CMP Upgrade This CMP Upgrade lists the changes to the Syllabus objectives, Core Reading and the ActEd material since last year that might realistically affect your chance of success in the exam. It is produced so that you can manually amend your 2016 CMP to make it suitable for study for the 2017 exams. It includes replacement pages and additional pages where appropriate. Alternatively, you can buy a full set of up-to-date Course Notes / CMP at a significantly reduced price if you have previously bought the full-price Course Notes / CMP in this subject. Please see our 2017 Student Brochure for more details.
This CMP Upgrade contains: all significant changes to the Syllabus objectives and Core Reading. additional changes to the ActEd Course Notes, Question and Answer Bank and
Series X Assignments that will make them suitable for study for the 2017 exams.
This section contains additional significant changes to the ActEd Course Notes. However, if you wish to have all the changes to the ActEd Course Notes, you will need to buy a full set of the up-to-date version (which you can do at a significantly reduced price if you have previously bought the full-price Course Notes / CMP in this subject). Chapter 10 Pages 9 and 10 Some of the ActEd text on net present values and internal rates of return on these pages has been amended for clarity. Replacement pages 9 to 14 are provided due to changes in subsequent page layout.
The most significant changes to the Q&A Bank are given below. Q&A All Parts Throughout the Q&A Bank, indicative marks for the first guess in “trial and error” style questions have been removed or reallocated for consistency with how marks are awarded in the exam. Q&A Part 2 Question 2.24 This new question has been added to the Q&A Bank. Replacement pages are provided. Q&A Part 4 Solution 4.8 The solution provided for parts (iii) and (iv) of this question has been updated. Replacement pages are provided. Q&A Part 5 Question 5.11 The number of marks for part (i) of this question has been increased from 2 to 3, and the number of marks for part (iii) of this question has been reduced from 4 to 3. Solution 5.11 The solution to this question has been expanded. Replacement pages are provided.
The most significant changes to the X Assignments are given below. All Assignments The following marking advice has been added at the start of each set of solutions: In “trial and error” questions, full marks should be awarded for obtaining the correct final answer whatever method is used (eg “table mode” on a calculator), so long as sufficient working is given. Assignment X2 Solution X2.9(i) The marks for this part of the solution have been reallocated, so that no mark is awarded for the choice of first guess. The numerical aspects of the solution are unchanged.
In addition to the CMP you might find the following services helpful with your study.
6.1 Study material
We also offer the following study material in Subject CT1:
● Online Classroom
● Flashcards
● MyTest
● Revision Notes
● ASET (ActEd Solutions with Exam Technique) and Mini-ASET
● Mock Exam A
● Additional Mock Pack. For further details on ActEd’s study materials, please refer to the 2017 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.
6.2 Tutorials
We offer the following tutorials in Subject CT1:
a set of Regular Tutorials (lasting two or three full days)
a Block Tutorial (lasting two or three full days)
a Revision Tutorial (lasting one full day)
a set of Live Online Tutorials (lasting three full days). For further details on ActEd’s tutorials, please refer to our latest Tuition Bulletin, which is available from the ActEd website at www.ActEd.co.uk.
You can have your attempts at any of our assignments or mock exams marked by ActEd. When marking your scripts, we aim to provide specific advice to improve your chances of success in the exam and to return your scripts as quickly as possible. For further details on ActEd’s marking services, please refer to the 2017 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.
6.4 Feedback on the study material
ActEd is always pleased to get feedback from students about any aspect of our study programmes. Please let us know if you have any specific comments (eg about certain sections of the notes or particular questions) or general suggestions about how we can improve the study material. We will incorporate as many of your suggestions as we can when we update the course material each year. If you have any comments on this course please send them by email to [email protected].
In this graph, the dotted line represents Project R and the solid line represents Project S. In the graph above, as is the case for most projects in the real world, the net present value decreases as the risk discount rate (i) increases. The reason for this is that in Projects R and S the income occurs later in time than the outgo, meaning it is discounted for a longer period, so its present value is more affected by the change in the risk discount rate. Therefore, as the risk discount rate increases, the present value of the project’s income reduces by more than the present value of the project’s outgo, and the net present value falls.
