Chapter 1
Cashflow models
1.1
Common Examples resulting in cashflow
scenarios
Throughout this course we shall be dealing with investment type of problems
that assume a series of cashflows Ct at some time t where
(
Ct > 0 implies income
Ct < 0 implies expense
There are many issues to consider here:
ˆ Do we know with certainty the cashflow time t? This is not so obvious.
Consider for example a retirement annuity promising a retiree some
payments in future after retirement “at the end of each month”. This
contract may be viewed as one with known payment dates t. Rent
contracts are also examples of contracts with known payments dates.
But its not all that easy for some contracts. Can you discuss these?
ˆ The size of payment Ct may be known or unknown. We can again give
examples of these in real life. This is left for you to explore. Your
ActEd notes has many examples (will summarize them at the end) of
Insurance type contracts which sometimes lead to unkown payments
dates or payment sizes and other non-insurance type contracts which
may also result in known or unknown payment dates nor sizes. Ask
yourself the type of contracts whose periodic payments may be known
with certainty or may be uncertain.
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ˆ There is usually an expected final payment date of most contracts. In
a rent contract, the investor (landlord) may receive rent up to and
including the day when the property is sold. This date may not be
known in advance which means an uncertain terminal date. When the
property is never sold, this contract is a perpetuity. The size of the
last payment may also be certain or uncertain. Government bonds are
examples of contracts with known terminal (redemption) dates and
usually a known last payment called the Redemption. We shall talk
more about this later.
Therefore common questions to expect are those that give you a given investments and ask whether the timing of cashflows are known or unknown,
whether the cashflows themselves are known or unknown and whether there
is a final Redemption and if so, whether its known or unknown. For this
course, our cashflows are assumed to be known in both size and timing (except otherwise stated). But as you progress your degree to other courses like
Contingencies, these assumptions get relaxed. Your ActEd Notes have very
good examples of contracts that can be used for your discussion for both
Insurance and non-Insurance contracts.
The following make part of the course’s common assumptions (we allow
for variations in exceptional cases):
ˆ The timing t of each cashflow is known with certainty or is uncertain,
usually measured in years from a common starting point t = 0. Where
the timing is random, it leads to complicated calculations beyond the
scope of the course.
ˆ The amount Ct , also known as size of cashflow, is either known with
certainty or unknown.
Example 1.1.1 Describe the cashflows involved for the issuer of an interestonly loan, referring to certainty of payment, timing of payment, sign of the
payments (+’ve or –’ve) and relative magnitude of cashflows.
Ignore the possibility of default in your answer.
Solution 1.1.1 The issuer of the loan has a
ˆ large –‘ve cashflow (outflow) initially or t=0
ˆ timing and amount certain
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In return, the issuer
ˆ receives regular +’ve cashflow (of interest payments)
ˆ relatively small amounts compared to loan amount
ˆ the timing of which are certain and already known
ˆ Amounts may be fixed (known in advance) or variable (unknown in
advance).
Finally,
ˆ large +’ve cashflow (inflow)
ˆ amount is certain = the initial amount of the loan
ˆ timing known or unknown depending on whether early repayment is
allowed.
Note that your official Financial Calculator HP12C allows you to enter
the cashflows using the Tab CTi , i ≥ 0. A special tutorial is reserved for you
to learn how to do this for common cases.
Below we look at some cases and their corresponding cashflows. It is a
common practice in this course to use a timeline to give you a rough picture
of the timing and sizes of cashflows. Most cases involve multiple cashflows
which one can easily visualize with the aid of a timeline. It will be to your
advantage to create yours for each problem.
Here are some familiar investments where you can have cashflows. In
each case discuss whether the cashflows and timing are certain or uncertain.
To guide you, lets see the example below:
A fixed interest security e.g coupon paying bond.
DN + RN
t0 = 0
DN
DN
6
6
t1
t2
6
......
tn = T
?
PN
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There are multiple transaction dates: at purchase the price PN, then
coupon payments DN, usually quoted as annual payments payable semiannually and then redemption RN if the bond has known redemption dates
and is not a perpetuity. The nominal value N is also called “par value”.
Therefore
ˆ If R = 1 the bond is redeemed at par since RN = N or if P = 1 the
bond is priced at par since PN = N.
