however, an investor will probably have to pay a higher rate of interest (
D
i , say) on borrowings than the rate (
S
i , say) they receive on investments.
In this case accumulation of net cash flows must be calculated from first principles, where the
rate of interest applied depends on whether the investor's account is in surplus (in which case
use
S
i ), or in deficit (in which case use
D
i ).
We now consider the discounted payback period when it is assumed that the investor can
borrow funds at the rate
D
i and invest funds at the rate
S
i . For simplicity, we will just work
with discrete cash flows for this example.
In this case, the discounted payback period
1
t is the smallest value of t such that:
0 ) 1 ( ) ( ≥ + = ∑
≤
−
t h
h t
D h
i c t AV
If the project is viable, the accumulated profit when the project ends at time T is:
t T
S
t t
t
t T
S
t t
t t
D t
t T
S
t t
t
t T
S
i c i i c i c i t AV P
−
>
−
≤
− −
>
−
+ + + + = + + + =
∑ ∑ ∑
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 )( (
1
1
1
1
1
1
1
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 10
The profit consists of two components:
• the accumulated value of any cash flows to time
1
t (accumulated at the rate of interest
D
i
), accumulated a further
1
t T − years at the rate of interest
S
i . This is because the balance
is in deficit prior to time
1
t , so interest is charged at
D
i . Following time
1
t the balance is
in surplus, so interest is received at the rate
S
i
• any additional net cash flows after time
1
t accumulated to time T at the rate of interest
S
i .
Other things being equal, a project with a shorter discounted payback period is preferable to a
project with a longer discounted payback period because it will start producing profits earlier.
EXAMPLE
A project consists of a payment of $7000 at time 0 in return for an income stream of $1150
per annum (in arrears) for 10 years. Find the discounted payback period and accumulated
profit for Project A using a borrowing rate of 6% p.a. and an investment rate of 3% p.a.
Solution
( )
0.06
7000(1.06) 1150 0
t
t
AV t s = − + ≥
Setting 0 ) ( = t AV and solving for t
0.06
0.06
0.06
7000(1.06) 1150 0
7000(1.06) 1150
7000 (1.06 1) 1 (1.06)
6.086957 (1.06) (1.06)
1150 0.06
t
t
t
t
t t
t t
t
s
s
s
i
−
− −
− + =
− = −
− −
= = = =
(1.06) 0.634626 7.80
t
t
−
= ⇒ = years
Since payments are made yearly, the first discrete time when the accumulated net cash flows
exceeds 0 is 8 years.
So, the discounted payback period in this example is 8 years.
We now calculate the accumulated value after 8 years, and use this figure to calculate the
accumulated profit after 10 years:
( )
8
80.06
8 7000(1.06) 1150 $225.15 AV s = − + =
Profit =
2
225.15 1.03 1150 1.03 1150 $2,573.36 × + × + =
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 11
代写 Capital Budgeting STAT2032/6046 Financial Mathematics
OTHER CONSIDERATIONS
At the simplest level, for projects involving similar amounts of money and with similar time
horizons, the project that results in the highest accumulated profit will be the most
favourable.
This is equivalent to selecting the project with the highest net present value. The internal rate
of return provides a useful secondary criteria.
Where external borrowing is involved, the accumulated profit must be calculated directly by
looking at the cash flows and taking into account the precise conditions of the loan. The
discounted payback period can provide a useful secondary criteria.
In practice, it may not be straightforward to decide between investment projects purely on the
basis of net present values, internal rates of return, or discounted payback periods.
Other considerations include:
• Cash flows
• Borrowing requirements
• Resources
• Risk
• Investment conditions
• Cost vs benefit
• Indirect benefits
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 12
MEASURING INVESTMENT PERFORMANCE
It is often necessary to be able to measure the investment performance of a fund over a
period.
Fund value will go up or down as a result of changes in:
• Income generated by the fund – interest payments, dividends, rental payments earned
by the fund’s assets.
• Changes in market value (capital gains/losses) – price investors are prepared to pay
for assets in the fund will vary from day to day.
• “New money“ - extra money paid into the fund that was not generated by the fund
itself (withdrawals are negative “new money”).
In investment performance calculations it is important to distinguish between money
generated by the fund and “new money”. Otherwise there is a danger of double counting or
under counting.
MONEY-WEIGHTED RATE OF RETURN (MWRR)
The internal rate of return is sometimes referred to as the money-weighted rate of return for
the transaction. This terminology is more likely to be used in the context of a transaction of
one year duration, when measuring the performance of an investment fund.
“The money-weighted rate of return is the interest rate satisfying the equation of value
incorporating the initial and final fund values and the intermediate cash flows.”