Question 10.4
The cashflows Ct (where the time t is measured in years and the amounts are in £000)
for two business ventures are as follows: Venture 1: C0 100 , C1 40 , C2 50 , C3 120
Venture 2: C1 45 , C3 25 , C4 25 , C5 25
Calculate the accumulated profit at time 5 and the net present value for each of these ventures using a risk discount rate of 15% per annum.
In economics and accountancy the yield per annum is often referred to as the “internal rate of return” (IRR) or the “yield to redemption”. The latter term is frequently used when dealing with fixed interest securities, for which the “running” (or “flat”) yield is also considered. We will leave the definition of the running yield until later in the course. The internal rate of return for an investment project is the effective rate of interest that equates the present value of income and outgo, ie it makes the net present value of the cashflows equal to zero. If all the payments for the project were transacted through a bank account that earned interest at the same rate as the internal rate of return, the net proceeds at the end of the project (ie the accumulated profit) would be zero. A higher internal rate of return indicates a more “profitable” project. For most projects, there will be a unique solution to the equation defining the internal rate of return, since the quantity “ PV payments in PV payments out ” generally
decreases as i increases, as we saw in the graph on the previous page. The internal rate of return need not be positive. A zero return implies that the investor received no return on investment and if the yield is negative then the investor lost money on investment. It is difficult, however, to find a practical interpretation for a yield less than –1, and so if there is not a solution to the equation greater than –1, the yield is undefined. In some cases, it is possible for there to be more than one solution. In such cases the smallest positive solution is usually used (or the most negative solution greater than –1, if there aren’t any positive solutions). Some authors however have stated that the yield is undefined if there is not a unique solution greater than –1. You are unlikely to come across many such examples whilst studying Subject CT1. Also, if there are only inflows of cash (ie no outflow), the internal rate of return will be infinite. Usually, the equation of value cannot be solved directly to find the interest rate. In these cases, an approximate solution can be found by first estimating an approximate value using the methods we looked at earlier. A more accurate value can then be found using linear interpolation by calculating the net present value for interest rates close to the initial estimate.
Example Find the internal rate of return for Project R. Solution We need to find the interest rate i that satisfies the equation of value:
150 250 250 1 000 02 3v v v, We already know that:
At 20%: 150 250 250 1 000 4682 3v v v, . Since the outgo precedes the income, the value of i must be greater than this. So we need to try a higher rate:
At 25%: 150 250 250 1 000 2 002 3v v v, . 46.8 2.00 0 20% 25% i We can approximate i by extrapolating (linearly) using these two values:
i
20%0 468
2 00 46825% 20%) 252%
.
. .( .
Question 10.5
Find the internal rate of return for Project S and hence determine which project is more favourable using this criterion.
The practical interpretation of the net present value function ( )NPV i and the
yield is as follows. Suppose that the investor may lend or borrow money at a fixed rate of interest i1 . Since, from Equation (1.2), 1( )NPV i is the present value
at rate of interest i1 of the net cashflows associated with the project, we
conclude that the project will be profitable if and only if: NPV i( )1 0
Also, if the project ends at time T, then the profit (or, if negative, loss) at that time is (by Expression (1.1)):
NPV i i T( )( )1 11
Let us now assume that, as is usually the case in practice, the yield i0 exists and
( )NPV i changes from positive to negative when i i 0 . Under these conditions it
is clear that the project is profitable if and only if: 1 0i i<
ie the yield exceeds that rate of interest at which the investor may lend or borrow money. 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, or hurdle, rate of return may be set for assessing whether a project is likely to be sufficiently profitable.