ˆ If R < 1 the bond is redeemed below par since RN < N or if P < 1
the bond is priced below par since P N < N
ˆ If R > 1 the bond is redeemed above par since RN > N or if P > 1
the bond is priced above par since P N > N
We consider as illustrations the following British government stocks:
(a) 9% Treasury 1994
This stock, which was issued in 1969, bears interest at 9% per annum, payable half-yearly on 17 May and 17 November. The stock was
redeemable at par on 17 November 1994.
(b) 3% British Gas 1990-5
This stock was issued in 1949 following the nationalisation of the gas
industry. It bears interest at 3% per annum, payable half-yearly on 1
May and 1 November, and was redeemable at par on any interest date
between 1 May 1990 and 1 November 1995 (inclusive) at the option
of the government. Since the precise redemption date is not predetermined, but may be chosen between certain limits by the borrower, this
stock is said to have ‘optional redemption dates’.
(c) 3 21 % War Loan
This stock was issued in 1932 as a conversion of an earlier stock, issued
during the 1914-1918 war. Interest is at 3 12 % per annum, payable halfyearly on 1 June and 1 December. This stock is now redeemable at
par on any interest date the government chooses, there being no final
redemption date. The stock may therefore be considered as having
optional redemption dates, the second of them being infinity.
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1.1.1
Securities from Financial Contracts
1. A zero-coupon bond
Only two notable transactions, one at purchase where the investor pays
the bond price PN and the other at redemption or maturity where the
bond issuer pays the redemption amount RN. Logically RN > P N
otherwise the investment will be worthless. The amount N is called the
nominal amount which for the purpose of this course, takes a default
value of N = R100 in line with the UK system which uses N = £100.
Here P is given as a percentage of nominal and R, the redemption, is
also a percentage.
2. Index linked security (e.g inflation linked or prime linked...)
These will be dealt with in Chapter 4 in detail. The coupon payments
and redemption are adjusted to inflation to compensate the investor
for the negative effects of inflation.
3. Cash on deposit (e.g call deposit (unknown amount to be withdrawn
and unknown time of withdrawal) and term deposits e.g ABSA 32-day
Notice Deposit, or CD’s).
Term deposits are similar to zero coupon bonds in terms of them having
only two dates for transactions. Call deposits are similar to American
options which you shall study later in your degree.
4. Equities (ordinary shares) are generally not fixed interest securities and
are usually considered perpetuities
5. an annuity certain (payment independent of any life)
6. An interest only loan
7. Property investment
8. A repayment loan (or mortgage)
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1.1.2
Securities from Insurance Contracts
1. A pure endowment (lump-sum payment at a certain pre-determined
period if the insured survives to that date and nothing if the insured
dies before the date)
2. An endowment assurance (payment to policy holder at maturity if alive
or at death if occurs before maturity)
3. A term assurance (payment if death occurs before a pre-specified time.
Nothing if insured survives to the term)
4. A contingent annuity (the contingency is usually death, annuity payments are to the beneficiary upon death of the policy holder)
ˆ single life annuity (payments stop at death of annuitant )
ˆ joint life annuity (payments continue after death to annuitant’s
spouse)
ˆ reversionary annuity (payment of the annuity at death of insured
i.e one of the 2 lives dies)
5. A car insurance policy
6. A health cash plan
1.2
Using Your Financial Calculator to Enter
Cashflows
Note that your official Financial Calculator HP12C allows you to enter the
cashflows using CTi , i ≥ 0. First you must familiarize yourself with how
your calculator understands timelines and cashflows. Use your pdf file of the
manual and read Section 3 , Subsection “Financial Calculations and the Cash
Flow Diagram” on page 44 . After that then learn how to enter cashflows as
explained in Section 4 page 72.
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1.3
Excel
Entering cashflows in Excel is as easy as you enter any numbers on a worksheet. Care must be taken on the timing depending on choice of basic time
unit (e.g monthly cashflows or yearly cashflows or half-year cashflows.)
As Practice exercise, consider the examples from page 74 of your Financial
Calculator pdf manual and practice entering cashflows on the calculator and
in Excel (create an excel worksheet during your free time in the student labs).
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