Note that the equation of value used in calculating the MWRR only takes account of new
money. Any cash flows generated as investment income by the fund itself must be ignored.
This is because the equation of value we are setting up is to value the rate of return that
generated the income and changes in market value.
EXAMPLE
The market value of a small pension fund’s assets was $2.7m on 1 January 2000 and $3.1 m
on 31 December 2000. During 2000 the only cash flows were:
• Bank interest and dividends totalling $125,000 received on 30 June
• A lump sum retirement benefit of $75,000 paid on 1 May
• A contribution of $50,000 paid to the fund on 31 December
Show that the MWRR is 16%.
Solution
The only “new money” is the lump sum retirement benefit of $75,000 and the contribution of
$50,000.
The bank interest and dividends lead to the growth in the value of the fund, which is what the
MWRR i is measuring. These payments are already “absorbed” in the value of i . Including
them as cash flows in the equation would result in double counting. Cash flows in respect of
new money, on the other hand, are not reflected in the value of i , and so these must be
included as extra terms in the equation of value.
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 13
The equation of value using compound interest (in $’000) is:
8/12
3,100 2,700(1 ) 75(1 ) 50 i i = + − + +
Evaluating the LHS at interest rates either side of 16%, we find:
15.95% 3,097.9 3,100
16.05% 3,100.5 3,100
i LHS
i LHS
= ⇒ = <
= ⇒ = >
Therefore, the MWRR is 16.0% (to the nearest 0.1%).
TIME-WEIGHTED RATE OF RETURN (TWRR)
The MWRR is sensitive to the amounts and timing of the net cash flows. If, for example, we
are assessing the skill of a fund manager, the MWRR is not ideal, as the fund manager does
not control the timing or amount of the cash flows – he or she is merely responsible for
investing the positive net cash flows and realising cash to meet the negative net cash flows.
For example, if a large amount of money was invested just after a market boom, the MWRR
is likely to be lower than if the money was invested just before the boom. This is not a
reflection of the skill of the fund manager, simply a result of the timing of the cash flows.
An alternative to the money-weighted rate of return for measuring investment fund
performance which tries to eliminate this effect is the time-weighted rate of return (TWRR).
The TWRR is found by calculating “growth factors” reflecting the change in the value of the
fund between the times of consecutive cash flows, then by combining these growth factors to
come up with an overall rate of return for the whole period.
“The time-weighted rate of return is found from the product of the growth factors
between consecutive cash flows”
The time-weighted rate of return for a fund over a one-year period is found by first
identifying the time points at which cash flows occur in the fund. These are the time points at
which new contributions are added, or withdrawals are made.
Assume that these time points are:
1 2
0 ... 1
n
t t t < < < < = . For each successive time interval
代写 Capital Budgeting STAT2032/6046 Financial Mathematics
[ ]
1 , k k
t t
−
, a periodic rate of return
k
i , and corresponding growth factor 1
k
k k
k
B
G i
A
= + = is
calculated where,
k
A is the fund value at time
1 k
t
−
after all transactions are completed, and
k
B is the fund value at time
k
t after interest is credited but just before the cash flow due at
time
k
t occurs. The aim of this is to identify the change in fund value due to interest credited
and capital value change only.
The growth factors are compounded over the full-year to produce the annual growth factor
1 2 1 2
.... (1 )(1 )...(1 ) (1 )
n n
G GG G i i i TWRR = = + + + = +
代写 Capital Budgeting STAT2032/6046 Financial Mathematics
Topic 5 Capital Budgeting
STAT2032/6046 – Financial Mathematics 14
A disadvantage of the TWRR is that it requires the fund values at all the cash flow dates.
EXAMPLE
Using the same example as that used for the MWRR, find the TWRR. Use the additional
information that the fund value at 30 April was $3m.
Solution
First we need to identify the time points at which cash flows occur in the fund. In this
example we know the final fund value ($3.1m). The timing of “new money” is:
1 May $75,000 lump sum benefit paid
31 December $50,000 contribution paid
The progress of the fund (in $’000) was as follows:
1 January to 30 April Fund value increased from 2,700 to 3,000
1 May Cash flow of –75
1 May to 30 December Fund value increased from 3,000-75 = 2,925 to 3,100-50 =
3,050
31 December Cash flow of +50, taking fund value to 3,100
So, during the period from 1 January to 30 April the fund grew by a factor of:
3,000
1.111
2,700
=
During the period from 1 May to 30 December the fund grew by a factor of
3,100 50
1.043
3,000 75
−
=
−
Therefore, the TWRR for the year is (1.111)(1.043) 1 15.9% − =
代写 Capital Budgeting STAT2032/6046 Financial Mathematics