1.4 The comparison of two investment projects
Suppose now that an investor is comparing the merits of two possible investments or business ventures, which we call projects A and B respectively. We assume that the borrowing powers of the investor are not limited. There are therefore no restrictions on how much the investor can borrow. Let NPV iA( ) and NPV iB ( ) denote the respective net present value functions and
let iA and iB denote the yields (which we shall assume to exist). It might be
thought that the investor should always select the project with the higher yield, but this is not invariably the best policy. A better criterion to use is the profit at time T (the date when the later of the two projects ends) or, equivalently, the net present value, calculated at the rate of interest i1 at which the investor may lend
or borrow money. This is because A is the more profitable venture if: NPV i NPV iA B( ) ( )1 1
The fact that i iA B may not imply that NPV i NPV iA B( ) ( )1 1 is illustrated in the
following diagram. Although iA is larger than iB , the NPV(i) functions “cross-
over” at i . It follows that NPV i NPV iB A( ) ( )1 1 for any i i1 , where i is the
cross-over rate. There may even be more than one cross-over point, in which case the range of interest rates for which project A is more profitable than project B is more complicated.
The following graph is similar to the graph shown on page 9, although it has been extended to show the yields for the two projects.
We now give a final example for this section.
Worked Example An investor is considering whether to invest in either or both of the following loans: Loan X: For a purchase price of £10,000, the investor will receive £1,000 per annum payable quarterly in arrears for 15 years. Loan Y: For a purchase price of £11,000, the investor will receive an income of £605 per annum, payable annually in arrears for 18 years, and a return of his outlay at the end of this period. The investor may lend or borrow money at 4% per annum. Would you advise the investor to invest in either loan, and, if so, which would be the more profitable?
and the yield is found by solving the equation NPV iX ( ) 0 , or a154 10
|( ) , which
gives iX 5 88%. .
This is easily checked by calculating a15
4|
( ) @ 5.88%:
av
i154
15
4
15
1 4
1 1 10588
4 10588 110 00|
( )( ) /
.
( . ).
as required
For loan Y we have:
NPV i a vY ( ) , ,| 11000 605 110001818
and the yield (ie the solution of NPV iY ( ) 0 ) is iY 5 5%. .
Question 10.6
Check that iY 55%. solves the equation of value.
The rate of interest at which the investor may lend or borrow money is 4% per annum, which is less than both iX and iY , so we compare NPVX ( . )0 04 and
NPVY ( . )0 04 .
Now NPVX ( . ) £1,0 04 284 and NPVY ( . ) £2,0 04 089 , so it follows that, although
the yield on loan Y is less than on loan X, the investor will make a larger profit from loan Y. We should therefore advise him that an investment in either loan would be profitable, but that, if only one of them is to be chosen, then loan Y will give the higher profit. The above example illustrates the fact that the choice of investment depends very much on the rate of interest i1 at which the investor may lend or borrow
money. If this rate of interest were 5¾%, say, then loan Y would produce a loss to the investor, while loan X would give a profit.
A training company is planning to expand into another country. The set up costs (which are paid out at the start) are expected to be $50,000. Rent and salaries totalling $120,000 pa are to be paid monthly in arrears for the first two years. After two years, the total monthly payment for rent and salaries increases each month by $100. The expected sales of courses and materials during the first three years of business are shown in the table below:
The income is to be payable monthly in advance but, in the first year, no income is received until the beginning of the 8th month. After the first three years, sales of both materials and courses are expected to grow at a rate of 0.5% per month compound. (i) Assuming that the business continues indefinitely, calculate the net present value
of this project at an effective rate of interest of 8% pa. [7] (ii) Show that the discounted payback period for the project, at an effective rate of
8% pa, is less than 2 years. Hence calculate the discounted payback period. [5] (iii) Calculate the accumulated profit for this project after 5 years, using an annual
effective rate of interest of 8% pa. [3] (iv) A year after setting up the project, it is sold. The original investors earned an
effective annual rate of return of 9% pa. Calculate the sale price. [4] [Total 19]
Company XYZ is considering investing in a capital project. The costs for the project will be £5 million at the beginning of the first year, and
£ ( )20.5 1.04k-¥ million payable at the beginning of year k , 2, 3, ..., 10k = .
The returns will be £2 million at the end of each of the first 5 years, increasing by £250,000 at the end of each of years 6 to 10. Company XYZ decides to use a risk discount rate of 12% pa to assess this project. (Note that a risk discount rate is an annual effective interest rate.) (i) Calculate the net present value of the project. [7] (ii) Calculate the discounted payback period for the project. [4] [Total 11] Question 2.24
An actuarial student takes out a mortgage for £250,000 with a term of 25 years. The mortgage is repayable in level instalments made monthly in arrears. Interest is charged at a rate of 6% pa effective. (i) Calculate the monthly repayment. [2] (ii) (a) Calculate the capital repaid in the fourth year. (b) Calculate the interest element of the 49th repayment. [4] After completing her exams, six years after taking out the mortgage, the newly-qualified actuary reviews her finances and realises that she can afford to make repayments at twice the rate calculated in (i). (iii) Calculate the length of time by which this course of action reduces the remaining
term of the loan. [4] (iv) Calculate the amount of the final repayment and hence the interest saved by the
actuary if she follows this course of action. [5] [Total 15]
So the net present value of the project is: 12.7193 8.0421 4.677NPV PVr PVc= - = - = ie approximately £4,677,000. [1] (ii) Discounted payback period The DPP is the first point in time for which the net present value of the project is positive. At time 4:
2 2 3 3 4
4
5 0.5 0.5 1.04 0.5 1.04 0.5 1.04
0.5 (1 (1.04 ) )5 5 1.6033 6.6033
1 1.04
PVc v v v v
v v
v
= + + ¥ + ¥ + ¥
-= + = + =-
[1]
Alternatively, this can be evaluated using an annuity. 42 2 3.0373 6.0747PVr a= = ¥ = [1]
and: 6.0747 6.6033 0.529 0NPV PVr PVc= - = - = - < [½ ] At time 5:
Alternatively, working in terms of accumulated values: At time 4, 10.3905AVc and 9.5587AVr , so 9.5587 10.3905 0.832AV . At time 5, 12.2223AVc and 12.7057AVr , so 12.7057 12.2223 0.483AV . Solution 2.24
(i) Monthly repayment Let M be the monthly repayment. The equation of value is:
(12)25
250,000 12Ma [1]
Using (12)25
13.1312a , this gives:
£1,586.55M [1]
(ii)(a) Capital repaid in the fourth year The capital outstanding at the start of the fourth year is calculated (prospectively) as:
(12)22
12 12 1,586.55 12.36924 £235,493.04Ma [1]
The capital outstanding at the end of the fourth year is calculated (prospectively) as:
(12)21
12 12 1,586.55 12.08419 £230,065.97Ma [1]
The capital repaid in the fourth year is therefore: 235,493.04 230,065.97 £5,427 [1]
(ii)(b) Interest element in the 49th repayment To calculate the interest element in the 49th repayment, the capital outstanding immediately after the previous (ie 48th) repayment is needed. The 48th repayment is made at the end of four years, so the capital outstanding at that time is £230,065.97 from (ii)(a).
So, the interest element in the 49th repayment is given by:
112230,065.97 (1.06 1) £1,120 [1]
(iii) Reduction in payment term After six years, when the student has qualified, the remaining term is 19 years. The capital outstanding at this point is:
(12)19
12 12 1,586.55 11.46174 £218,215.42Ma [1]
If the actuary makes monthly repayments at twice the original rate, the equation of value is:
(12)218,215.42 12 2 1,586.55n
a
where n is the reduced payment term. So:
(12)(12)
15.73087
n
n
va
i
[1]
1 0.33474
0.66526
n
n
v
v
Taking logs of both sides: ln(1.06) ln(0.66526)n
6.9949n [1]
Therefore, the final repayment will be made 7 years after the increased payments commence. This means the payment term is shortened by 12 years. [1] (iv) Final repayment amount and total interest saved Now assume that the actuary makes twice the original monthly repayments, and let P be the amount of the final repayment made.
The total interest paid is equal to the difference between the total repayments made and the total capital to be repaid. Hence, if the actuary is making twice the original monthly repayments, the total interest paid after the end of the sixth year is:
If the actuary continues making only the original repayments, the total interest paid after the end of the sixth year is: 12 1,586.55 19 218,215.42 £143,517.98 [1]
Hence, the total interest saved by following the new course of action is: 143,517.98 48,129.54 £95,388 [1]
Alternatively, calculate the total interest paid over the whole term of the loan, under each of the repayment schedules. Where only the original repayments are made, the total interest is: 12 1,586.55 25 250,000 £225,965.00
Where twice the original repayments are made after six years, the total interest is:
(iii) The first annuity payment (made at time 1) is 1,000. The second annuity payment (made at time 2) is 1,000 1.05¥ , and so on, with the tth annuity payment (made at
time t) being 11,000 1.05t-¥ . So, here we need to find:
20 201 1
1 1
20 201 1
1 1
1,000 1.05 1,000 1.05
1.05 1.05
t t t t
t t
t t t t
t t
DMT t v v
t v v
- -
= =
- -
= =
= ¥ ¥ ¥ ¥ ¥
= ¥ ¥ ¥
 Â
  [1]
The denominator is a geometric series:
( )
201 2 19 20
1
20
1.05 1.05 1.05
1 (1.05 )
1 1.05
12.1121
t t
t
v v v v
v v
v
-
=¥ = + + +
-=
-
=
Â
[1] The numerator can be evaluated as follows:
20 20 20
1
1 1 1
1 11.05 1.05
1.05 1.05t t t t t
t t t
t v t v t V-
= = =¥ ¥ = ¥ ¥ = ¥Â  Â
where 1 1.05 1.1
1 4.7619%1 1.1 1.05
V II
= = fi = - =+
. So, the numerator is:
( )
2020
20 @ 4.7619%
1 (1 )20 (1 )
1 1 / (1 )1.05 1.05
108.7075
I
II
I IIa
I
--
=
Ê ˆ- + - ¥ +Á ˜+Á ˜=Á ˜Á ˜Ë ¯
= [1] So the discounted mean term is 108.7075 12.1121 8.98 years= . [1]
Purpose of models Both types are used to project the accumulated value of flows of money. [½ ] Assumptions Deterministic models assume that future rates of return are fixed. Stochastic models assume that future rates of return are random variables. [½ ] Results obtained For a given assumed set of future rates of return, a deterministic model will give a single definite answer. For a given assumption about the statistical distribution of future rates of return, a stochastic model will give a statistical distribution describing a range of possible answers. [1] Risk and uncertainty Stochastic interest rate models make allowance for uncertainty by enabling the probability that the actual value will lie in a given range to be calculated. Deterministic models make allowance for uncertainty by carrying out calculations based on different sets of assumptions (eg by including contingency margins in the assumptions). [1]
(iii) If we exclude the progress of the subfund, the main fund grows from 7 to 7.5 during the third quarter, and from 8.7 to 9 in the last quarter. So the equation for the money-weighted rate of return is now:
¾ ½ ¼4(1 ) 3(1 ) 1(1 ) 1.2(1 ) 9i i i i+ + + - + + + = [1]
Using a first order binomial approximation to obtain a first guess:
4(1 ) 3(1 ¾ ) 1(1 ½ ) 1.2(1 ¼ ) 9
6.05 1.8 29.8%
i i i i
i i
+ + + - + + + ª
fi ª fi ª
When 30%i = , ¾ ½ ¼4(1 ) 3(1 ) 1(1 ) 1.2(1 ) 8.9936i i i i+ + + - + + + = .
When 30.5%i = , ¾ ½ ¼4(1 ) 3(1 ) 1(1 ) 1.2(1 ) 9.0231i i i i+ + + - + + + = .
Using linear interpolation, the money-weighted rate of return is:
