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o13
Introduction to Actuarial Science
Matthias Winkel1
University of Oxford
MT 2002
1Departmental lecturer at the Department of Statistics, supported by the Institute of Actuaries
o13
Introduction to Actuarial Science16 lectures MT 2002 and 16 lectures HT 2003
Aims
This course is supported by the Institute of Actuaries. It is designed to give the under-graduate mathematician an introduction to the financial and insurance worlds in whichthe practising actuary works. Students will cover the basic concepts of risk managementmodels for mortality and sickness, and for discounted cash flows. In the final examina-tion, a student obtaining at least an upper second class mark on paper o13 can expect togain exemption from the Institute of Actuaries’ paper 102, which is a compulsory paperin their cycle of professional actuarial examinations.
Synopsis
Fundamental nature of actuarial work. Use of generalised cash flow model to describefinancial transactions. Time value of money using the concepts of compound interestand discounting. Present values and the accumulated values of a stream of equal orunequal payments using specified rates of interest and the net present value at a realrate of interest, assuming a constant rate of inflation. Interest rates and discount ratesin terms of different time periods. Compound interest functions, equation of value, loanrepayment, project appraisal. Investment and risk characteristics of investments. Simplecompound interest problems. Price and value of forward contracts. Term structure ofinterest rates, simple stochastic interest rate models. Single decrement model, presentvalues and the accumulated values of a stream of payments taking into account theprobability of the payments being made according to a single decrement model. Annuityfunctions and assurance functions for a single decrement model. Liabilities under a simpleassurance contract or annuity contract.
Reading
All of the following are available from the Publications Unit, Institute of Actuaries, 4Worcester Street, Oxford OX1 2AW
• Subject 102: Financial Mathematics. Core reading 2003. Faculty & Institute ofActuaries 2002
• J J McCutcheon and W F Scott, An Introduction to the Mathematics of Finance,Heinemann 1986
• P Zima and R P Brown, Mathematics of Finance, McGraw-Hill Ryerson 1993
• H U Gerber, Life Insurance Mathematics, Springer 1990
• N L Bowers et al, Actuarial mathematics, 2nd edition, Society of Actuaries 1997
Contents
1 Introduction 7
1.1 The actuarial profession . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 The generalised cash flow model . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Actuarial science as examples in the generalised cash flow model . . . . . 9
2 The theory of compound interest 11
2.1 Simple versus compound interest . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Time-dependent interest rates . . . . . . . . . . . . . . . . . . . . . . . . 13
3 The valuation of cash flows 15
3.1 Accumulation factors and consistency . . . . . . . . . . . . . . . . . . . . 15
3.2 Discounting and the time value of money . . . . . . . . . . . . . . . . . . 16
3.3 Continuous cash flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Constant discount rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Fixed-interest securities and Annuities-certain 19
4.1 Simple fixed-interest securities . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Securities above/below/at par . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 pthly paid interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Securities with pthly paid interest . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Annuities-certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.6 Perpetuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.7 Annuities and perpetuities payable pthly and continuously . . . . . . . . 22
5 Mortgages and loans 23
5.1 Loan repayment schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Equivalent cash flows and equivalent models . . . . . . . . . . . . . . . . 24
5.3 Fixed, discount, tracker and capped mortgages . . . . . . . . . . . . . . . 26
6 An introduction to yields 27
6.1 Flat rates and APR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 The yield of a cash flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3 General results ensuring the existence of the yield . . . . . . . . . . . . . 29
3
4 Contents
7 Project appraisal 31
7.1 A remark on numerically calculating the yield . . . . . . . . . . . . . . . 31
7.2 Comparison of investment projects . . . . . . . . . . . . . . . . . . . . . 32
7.3 Investment projects and payback periods . . . . . . . . . . . . . . . . . . 32
7.4 Funds and weighted rates of return . . . . . . . . . . . . . . . . . . . . . 33
8 Taxation and inflation 35
8.1 Fixed interest securities and running yields . . . . . . . . . . . . . . . . . 35
8.2 Income tax and capital gains tax . . . . . . . . . . . . . . . . . . . . . . 36
8.3 Inflation indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9 Inflation models and real interest 39
9.1 Modelling inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
9.2 Constant inflation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3 Inflation adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10 Uncertain payment and probabilistic models 43
10.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
10.2 Notation and introduction to probability . . . . . . . . . . . . . . . . . . 43
10.3 Fair premiums and risk under uncertainty . . . . . . . . . . . . . . . . . 45
11 Corporate bonds and uncertain payment 47
11.1 Uncertain payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
11.2 Pricing of corporate bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 49
12 Uncertain investment projects and risk 51
12.1 Pricing of equity shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
12.2 Examples: Comparison of investment projects . . . . . . . . . . . . . . . 52
12.3 Individual risk models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
12.4 Pooling reduces risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
13 Life insurance: the single decrement model 55
13.1 Uncertain cash flows in life insurance . . . . . . . . . . . . . . . . . . . . 55
13.2 Conditional probabilities and the force of mortality . . . . . . . . . . . . 56
13.3 The curtate future lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 57
13.4 Insurance types and examples . . . . . . . . . . . . . . . . . . . . . . . . 58
14 Life insurance: premium calculation 59
14.1 Residual lifetime distributions . . . . . . . . . . . . . . . . . . . . . . . . 59
14.2 Actuarial notation for life products . . . . . . . . . . . . . . . . . . . . . 60
14.3 Lifetables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
14.4 Life annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.5 Multiple premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Contents 5
15 Some elements of General Insurance 6315.1 Premium principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6315.2 The Central Limit Theorem and an example . . . . . . . . . . . . . . . . 64
16 Summary: it’s all about Equations of Value 6716.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
16.1.1 Basic notions used throughout the course . . . . . . . . . . . . . . 6716.1.2 Deterministic applications . . . . . . . . . . . . . . . . . . . . . . 6816.1.3 Applications with uncertaincy . . . . . . . . . . . . . . . . . . . . 68
16.2 Equations of value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6916.3 Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6916.4 Hilary Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.5 Assignment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A A 1967-70 Mortality table 71
Lecture 1
Introduction
This introduction is two-fold. First, we give some general indications on the work of anactuary. Second, we introduce cash flow models as the basis of this course and a suitablemeans to describe and look beyond the contents of this course.
1.1 The actuarial profession
Actuarial Science is an old discipline. The Institute of Actuaries was formed in 1848, (theFaculty of Actuaries in Scotland in 1856), but the profession is much older. An importantroot is the construction of the first life table by Sir Edmund Halley in 1693. However,this does not mean that Actuarial Science is oldfashioned. The language of probabilitytheory was gradually adopted between the 1940s and 1970s. The development of thecomputer has been reflected and exploited since its early days. The growing importanceand complexity of financial markets currently changes the profession.
Essentially, the job of an actuary is risk assessment. Traditionally, this was insurancerisk, life insurance and later general insurance (health, home, property etc). As typicallyenourmous amounts of money, reserves, have to be maintained, this naturally extendedto investment strategies including the assessment of risk in financial markets. Today,the Faculty and Institute of Actuaries claim in their slogan yet more broadly to make“financial sense of the future”.
To become an actuary in the UK, one has to pass nine mathematical, statistical,economic and financial examinations (100 series), an examination on communicationskills (201), an examination in each of the five specialisation disciplines (300 series) andfor a UK fellowship an examination on UK specifics of one of the five specialisationdisciplines. This whole programme takes normally at least three or four years after amathematical university degree and while working for an insurance company.
This course is an introductory course where important foundations are laid and anoverview of further actuarial education and practice is given. The 101 paper is coveredby the second year probability and statistics course. An upper second mark in theexamination following this course normally entitles to an exemption from the 102 paper.Two thirds of the course concern 102, but we also touch upon material of 103, 104, 105and 109.
7
8 Lecture 1: Introduction
1.2 The generalised cash flow model
The cash flow model systematically captures cash payments either between differentparties or, as we shall focus on, in an in/out way from the perspective of one party. Thiscan be done at different levels of detail, depending on the purpose of an investigation,the complexity of the situation, the availability of reliable data etc.
Example 1 Look at the transactions on a worker’s monthly bank statement
Date Description Money out Money in01-09-02 Gas-Elec-Bill £21.3704-09-02 Withdrawal £100.0015-09-02 Telephone-Bill £14.7216-09-02 Mortgage Payment £396.1228-09-02 Withdrawal £150.0030-09-02 Salary £1022.54
Extracting the mathematical structure of this example we define elementary cash flows.
Definition 1 A cash flow is a vector (tj, cj)1≤j≤m of times tj ≥ 0 and amounts cj ∈ IR.Positive amounts cj > 0 are called inflow. If cj < 0, then −cj is called outflow.
Example 2 The cash flow of Example 1 is mathematically given by
j tj cj
1 1 -21.372 4 -100.003 15 -14.72
j tj cj
4 16 -396.125 28 -150.006 30 1022.54
Often, the situation is not as clear as this, and there may be uncertainty about thetime/amount of a payment. This can be modelled using probability theory.
Definition 2 A generalised cash flow is a random vector (Tj, Cj)1≤j≤M of times Tj ≥ 0and amounts Cj ∈ IR with a possibly random length M ∈ IN.
Sometimes, in fact always in this course, the random structure is simple and the timesor the amounts are deterministic, or even the only randomness is that a well specifiedpayment may fail to happen with a certain probability.
Example 3 Future transactions on a worker’s bank account
j Tj Cj Description1 1 -21.37 Gas-Elec-Bill2 T2 C2 Withdrawal?3 15 C3 Telephone-Bill
j Tj Cj Description4 16 -396.12 Mortgage payment5 T5 C5 Withdrawal?6 30 1022.54 Salary
Here we assume a fixed Gas-Elec-Bill but a varying telephone bill. Mortgage paymentand salary are certain. Any withdrawals may take place. For a full specification of thegeneralised cash flow we would have to give the (joint!) laws of the random variables.
o13 Lecture Notes: Introduction to Actuarial Science 9
This example shows that simple situations are not always easy to model. It is animportant part of an actuary’s work to simplify reality into tractable models. Sometimes,it is worth dropping or generalising the time specification and just list approximateor qualitative (’big’, ’small’, etc.) amounts of income and outgo. Cash flows can berepresented in various ways as the following more relevant examples illustrate.
1.3 Actuarial science as examples in the generalised
cash flow model
Example 4 (Zero-coupon bond) Usually short term investments with interest paidat the end of the term, e.g. invest £1000 for ninety days for a return of £1010.
j tj cj
1 0 -10002 90 +1010
Example 5 (Government bonds, fixed interest securities) Usually long term in-vestments with annual or semi-annual coupon payments (interest), e.g. invest £10000 forten years at 5% p.a. The government borrows money from investors.
−£10000 +£500 +£500 +£500 +£500 +£10500
0 1 2 3 9 10
Alternatively, interest and redemption value may be tracking an inflation index.
Example 6 (Corporate bonds) They work the same as government bonds, but theyare not as secure. Rating of companies gives an indication of security. If companies gobankrupt, invested money is often lost. One may therefore wish to add probabilities tothe positive cash flows in the above figure. Typically, the interest rate in corporate bondsis higher to allow for this extra risk of default that the investor takes.
Example 7 (Equities) Shares in the ownership of a company that entitle to regulardividend payments of amounts depending on the profit of the company and decisions atits Annual General Meeting of Shareholders. Equities can be bought and sold (through astockbroker) on stock markets at fluctuating market prices. In the above figure (includingdefault probabilities) the inflow amounts are not fixed, the term at the discretion of theshareholder and the final repayment value is not fixed. There are advanced stochasticmodels for stock price evolution. A wealth of derivative products is also available, e.g.forward contracts, options to sell or buy shares, also funds to spread risk.
Example 8 (Annuity-certain) Long term investments that provide a series of regularannual (semi-annual or monthly) payments for an initial lump sum, e.g.
−£10000 +£1400 +£1400 +£1400 +£1400 +£1400
0 1 2 3 9 10
10 Lecture 1: Introduction
Example 9 (Interest-only loan) Formally in the cash flow model the inversion of abond, but the rights of the parties are not exactly inverted. Whereas the bond investorcan usually redeem early with only minor restrictions, the lender of a loan normally hasto obey stricter rules, for the benefit of the borrower.
Example 10 (Repayment loan) Formally in the cash flow model the inversion of anannuity-certain, but with differences in the rights of the parties as for interest-only loans.
Example 11 (Life annuity) The only difference to Annuity-certain is the term of pay-ments. Instead of having a fixed term life annuities terminate on death of the holder.Risk components like his age, health and profession when entering the contract deter-mine the amount of the initial deposit. They are usually issued by insurance companies.Several modifications exist (minimal term, maximal term, payable from the death of oneperson to a second person for their life etc.).
Example 12 (Term assurance) They pay a lump sum on death (or serious illness) formonthly or annual premiums that depend on age and health of the policy holder whenthe policy is underwritten. A typical assurance period is twenty years, but age limits ofsixty-five or seventy years are common. The amount can be reducing in accordance withan outstanding mortgage. There is no cash in value at any time.
Example 13 (Endowment assurance) They have the same conditions as term assur-ances but also offer payment in the event of survival of the term. Due to this they aremuch more expensive. They increase in value and can be sold early if needed.
Example 14 (Property insurance) They are one class of general insurance (othersare health, building, motor etc.). For regular premiums the insurance company replacesor refunds any stolen or damaged objects included in the policy. From the provider’spoint of view, all policy holders pay into a pool for those that have claims. Claim historyof policy holders affects their premium.
-time
6pool
��
���
T1
���
T2
��
��
T3
��
��
T4
��
��
T5
��
T6
��
R
�
A branch of an insurance company is said to suffer technical ruin if the pool runs empty.
Example 15 (Appraisal of investment projects) E.g., consider the investment intoa building project. An initial construction period requires certain negative cash flows,the following exploitation (e.g. letting) essentially positive cash flows, but maintenancehas to be taken into accout as well. Under what circumstances is the project profitable?
More generally, one can assess whole companies on their profitability. The importantand difficult first step is estimating the in- and outflows. This should be done by anindependent observer to avoid manipulation. It is common practice to compare average,optimistic and pessimistic estimations reflecting an implicit underlying stochastic model.
Lecture 2
The theory of compound interest
Quite a few problems addressed and solved in this course can be approached in an intuitiveway. However, it adds to clarity and understanding to specify a mathematical model inwhich the concepts and methods can be discussed. The concept of cash flows seen inthe last lecture is one part of this model. In this lecture, we shall construct the basiccompound interest model in which interest of capital investments under varying interestrates can be computed. This model will play a role during the whole course, with suitableextensions from time to time.
Whenever mathematical models are used, reality is only partially represented. Im-portant parts of mathematical modelling are the discussions of model assumptions andparameter specification, particularly the interpretation of model results in reality. It isinstructive to read these lecture notes with this in mind.
2.1 Simple versus compound interest
Consider an investment of C for t time units at the end of which S = C + I is returned.Then we call t the term, I the interest and S the accumulated value of the initial capitalC. One might want to call i = I/tC the interest rate per unit time, but there are differenttypes of interest rates that need to be distinguished, so we have to be more precise.
Definition 3 I = Isimp(i, t, C) = tiC is called simple interest on the initial capital C ∈ IRinvested for t ∈ IR+ time units at the (effective) interest rate i ∈ IR+ per unit time.
S = Ssimp(i, t, C) = C + Isimp(i, t, C) = (1 + ti)C
is called the accumulated value of C after time t under simple interest at rate i.
Interest rates always refer to some time unit. The standard choice is one year, but itsometimes eases calculations to choose one month or one day. The definition reflects theassumption that the interest rate does not vary with the initial capital nor the term.
The problem with simple interest is that splitting the term t = t1 + t2 and reinvestingthe accumulated value after time t1 yields
Ssimp(i, t2, Ssimp(i, t1, C)) = (1 + ti + t1t2i2)C > (1 + ti)C = Ssimp(i, t, C),
11
12 Lecture 2: The theory of compound interest
provided only 0 < t1 < t and i > 0. This profit by term splitting has the disadvantageouseffect that the customer who maximises his profit keeps reinvesting his capital for shortperiods to achieve interest on his interest, so-called compound interest. In fact, he wouldhave to choose infinitesimally small periods:
Proposition 1 For any given interest rate δ ∈ IR+ we have
supn∈IN,t1,...,tn∈IR+:t1+...+tn=t
Ssimp(δ, tn, . . . Ssimp(δ, t2, Ssimp(δ, t1, C)) . . .)
= limn→∞
Ssimp(δ, t/n, . . . Ssimp(δ, t/n, Ssimp(δ, t/n, C)) . . .)
= etδC
for all C ∈ IR and t ∈ IR+.
Proof: For the second equality we first establish
Ssimp(δ, t/n, . . . Ssimp(δ, t/n, Ssimp(δ, t/n, C)) . . .) =
(
1 +tδ
n
)n
C
by induction from the definition of Ssimp. Then we use the continuity and power expansionof the natural logarithm to see the existence of the limit and
log
(
limn→∞
(
1 +tδ
n
)n)
= limn→∞
n log
(
1 +tδ
n
)
= limn→∞
n
(
tδ
n+ O(1/n2)
)
= tδ.
Furthermore, the first equality follows from the observation
1 + tδ ≤∑
k≥0
(tδ)k
k!= etδ ⇒ (1 + t1δ)(1 + t2δ) . . . (1 + tnδ) ≤ et1δ+...+tnδ = etδ
for all t1 . . . tn ∈ IR+ with t1 + . . . + tn = t, and this inequality is preserved when we takethe supremum over all such choices of t1, . . . , tn. 2
We changed our notation for the interest rate from i to δ since this compounding ofinterest allows different quantities to be called interest rate. In particular, if we apply the“optimal strategy” of Proposition 1 to an initial capital of 1, the return after one timeunit is eδ = 1 + (eδ − 1) and i = eδ − 1 is a natural candidate to be called the interestrate per unit time. Note that then
etδ = (1 + i)t and δ =∂
∂t(1 + i)t
∣
∣
∣
∣
t=0
.
Definition 4 S = Scomp(i, t, C) = (1 + i)tC is called the accumulated value of C ∈ IRafter t ∈ IR+ time units under compound interest at the effective interet rate i ∈ IR+ perunit time. I = Icomp(i, t, C) = Scomp(i, t, C) − C is called compound interest on C aftern time units at rate i per unit time.
δ = log(1 + i) =∂
∂tIcomp(i, t, 1)
∣
∣
∣
∣
t=0
is called the force of interest.
o13 Lecture Notes: Introduction to Actuarial Science 13
Compound interest is the standard for long term investments. One might say, simpleinterest is oldfashioned, but it is still used for short term investments when the differenceto compound interest is relatively small. Interest cannot be paid continuously even ifthe tendency is to increase the frequency of interest payments (used to be annually, nowquaterly or even monthly). Within one such time unit, any interest is usually calculatedas simple interest and credited at the end of each time unit. Of course one could alsocredit compound interest at the end of time units only. However, for the investor, theuse of simple interest is an advantage, since
Proposition 2 Given an effective interest rate i > 0 and an initial capital C > 0,
0 < t < 1 ⇒ Isimp(i, t, C) > Icomp(i, t, C)
1 < t < ∞ ⇒ Isimp(i, t, C) < Icomp(i, t, C)
Proof: We compare accumulated values. The strict convexity of f(t) = (1 + i)t followsby differentiation. But then we have for 0 < t < 1
f(t) < tf(1) + (1 − t)f(0) = t(1 + i) + (1 − t) = 1 + ti = g(t)
and for t > 1
f(1) <1
tf(t) +
(
1 − 1
t
)
f(0) ⇒ f(t) > tf(1) + (1 − t)f(0) = g(t).
This completes the proof since Ssimp(i, t, C) = Cg(t) and Scomp(i, t, C) = Cf(t). 2
Rates quoted by banks are not always effective rates. Therefore, comparison of dif-ferent types of interest should be made with care. This statement will be supported forinstance by the discussion of nominal interest rates in the next lecture.
As indicated in the definitions, we shall relax our heavy notation in the sequel, e.g.Icomp(i, t, C) to Icomp or I, whenever there is no ambiguity.
Example 16 Given an effective interest rate of i = 4% per annum (p.a.). InvestingC = £1000 for t = 5 years yields
Isimp = tiC = £200.00 or Icomp =(
(1 + i)t − 1)
C = £216.65
2.2 Time-dependent interest rates
In the previous section we assumed that interest rates are constant over time. Suppose,we now let i = i(t) vary with discrete time t ∈ IN+. If we want to avoid odd effects byterm splitting, we should define the accumulated value at time n for an investment of Cat time 0 as
(1 + i(n − 1)) . . . (1 + i(1)) . . . (1 + i(0))C.
14 Lecture 2: The theory of compound interest
When passing from integer terms to non-integer terms, it turns out that instead of spec-ifying i, we had better specify the force of interest δ = δ(t) which we saw to have a localmeaning as the derivative of the compound interest function.
More explicitly, if δ is piecewise constant, then the iteration of Scomp along the suc-cessive subterms tj ∈ IR+ at constant forces of interest δj yields the return of an initialamount C
S = eδntn . . . eδ2t2eδ1t1C (1)
and we can see this as the exponential Riemann sum defining exp(∫ t
0δ(s)ds).
Definition 5 Given a time dependent force of interest δ(t), t ∈ IR+, that is (locally)Riemann integrable, we define the accumulated value at time t ≥ 0 of an initial capitalC ∈ IR under a force of interest δ as
S = C exp
{∫ t
0
δ(s)ds
}
.
I = S − C is called the interest of C for time t under δ.
δ(t) can be seen as defining the environment in which the value of invested capitalevolves. We will see in the next lecture that this definition provides the most general(deterministic) setting, under some weak regularity conditions and under a consistencycondition (consistency under term splitting), in which we can attribute time values tocash flows (tj, cj)j=1,...,m.
Local Riemann integrability is a natural assumption that makes the expressions mean-ingful. In fact, for practical use, only (piecewise) continuous functions δ are of importance,and there is no reason for us go beyond this.
In support of this definition, we conclude by quoting some results from elementarycalculus and Riemann integration theory. They show ways to see our definition as theonly continuous (in δ!) extension of the natural definition (1) for piecewise continuousforce of interest functions.
Lemma 1 Every continuous function f : [0,∞) → IR can be approximated locally uni-formly by piecewise constant functions fn.
This naturally extends to functions that are piecewise continuous with left and rightlimits on the discrete set of discontinuities. Furthermore, without any continuity assump-tions, we have the convergence of integrals:
Lemma 2 If fn → f locally uniformly for Riemann integrable functions fn, n ∈ IN, then∫ t
0
fn(s)ds →∫ t
0
f(s)ds
locally uniformly as a function of t ≥ 0.
Also, although the approximation by upper and lower Riemann sums is not locallyuniform, in general, the definition of Riemann integrability forces the convergence of theRiemann sums (which are integrals of approximations by piecewise constant functions).
Lecture 3
The valuation of cash flows
This lecture combines the concepts of the first two lectures, cash flows and the compoundinterest model by valuing the former in the latter. We also introduce and value continuouscash flows.
3.1 Accumulation factors and consistency
In the previous lecture we defined an environment for the evolution of the accumulatedvalue of capital investments via a time-dependent force of interest δ(t), t ∈ IR+. Thecentral formula gives the value at time t ≥ 0 of an initial investment of C ∈ IR at time 0:
S(0, t) = C exp
{∫ t
0
δ(s)ds
}
=: C A(0, t) (1)
where we call A(0, t) the accumulation factor from 0 to t. It is the factor by whichcapital invested at time 0 increases until time t. We also introduce A(s, t) as the factorby which capital invested at time s increases until time t. The representation in termsof δ is intuitively obvious, but we can also derive this from the following important termsplitting consistency assumption
A(r, s)A(s, t) = A(r, t) for all t ≥ s ≥ r ≥ 0. (2)
Proposition 3 Under definition (1) and the consistency assumption (2) we have
A(s, t) = exp
{∫ t
s
δ(r)dr
}
for all t ≥ s ≥ 0. (3)
The accumulated value at time s of a cash flow c = (tj, cj)1≤j≤m, (tj ≤ s, j = 1, . . . , m)is given by
AV als(c) :=
m∑
j=1
A(tj, s)cj. (4)
Proof: For the first statement choose r = 0 in (2), apply (1) and solve for A(s, t).By definition of A(s, t), any investment of cj at time tj increases to A(tj, s)cj by time
s. The second statement now follows adding up this formula over j = 1, . . . , m. 2
15
16 Lecture 3: The valuation of cash flows
Accumulation factors are useful since they allow to move away from the referencetime 0. Mathematically, this is not a big insight, but as a concept and notationally, ithelps to value and nicely represent more complex structures.
We can strengthen the first part of Proposition 3 considerably as follows.
Proposition 4 Suppose, A : [0,∞)2 → (0,∞) satisfies the consistency assumption (2)and is continuously differentiable in the second argument for every fixed first argument.Then there exists a continuous function δ such that (3) holds.
Proof: First note that the consistency assumption for r = s = t implies A(t, t) = 1 forall t ≥ 0. Then define
δ(t) := limh↓0
A(t, t + h) − 1
h= lim
h↓0
A(0, t + h) − A(0, t)
hA(0, t),
the second equality by the consistency assumption. Now define g(t) = A(0, t), f(t) =log(A(0, t)), then we have
δ(t) =g′(t)
g(t)= f ′(t) ⇒ log(A(0, t)) = f(t) =
∫ t
0
δ(s)ds
which is (1) and by the preceding proposition, the proof is complete. 2
This result shows that the concrete and elementary consistency assumption naturallyleads to our models specified in a more abstract way by a time-dependent force of interest.
Corollary 1 For any (locally) Riemann integrable δ, the accumulated value h(t) =S(0, t) is the unique (continuous) solution to
h′(t) = δ(t)h(t), h(0) = C.
Proof: This can be seen as in the preceding proof, since h(t) = Cg(t). 2
3.2 Discounting and the time value of money
So far our presentation has been oriented towards calculating returns for investments. Wenow want to realise a certain return at a specified time t, how much do we have to investtoday? The calculation of such present values of future returns is called discounting, andthe inversion of (1) yields a (discounted) present value of
C =S
A(0, t)= S exp
{
−∫ t
0
δ(s)ds
}
=: S V (0, t) =: S v(t) (5)
for a return S at time t ≥ 0, and more generally a discounted value at time s ≤ t of
Cs =S
A(s, t)= S exp
{
−∫ t
s
δ(r)dr
}
=: S V (s, t) = Sv(t)
v(s). (6)
V (s, t) is called the discount factor from t to s, v(t) the (discounted) present value of 1.
o13 Lecture Notes: Introduction to Actuarial Science 17
Proposition 5 The discounted value at time s of a cash flow c = (tj, cj)1≤j≤m, (tj ≥ s,j = 1, . . . , m) is given by
DV als(c) =m∑
j=1
cjV (s, tj) =1
v(s)
m∑
j=1
cjv(tj). (7)
Proof: This follows adding up (6) over all in- and outflows (tj, cj), j = 1, . . . , m. 2
The restriction to tj ≥ s is mathematically not necessary, but eases interpretation.Our question was how much money we have to put aside today to be able to makefuture payments. If some of the payments happened in the past, particularly inflows, itis essential that the money remained in the system to earn the appropriate interest. Butthen we have the accumulated value of these past payments given by Definition 5.
Definition 6 The time-t value of a cash flow c is denoted by
V alt(c) = AV alt(c[0,t]) + DV alt(c(t,∞)).
This simple formula (with (4) and (7)) is central in investment and project appraisalthat we discuss later in the course.
We conclude this section by a corollary to Propositions 3 and 5.
Corollary 2 For all s ≤ t we have V alt(c) = V als(c)A(s, t) = V als(c)v(s)
v(t).
3.3 Continuous cash flows
When many small inflows (or outflows) accumulate regularly spread over time, it ispractically useful and mathematically natural to consider a continuous approximation,continuous cash flows. We have seen this in Example 14 when a premium pool wasassumed to increase continuously by regular premium payments.
Definition 7 A continuous cash flow is a (locally) Riemann integrable function c : IR+ →IR. c(t) is also called the payment rate at time t.
The interpretation is that the total payment between s and t is∫ t
sc(r)dr, although
this is ignoring the time value of money.
Proposition 6 Under a force of interest δ(·), a continuous cash flow c up to time tproduces an accumulated value of
AV alt(c) =
∫ t
0
A(r, t)c(r)dr =
∫ t
0
exp
{∫ t
r
δ(s)ds
}
c(r)dr.
The discounted value at time t of the post-t cash flow c is
DV alt(c) =
∫ T
t
V (t, r)c(r)dr =
∫ T
t
exp
{
−∫ r
t
δ(s)ds
}
c(r)dr
where T = sup supp(c) can be infinite provided the limit exists.We call V alt(c) = AV alt(c[0,t]) + DV alt(c(t,∞)) the value of c at time t. Corollary 2
still holds, also for mixtures of discrete and continuous cash flows.
18 Lecture 3: The valuation of cash flows
Proof: Define h(t) = St. First we note that h(0) = 0 and by Corollary 1
h(t) =
∫ t
0
h(s)δ(s)ds +
∫ t
0
c(s)ds.
Then, by differentiation h′(t) = h(t)δ(t) + c(t). Now h is the accumulated value, let’slook at the discounted time 0 values
η(t) = StV (0, t) = h(t) exp
{
−∫ t
0
δ(s)ds
}
which satisfies
η′(t) = h′(t)V (0, t) − h(t)δ(t)V (0, t) = c(t)V (0, t)
and integrating this yields η and then h as required.For the discounted value at time u note that the case u = 0 and c supported by [0, t]
is given by η(t) = C0. The general statement is obtained letting t tend to infinity andmultiplying by accumulation factors A(0, u) to pass from time 0 to time u. 2
3.4 Constant discount rates
Let us turn to the special case of a constant force of interest δ ∈ IR+. We have seen adescription of the model in terms of the effective interest rate i = eδ − 1 ∈ IR+. Theinterpretation of i is that an investment of 1 earns interest i in one time unit. The conceptof discounting can be approached in the same way.
Proposition 7 In a model with constant force of interest δ, an investment of v := 1 −d := e−δ yields a return of 1 after one time unit.
Proof: Just apply (5), the definition of V (t, t + 1) = e−δ. 2
Definition 8 d = 1 − e−δ is called the effective discount rate, v = e−δ the effectivediscount factor per unit time.
i, d and v are all expressed in terms of δ, and any one of them determines all theothers.
d is sometimes quoted if a bill due at at a future date contains a (simple, so-calledcommercial) early payment discount proportional to the number of days we pay in ad-vance. In analogy with simple interest this means, that a billed amount C due at time tcan be paid off paying C(1 − td).
Returning to the compound interest model, v is very useful in expressions like in thelast sections where now
v(s) = vs, V (s, t) = vt−s, A(s, t) = v−(t−s).
Example 17 For a return of £10,000 in 4.5 years’ time, how much do you have to investtoday at an effective interest rate of 5% p.a.?
v = e−δ =1
1 + i=
1
1.05, C = v4.510, 000 = 8028.75.
We can also say, that today’s value of a payment of £10,000 in 4.5 years’ time is £8028.75.
Lecture 4
Fixed-interest securities andAnnuities-certain
In this chapter we work out practical examples in the compound interest model, that areof central importance: securities and annuities. We put some emphasis on cases wherethe payment frequency is not unit time. Often, when choosing to the appropriate timeunit, there is no need for a continuous time model and we could switch to the naturallyembedded discrete time model. Also, all securities and annuities considered here areassumed to have no risk of default.
4.1 Simple fixed-interest securities
As seen in Example 5, an investment into the simplest type of a fixed-interest securitypays interest at rate j at the end of each time unit over an integer term n and repays theinvested money at the end of the term. The cash flow representation is
c0 = ((0,−C), (1, Cj), . . . , (n − 1, Cj), (n, C + Cj))
In practice, a security is a piece of paper (with coupon strips to cash the interest) thatcan change owner (sometimes under some restrictions). It is therefore useful to splitc0 = ((0,−C), c) into the inflows c and their purchase price at time 0. The intrinsicmodel for this security is the compound interest model with constant rate i = j. Thetrading price of the security is then DV alt(c), the discounted value of all post-t flows. Atany integer time t after the interest payment this value is C whereas the value increasesexponentially between integer times. Note that C is the fair price at time 0 in this modelbecause V al0(c0) = 0, where we recall V al = AV al + DV al.
4.2 Securities above/below/at par
More generally, one can consider securities with coupon payments at rate j different fromthe not necessarily constant interest rate of the market, or with rates increasing fromyear to year. In this case, the initial capital C and the repayment sum R do not coincide.
19
20 Lecture 4: Fixed-interest securities and Annuities-certain
And also interest may be paid on a third, so-called nominal amount N . In any givenmodel a security
c = ((1, Nj1), (2, Nj2), . . . , (n − 1, Njn−1), (n, R + Njn))
is then bought at DV al0(c), which for constant interest rates and constant coupon pay-ments jk = j, k = 1, . . . , n, is
DV al0(c) = jN∑
k=1n
vk + Rvn = jNv1 − vn
1 − v+ Rvn.
If DV al0(c) < N , we say that the security is below par or at a discount. If DV al0(c) > N ,we say that the security is above par or at a premium. If DV al0(c) = N , we say that thesecurity is at par.
If the security is not redeemed at par, the redemption price R is stated on the securityas a percentage P = 100R/N of the nominal amount N . Interest payments are alwayscalculated from the nominal amount. Redemption at par is the standard.
4.3 pthly paid interest
Another generalisation is to increase the frequency of interest payments. This is a veryimportant feature since virtually all British securities have semi-annual coupon paymentswhereas it is natural to work with annual unit time.
Before valuing securities, we introduce the notion of nominal interest rates that isused whenever interest is paid more than once per time unit. There is also an analoguefor discount rates.
Example 18 If a bank offers 8% interest per annum convertible quarterly, then it oftenmeans that it pays 2% interest per quarter. We check that an initial capital of £10000.00increases via £10200.00, £10404.00 and £10612.08 to £10824.32 in one year. We havecalled this an effective interest rate of 8.2432%. To distinguish, we call the rate of 8%given in the beginning, the nominal interest rate convertible quarterly.
The general concept is as follows.
Definition 9 Given an effective interest rate i ∈ IR+ and a frequency of p ∈ IN paymentsper time unit, we call i(p) such that
(
1 +i(p)
p
)p
= 1 + i, i.e. i(p) = p(
(1 + i)1/p − 1)
the nominal interest rate convertible pthly.
o13 Lecture Notes: Introduction to Actuarial Science 21
We can also see i(p) as the total amount of interest payable in equal instalments at theend of each pth subinterval. This formulation should be taken with care, however, sincewe add up payments made at different times, and their time values are not the same.Also, interest payments should not be mixed up with jumps in value for an investmentsince our models specify continuous increase in value by interest, and in fact the right oninterest is accumulated continuously just with the payment made later (in arrear).
For p → ∞ we obtain
Proposition 8 limp→∞ i(p) = δ, the force of interest.
Proof: By definition of i(p), this limit takes the derivative of t 7→ (1 + i)t at t = 0. Wehave done this in Definition 4 to introduce the force of interest. 2
4.4 Securities with pthly paid interest
A security of term n and (nominal=redemption) value N that pays interest at rate j(nominal) convertible pthly is the cash flow
c =
((
1
p,j
pN
)
,
(
2
p,j
pN
)
, . . . ,
(
n − 1
p,j
pN
)
,
(
n,j
pN + N
))
.
It can be bought at time 0 for DV al0(c) which in the compound interest model at constantinterest rate i is
DV al0(c) =
[
vn +j
p
pn∑
k=1
vk/p
]
N =
[
vn +j
pv1/p 1 − vn
1 − v1/p
]
N.
4.5 Annuities-certain
As we saw in Example 8, an annuity-certain of term n provides annual payments of someconstant amount X:
c = ((1, X), (2, X), . . . , (n − 1, X), (n, X)).
In the constant rate compound interest model the issue price can be given by
DV al0(c) = X
n∑
k=1
vk = Xv1 − vn
1 − v= X
1 − vn
i=: X an|
where the last symbol, or more precisely an|i to mention the interest rate, is read ’aangle n (at i)’. This is the first example of the peculiar actuarial notation that has beendevelopped over centuries.
Obviously, this formula also allows to calculate the payment amount X from a givencapital to be invested at time 0.
Actuaries also use the following notation for the accumulated value at time n
AV aln(c) = v−nDV al0(c) = X(1 + i)n − 1
i=: X sn|.
22 Lecture 4: Fixed-interest securities and Annuities-certain
4.6 Perpetuities
Perpetuities are annuities providing payments in perpetuity, i.e.
c = ((1, X), (2, X), . . . , (n, X), . . .).
We have
DV al0(c) = X∞∑
k=1
vk = X1
i=: X a∞|.
Note that the value of a perpetuity at integer times remains constant. It can thereforealso be seen as a fixed-interest security with an infinite term.
4.7 Annuities and perpetuities payable pthly and con-
tinuously
Also annuities and perpetuities with a higher frequency of interest payments may beconsidered. The actuarial symbols are
a(p)n|
=1
p
pn∑
k=1
vk/p =v
p
1 − vn
1 − v1/p=
1 − vn
i(p)
for the present value of payments of 1/p p times a year over a term n,
s(p)n|
= v−na(p)n|
=(1 + i)n − 1
i(p)
for the accumulated value at time n and
a(p)∞|
=1
p
∞∑
k=1
vk/p =v
p
1
1 − v1/p=
1
i(p)
for the corresponding perpetuity.If we pass to the limit of p to infinity, we obtain continuously payable annuities and
perpetuities, and formulas
an| =
∫ n
0
vtdt =vn − 1
log(v)=
1 − vn
δ
sn| = v−nan| =1 − vn
vnδ
a∞| =1
δ.
Note the similarity of all these expressions for ordinary, pthly payable and continuouslypayable annuities, and one could add more. They only differ in the appropriate interestrate i, i(p) or δ. This can be understood by an equivalence principle for interest payments:the payment of i at time 1 is equivalent (has the same value) to p equally spread paymentsof i(p)/p or an equally spread continuous payment at rate δ.
Lecture 5
Mortgages and loans
As we indicated in the Introduction, interest-only and repayment loans are the formalinverse cash flows of securities and annuities. Therefore, most of the last lecture can bereinterpreted for loans. We shall here only translate the most essential formulae and thenpass to specific questions and features arising in loans and mortgages, e.g. calculationsof outstanding capital, proportions of interest/repayment, discount periods and APR.
5.1 Loan repayment schemes
A repayment scheme for a loan of amount L at force of interest δ(·) is a cash flow
c = ((t1, X1), (t2, X2), . . . , (tn, Xn))
such that
L = DV al0(c) =n∑
k=1
v(tn)Xn. (1)
Condition (1) ensures that, in the model given by δ(·), the loan is repaid after the nthpayment since it means that the cash flow ((0,−L), c) has zero value at time 0, and thenby Corollary 2 at all times.
Example 19 A bank lends you £1000 at an effective interest rate of 8% p.a. initially,but due to rise to 9% after the first year. You repay £400 both after the first and half waythrough the second year and wish to repay the rest after the second year. The first twopayments are worth 400(1.08)−1 = 370.37 and 400(1.09)−1/2(1.08)−1 = 354.75 at time 0,hence the final payment 274.88 = 323.59(1.09)−1(1.08)−1, £323.59 after two years.
Often, the times tk are regularly spaced and passing to the appropriate unit time, wecan assume tk = k. Often, the interest rate is constant i say and the payments are levelpayments X. This is an inverse ordinary annuity and
L = X an|
allows to calculate X from L, n and i.
23
24 Lecture 5: Mortgages and loans
Let us return to the general case. In our example, we compared values at time 0 tocalculate the outstanding debt, an important quantity. In general we have the followingso-called retrospective formula.
Proposition 9 Given a loan (L, δ(·)) and repayments
cm = ((t1, X1), (t2, X2), . . . , (tm, Xm))
up to time t, the outstanding debt is
AV alt((0, L)) − AV alt(cm) = A(0, t)L −m∑
k=1
A(tk, t)Xk =: Lt.
Proof: The equation says that the time-t value of the loan minus the time-t values ofall previous payments is the outstanding loan. Hence, ((0,−L), cm, (t, Lt)) is a zero-valuecash flow as required. 2
Alternatively, for a given repayment scheme (satisfying (1)), one can also use thefollowing prospective formula.
Proposition 10 Given a loan (L, δ(·)) and a repayment scheme c, the outstanding debtat time t ∈ [tj, tj+1) is
Lt = DV alt(c) =n∑
k=j+1
V (t, tk)Xk. (2)
Proof: By assumption, ((0,−L), c) is a zero-value cash flow. If we call the right handside of (2) R, then ((t,−R), c(t,∞)) is a zero-value cash flow, and so is their difference.Hence R repays the loan at time t, Lt = R. 2
It is important to have both a mathematical understanding of the model that allowsto do explicit calculations, and a practical understanding to argue ’by general reasoning’.The two preceding proofs are more of the latter style, although still rigorous. A moremathematical proof can be given by induction over the number of payments.
Corollary 3 The jth payment of a loan repayment schedule c consists of Rj = Ltj−1−Ltj
capital repayment and Ij = Xj − Rj = Ltj−1(A(tj−1, tj) − 1) interest payment.
5.2 Equivalent cash flows and equivalent models
In practice, the embedded discrete time model is more important than our more generalcontinuous model. Often, cash flows can be simplified onto a discrete lattice. One methodto achieve discrete time models from continuous time models is to simplify cash flows bymoving them onto a lattice, i.e. replacing them by suitable cash flows that only containtransaction on a time grid.
Definition 10 Given a model δ(·), two cash flows c1 and c2 are called equivalent in δ(·)if V alt(c1) = V alt(c2) for one (all) t ∈ IR+.
o13 Lecture Notes: Introduction to Actuarial Science 25
In this sense, all repayment schemes of a loan in a given model are equivalent. Also, ina constant interest rate model, pthly interest payments at nominal rate i(p) are equivalentfor all p ∈ IN, and they are equivalent to continuous interest payments at rate δ. Notethat the equivalence of two cash flows depends on the model. In fact, two cash flows thatare equivalent in all models, are the same.
Proposition 11 a(p)n|
= ii(p) an|
Proof: By the definition of i(p), pthly level payments of i(p)/p are equivalent to paymentsi per time unit, that is pthly level payments of 1/p are equivalent to payments i/i(p) per
unit time. Extended over n time units, they define a(p)n|
and (i/i(p))an| respectively. 2
This gives an alternative approach to pthly payable annuities.
Definition 11 Given a cash flow c, two models δ1(·) and δ2(·) are called equivalent forc if δ1 − V alt(c) = δ2 − V alt(c) for all t ∈ supp(c), where supp(c) := {0, t1, t2, . . . , tn} fora discrete cash flow c = ((t1, C1), (t2, C2), . . . , (tn, Cn)).
Note that the equivalence of models depends on the cash flow. Two models that areequivalent for all cash flows coincide.
Proposition 12 Given a discrete cash flow c and a model δ1(·), a model δ2(·) is equiv-alent for c if and only if for all j = 0, . . . , n − 1
∫ tj+1
tj
δ1(s)ds =
∫ tj+1
tj
δ2(s)ds.
In particular, there is always a piecewise constant model δ3 equivalent to δ1 for c.
Proof: For a discrete cash flow the value of c at time tj is given by
V altj (c) =1
v(tj)
n∑
k=1
v(tk)ck.
Provided, this value coincides for j = 0, equality for all other j enforces v1(tj) = v2(tj)for all j = 1, . . . , n. Vice versa, if v1(tj) = v2(tj) for all j = 1, . . . , n, then values coincidefor all j = 0, . . . , n. Now by definition
v1(tj) = exp
{
−∫ tj
0
δ1(s)ds
}
= v2(tj)
for all j = 1, . . . , n if and only if the integrals of δ1 and δ2 from 0 to tj coincide for allj = 1, . . . , n if and only if the integrals from tj−1 to tj coincide for all j = 1, . . . , n.
For the second assertion just define for j = 1, . . . , n
δ3(s) =1
tj+1 − tj
∫ tj+1
tj
δ1(t)dt, tj ≤ t < tj+1.
2
26 Lecture 5: Mortgages and loans
Proposition 13 Piecewise constant models δ(·) can be represented by ij = eδj − 1 via
v(t) = (1 + i0)−t1(1 + i1)
−(t2−t1) . . . (1 + ij)−(t−tj ), tj ≤ t < tj+1
if δ(t) = δj for all tj < t < tj+1, j ∈ IN.
Proof: By definition of v(t)
v(t) = exp
{
−∫ t
0
δ(s)ds
}
= exp
{
−j−1∑
k=0
(tk+1 − tk)δk − (t − tj)δj
}
which transforms to what we need via 1 + ij = eδj . 2
Sometimes, evaluating a cash flow c in a model δ(·) can be conveniently carried outpassing first to an equivalent cash flow and then to an equivalent model.
5.3 Fixed, discount, tracker and capped mortgages
In practice, the interest rate of a mortgage is rarely fixed for the whole term and thelender has some freedom to change their Standard Variable Rate (SVR). Usually changesare made in accordance with changes of the UK base rate fixed by the Bank of England.However, to attract customers, an initial period has often some special features.
Example 20 (Fixed period) For an initial 2-10 years, the interest rate is fixed, usuallybelow the current SVR, the shorter the period, the lower the rate.
Example 21 (Capped period) For an initial 2-5 years, the interest rate can fall par-allel to the base rate or the SVR, but cannot rise above the initial level.
Example 22 (Discount period) For an initial 2-5 years, a certain discount on theSVR is given. This discount may change according to a prescribed schedule.
Example 23 (Tracking period) For an initial or the whole period, the interest ratemoves parallel to the UK or another base rate rather than following the lender’s SVR.
Whichever special features there may be, the monthly payments are always calculatedas if the current rate was valid for the whole term. Therefore, even if the rate is knownto change after an initial period, no level payments are calculated. The effect is that,e.g. a discount period leads to lower initial payments. With every change in interest rate(whether known in advance or reacting on changes in the base rate) leads to changesin the monthly payments. In some cases, there may be the option to keep the originalamount and extend the term.
Initial advantages in interest rates are usually combined with early redemption penal-ties that may or may not extend beyond the initial period. A typical penalty is 6 monthsof interest on the amount redeemed early.
Lecture 6
An introduction to yields
Given a cash flow representing an investment, its yield is the constant interest rate thatmakes the cash flow a fair deal. Yields allow to assess and compare the performance ofpossibly quite different investment opportunities as well as mortgages and loans.
6.1 Flat rates and APR
To compare different mortgages with different features, two common methods should bementioned, a bad one and a better one. The bad method is the so-called flat rate whichis the total interest per year of the loan per unit of initial loan, i.e.
F =
∑nj=1 Ij
tnL=
∑nj=1 Xj − L
tnL
where usually Xj = X, tn = n. This method is bad because it does not take into accountthat as time evolves, interest is paid only on the outstanding loan. One consequence isthat loans with different term but same interest rates can have very different flat rates.
The better method is to give the Annual Percentage Rate of Charge (APR). In caseof a fixed (effective) interest rate i, this is just i. If the interest rate varies, APR isthe constant rate under which the schedule exactly repays the loan, rounded to the nextlower 0.1%. We call this the yield and subject of the next section. Note that the constantinterest rate model is not equivalent for the schedule, in general, since we do not andcannot expect that values coincide at all payment dates.
Example 24 Given a mortgage of amount L over a term of 25 years with a discountperiod of 5 years at 3%, after which the SVR of currently 6% is payable. Although thisis not common practice, let us assume that level payments c = ((1, X), . . . , (25, X)) aremade over the whole term. Then
L = DV al0(c) = Xa5|3% + X(1.03)−5a20|6%
determines X, and the APR is essentially i such that
L = i-DV al0(c) = Xa25|i.
This cannot be solved algebraically, but numerically we obtain i ≈ 4.737%, hence APR =4.7%
27
28 Lecture 6: An introduction to yields
6.2 The yield of a cash flow
Suppose, we are offered a deal that involves transactions according to a cash flow c. Sofar we have learnt how to calculate the present value in a given model. This tells whetherthe deal is profitable (V al0(c) > 0) or not. However, in practice there does not exist atrue known model. Whoever offers us the deal, may have a model δ1 in mind that isdifferent from our model δ0 (because of different expectations of interest rate evolution,to mention the simplest influence). One can introduce a concept of utility to explain thecoexistence of different models (e.g. you may feel that the first £10,000 are worth morethan the second £10,000), but we shall not follow this route here. The essential questionremains: do we believe in one model δ0? Maybe it is prudent to compare different models.
Given a cash flow c, it is rather hopeless to look at the family of values for all time-dependent forces of interest δ(·), but focussing on constant interest rate models, i ∈ IR+,gives us a nice family of values that we denote ai = i-V al0(c). We can draw this as afunction of i to show us under what interest rate assumptions the deal c is profitable.
Before we pass to any definite statements, one remark on the class of cash flows thatwe look at. In that follows, it does not make much difference whether c is discrete,continuous or mixed, whether c is finite (i.e. has a last in- or outflow) or infinite (like e.g.perpetuities). Only, to reasonably include the infinite case, we assume in the sequel thatthe cash flows are finite values cash flows.
Proposition 14 Given a cash flow c the function i 7→ ai = i-V al0(c) is continuous on(−1,∞).
Proof: In the discrete case c = ((t1, C1), . . . , (tn, Cn))
ai = i-V al0(c) =
n∑
k=1
Ckv(tk) =
n∑
k=1
Ck(1 + i)−tk .
and this is clearly continuous in i for all i > −1. For a continuous-time cash flow c(s),0 ≤ s ≤ t we use the uniform continuity of i 7→ (1 + i)−s on compact intervals s ∈ [0, t]for continuity to be maintained after integration
ai = i-V al0(c) =
∫ t
0
c(s)v(s)ds =
∫ t
0
c(s)(1 + i)−sds.
2
Corollary 4 For any cash flow c and time t the function i 7→ i-V alt(c) is continuous on(−1,∞).
Often the situation is such that a deal is profitable if the interest rate is below a certainlevel, but not above, or vice versa. By the intermediate value theorem, this threshold isa zero of i 7→ ai, and we define
Definition 12 Given a cash flow c, if i 7→ ai = i-V al0(c) has a unique root on (−1,∞),we define the yield y(c) to be this root. If i 7→ ai does not have a root in (−1,∞) or hasmore than one root, we do not define the yield of this cash flow.
o13 Lecture Notes: Introduction to Actuarial Science 29
So far, we implicitly or explicitly assumed that i ≥ 0, both to ease intuition and tostreamline some minor technical issues. Particularly in this setting of yields, we shouldallow for the case that money invested in certain projects can result in a loss, whichamounts to a negative interest rate. All formulas remain the same, provided i > −1.In fact, i = −1 creates singularities in the discount factors because it corresponds to acomplete loss of money, so that the original value is no longer a fraction of the later value(zero).
The concept is useful only because of the continuity of i 7→ ai. If this function wasnot continuous, profitability could change without root.
The yield can be interpreted as the fixed interest rate under which the deal c is fair.We see in the next section that the yield exists for the majority of practical situations.The yield is also called the internal rate of return or the money-weighted rate of return.For the assessment of funds there are other weighted rates of return that we discuss in alater lecture.
Example 25 Suppose that for an initial investment of £1000 you obtain a payment of£400 after one year and 770 after two years. What is the yield of this deal? By definition,we are looking for zeros i ∈ (−1,∞) of
ai = −1000 + 400(1 + i)−1 + 800(1 + i)−2 = 0
⇐⇒ 1000(i + 1)2 − 400(i + 1) − 770 = 0
The solutions to this quadratic equation are i1 = −1.7 and i2 = 0.1. Since only thesecond zero lies in (−1,∞), the yield is y(c) = 0.1, i.e. 10%.
6.3 General results ensuring the existence of the yield
Proposition 15 If c has in- and outflows and all inflows of c precede all outflows of c(or vice versa), then the yield y(c) exists.
Proof: By assumption, there is (at least one) T ∈ IR+ such that all inflows are in [0, T )and all outflows in (T,∞). Then
pi = i-AV alT (c)
is positive strictly increasing in i with p−1 = 0 and p∞ = ∞ (by assumption there areinflows) and
ni = i-DV alt(c)
is negative strictly increasing with n−1 = −∞ (by assumption there are outflows) andn∞ = 0. Therefore
bi = pi + ni = i-V alT (c)
is strictly increasing from −∞ to ∞, continuous by Corollary 4, denote the unique rootby i0. Then i0 is also the unique root of ai = i-V al0(c) = (1+i)−T [i-V alT (c)] by Corollary2. 2
30 Lecture 6: An introduction to yields
This result applies to all typical capital investment and borrowing situations.
Corollary 5 In the situation of the Proposition 15 with inflows preceding outflowsy(c) > 0 if and only if 0V al0(c) < 0. If outflows precede inflows y(c) > 0 if and only if0-V al0(c) > 0.
Proof: In the first setting assume y(c) > 0, then by the monotonicity of i 7→ bi we haveb0 < 0 and therefore 0-V al0(c) = a0 = (1 + i)−T b0 < 0.
If conversely a0 = 0-V al0(c) < 0 then also b0 < 0 and by the intermediate valuetheorem with b0 < 0 and b∞ = ∞ the root lies between 0 and ∞.
The second setting is analogous with the obvious changes in signs. 2
Proposition 16 If t 7→ 0-AV alt(c) (corresponds to zero-interest!) changes sign preciselyonce, then there is a unique positive root to i 7→ ai.
Proof: Let T be the time of the sign change (any choice if it happens around an intervalof zeros). W.l.o.g. this sign change is from plus to minus. Then 0-V al0(c) < 0, henceb0 = 0-V alT (c) < 0. Now see what happens for
bi = i-V alT (c)
when i tends to infinity. We split again into past and future:
pi = i-AV alT−(c)
is positive and increases strictly to infinity as i tends to infinity (more and more intereston a changing but positive balance).
ni = i-DV alT−(c)
is negative and increases strictly to zero (more and more discount on a changing butnegative balance). Therefore bi strictly increases to infinity as i tends to infinity. Sinceb0 < 0, there must be a zero of i 7→ bi in (0,∞). 2
It is often useful to refer to this unique positive root as the yield even if the existenceof negative roots cannot be excluded. A practical example is, if a mortgage balance isincreased during the term (e.g. to finance major refurbishing). The proposition doesnot apply in this general situation, but often it does. As a slight modification one mightreplace the zero interest lower bound by the minimal interest rate payable throughoutthe term. A similar result holds and establishes a yield for this situation:
Corollary 6 If (under interest rate i0) t 7→ i0-AV alt(c) changes sign precisely once, theni 7→ ai has a unique root i ≥ i0.
Proof: To adapt the proof of Proposition 16, replace 0-V al·(c) by i0-V al·(c) throughout.2
In the next lecture we shall see some more applications of yield calculations.
Lecture 7
Project appraisal
Last lecture we introduced the yield of cash flows and indicated how it can be used toassess investment projects. We make this more precise here and discuss methods that areused in practice for specific types of projects. These include calculating payback periodsfor business projects and weighted rates of return for investment funds.
7.1 A remark on numerically calculating the yield
Let c be a cash flow. Suppose Proposition 15 guarantees the existence of the yield y(c).Remember, this means that f(i) = i-V al0(c) is continuous, strictly monotone and takesvalues of different signs at the boundaries of (−1,∞).
Interval splitting allows to trace the root of f : (l0, r0) = (−1,∞), make successiveguesses in ∈ (ln, rn), calculate f(in) and define
(ln+1, rn+1) := (in, rn) or (ln+1, rn+1) = (ln, in)
such that the values at the boundaries f(ln+1) and f(rn+1) are still of different signs.
The challenge is to make good guesses. Bisection
in = (ln + rn)/2
(once rn < ∞) is the ad hoc way, linear interpolation
in =rnf(rn) − lnf(ln)
f(rn) − f(ln)
an efficient improvement. There are more efficient variations of this method using somekind of convexity property of f , but we don’t go beyond these indications here.
31
32 Lecture 7: Project appraisal
7.2 Comparison of investment projects
First, since the notation i-V al0(c) is getting a bit heavy, we introduce new (actuallymore standard) notation. For a cash flow c, or cash flows cA and cB representing twoinvestment projects
NPV (i) := i-V al0(c), NPVA(i) := i-V al0(cA) and NPVB(i) := i-V al0(cB).
These quantities are called the Net Present Values of the underlying cash flows at interestrate i. ’Net’ may seem a bit odd in our context, but refers to the fact that both in- andoutflows have been incorporated. We have done this from the very beginning.
Now to compare projects A and B that satisfy the conditions of Proposition 15 say,one can calculate their yields yA and yB. Suppose you have first the outflows and thenthe inflows, then each project is profitable if its yield exceeds the market interest rate.In particular if the market interest rate is in (yA, yB), say, project B is profitable, projectA is not. But this does not mean that project B is more profitable than project A (i.e.NPVB(i) > NPVA(i) for all lower interest rates as the following figure shows.
-market interest rate i
6Net Present Value
iX yB yA
NPVB(i)
NPVA(i)
iX is called a cross-over rate and can be calculated as the yield of cA − cB if it is unique.For interest rates below iX , project B is more profitable than project A, although itsyield is smaller. A decision for one or the other project (or against both) now clearlydepends on the expectations on interest rate changes.
7.3 Investment projects and payback periods
For a profitable investment project in a given interest rate model δ(·) we can study theevolution of the accumulated value. With outflows preceding inflows, this accumulatedvalue will first be negative, but tend to the positive terminal value (which is positivesince the project is profitable). Of interest is the time when it becomes first positive.
Definition 13 Given a model δ(·) and a profitable cash flow c with outflows precedinginflows. We define the discounted payback period
T+ = inf{t ≥ 0 : AV alt(c) ≥ 0}.
o13 Lecture Notes: Introduction to Actuarial Science 33
-time t
6balance AV alt(c)
T+
If rather than investing existing capital, you finance the investment project by takingout loans using any inflows for repayment, T+ is the time when your account balancechanges from negative to positive. Therefore, you have repaid your debt and all remaininginflows are simply profit.
Obviously, before time T+ you pay interest, after time T+ you receive interest. If wenow specify a model for an account, say, to keep it simple, that has different forces ofinterest δ− when the balance is negative and δ+ when the balance is positive, then thedefinition of T+ only depends on δ− and the profit can be calculated from δ−-AV alT+(c)and δ+-DV alT+(c).
7.4 Funds and weighted rates of return
Funds are pools of money into which people pay for different reasons and different benefits.They may be simply investment opportunities. In this case they usually clearly state whatproportions they invest into securities and/or equities of certain countries or branchesof the economy with some freedom remaining. Alternatively they may consist of thereserves of pension schemes or contain foundation capital. In any of the cases there is afund manager who adapts the investments to current market situations and releases anymoney withdrawn from the fund.
An important issue is to assess the performance of a fund, for instance to check thatthe fund manager does a good job. Also, in the case of an investment fund, investorsneed to know the exact value of their capital invested in the fund.
Definition 14 Let F0 be the initial amount of a fund, c the cash flow describing its in-and outflows between times 0 and T , FT the amount at time T . The money weightedrate of return of the fund between times 0 and T is defined to be the yield y(c0T ) of thecash flow
((0, F0), c, (T,−FT )).
But this is not the yield for the investor:
Example 26 Suppose there is a single investor who invests £100 into a fund at time 0and receives £150 at time 4. His yield is easily calculated from the equation of value
100 − 150(1 + y1)−4 = 0 ⇒ y1 ≈ 10.668%.
34 Lecture 7: Project appraisal
Suppose, in between a second investor invested £100 at time 1 when the value of theinvestment of the first investor so happened not to have changed. At time 2 however, thefund had lost 25%, and the second investor decides to withdraw his share at time 3 whenagain, the value of the investment did not change again. He obtained therefore £75. Asnoted above, the value of the first investor’s investment rises to £150 by time 4 which isan increase of 100% since time 3. Now, we solve the yield equation
100 + 100(1 + i)−1 − 75(1 + i)−3 − 150(1 + i)−4 = 0
to obtain (numerically) a yield of y1+2 ≈ 3.789%.This value is much lower since it takes into account the loss that the second investor
experienced.
By the way, to assess the fund manager, the money weighted rate of return is notquite fair either since he has no influence on the in- and outflows. If in- and outflowsform a discrete cash flow, we can calculate the yield between any two successive flows.In these periods, the evolution reflects the skills of the fund manager.
Definition 15 Given a fund of amount Ft at time t ∈ [0, T ] with in- and outflowsaccording to c = ((t1, C1), . . . , (tn, Cn)), we define the time weighted rate of return to bethe value i ∈ (−1,∞) such that
(1 + i)T =Ft1−
F0+
Ft2−
Ft1+. . .
Ftn−
Ftn−1+
FT−
Ftn+.
Proposition 17 The time weighted rate of return is the yield between 0 and T for aninvestor who invested at time 0.
Proof: The statement is true if there is no flow. We now interpret the factors Ftj−/Ftj−1+
as accumulation factors. An initial capital of C in the fund accumulates to
A1 = CFt1−
F0+
by time t1. The fact that other investors pay into or withdraw from the fund, does notchange our investor’s accumulated value A1. We can therefore just repeat the argumentand multiply by the next ratio to calculate the accumulated value A2 at time t2, and soon, until tn and eventually T where we get an accumulated value of
S = CFt1−
F0+
Ft2−
Ft1+. . .
Ftn−
Ftn−1+
FT−
Ftn+.
Now the yield of this investment cT = ((0,−C), (T, S)) is defined as i such that
0 = NPV (i) = −C + (1 + i)−T = −C + (1 + i)−T CFt1−
F0+
Ft2−
Ft1+. . .
Ftn−
Ftn−1+
FT−
Ftn+
and this is the equation for the time weighted rate of return. 2
The time weighted rate of return has its name since it is an average of yields betweenflows that takes into account the lengths of these time periods. The money weighted rateof return gives more weight to times when the fund amount is big. So far, we have onlypointed out where the money weighted rate of return is not suitable. We should stressthat it does reflect the (money weighted, as it should be) average yield of investors.
Lecture 8
Taxation and inflation
In practice, taxation causes complications. From our model’s point of view this is notcrucial. The changes are mainly to interest rates and amendments at the end of calcula-tions. Inflation is more fundamental. We have to extend our model to include a properrepresentation. In this lecture we only discuss inflation indices and their use. The moreformal model extension is subject of the next lecture.
8.1 Fixed interest securities and running yields
We discussed in Lecture 4 the pricing of fixed interest securities given a model. Clearlythese formulas can be used as yield equations to determine the yield given the price.
In the context of securities, the use precise terminology is important, and the followingis to be noted.
Definition 16 Given a security, the yield y(c) of the underlying cash flow
c = ((0,−NP0), (1, ND), . . . , (n − 1, ND), (n, ND + R))
is called the yield to redemption.
If the security is traded for Pk per unit nominal at time k, then the ratio D/Pk ofdividend (coupon) rate and price per unit nominal is called the running yield of thesecurity at time k.
For equities the running yield is the analogue with dividend instead of coupon rates.
The price Pk determines the current capital value of the security, and the runningyield then expresses the rate at which interest is paid on the capital value.
This distinction of yield to redemption and running yield is related to the notions ofinterest income and capital gains relevant for taxation. Coupon and dividend paymentsare considered income, whereas any profit due to different purchase and redemption pricesis considered capital gain. The yield to redemption takes into account both income andcapital gains (or losses), whereas the running yield only contains the income part.
35
36 Lecture 8: Taxation and inflation
Example 27 Given a typical 6% security (payable semi-annually) with redemption datethree years from now that is currently traded above par at 105%. We calculate therunning yield as 6/105 ≈ 5.7%. The yield to redemption is the solution of
0 = i-V al0((0,−105), (0.5, 3), (1, 3), (1.5, 3), (2, 3), (2.5, 3), (3, 103))
= −105 + 6a(2)
3|i+ (1 + i)−3100
and numerically, we calculate a yield to redemption of ≈ 4.3%. Clearly, the difference isdue to the capital loss.
8.2 Income tax and capital gains tax
Example 28 The holder of a savings account at 2.5% gross interest is usually liable toincome tax on savings at a rate of t1 = 20%, reducing his interest rate to 2% net.
Example 29 An investor who buys equities for C and sell for S > C within a year, pays40% capital gains tax of S − C.
The taxation legislation is complex and not subject of this course. We always placeourselves in situations where we assume to be given tax rates and whether or not aninvestor and his investment are liable to these taxes. Nevertheless, we need to distinguishthe two taxes in certain situations, the most important and straightforward case is fixed-interest securities.
In general terms, income tax is applied to interest income that is typically payableregularly. Capital gains tax is payable on sale or redemption of equities, securities or, inprinciple any other financial product. The difference between sales and purchase priceis called the capital gain , for obvious reasons. If it is positive, tax is applied. If it isnegative, no tax is payable. Under certain restrictions, one may offset capital losses Lagainst other taxable capital gains G so as to pay tax only on G − L if G > L.
Example 30 If the holder of a fixed-interest security is liable to income tax at ratet1 and capital gains tax at rate t2, in principle, and if the fixed-interest security is notexempt from any of these taxes, then the liabilities are as follows.
Then income tax is applied to the coupon payments of ND at times k = 1, . . . , n, atrate D on the nominal value N , reducing the payments to ND(1 − t1).
If held for the whole term, then the difference of redemption price R and purchaseprice A is subject to capital gains tax, reducing the redemption proceeds to R−t2(R−A)+.
If not held for the whole term but sold at time k for Pk per unit nominal, capitalgains tax reduces the sales proceeds PkN to PkN − t2(PkN − A)+.
It can be argued that capital gains tax on (R − A)+ is not quite fair, particularlyif purchase and sale are far apart, because of inflation. In fact, there is an adjustmentallowing for inflation.
o13 Lecture Notes: Introduction to Actuarial Science 37
8.3 Inflation indices
Inflation means goods are getting ”more expensive”. An inflation index simply recordsthese prices for a particular good or a basket of goods. The most commonly used index isthe Retail Price Index (RPI). Its basket contains virtually everything, from different kindsof bread over salaries and houses to electricity and gas, whatever an average Englishmanis likely to spend, weighted in a way to express statistical relevance.
Year 1996 1997 1998 1999 2000 2001 2002Retail Price Index in January 150.2 154.4 159.5 163.4 166.6 171.1 173.3Annual Inflation rate 2.8% 3.3% 2.4% 2.0% 2.7% 1.3%
where the inflation rates are calculated e.g. as e2001 = 1 + 1.3% = 173.3/171.3. Wewill see later how these inflation rates can be incorporated into our interest models.Theoretically, ek is the interest rate you earn by buying the basket at time k and sell itat time k + 1. (In practice, many goods in the basket don’t allow this.)
Although we will probably exclusively work with the RPI, it should be noted thatthere are other important indices that are worth mentioning. Also, prices for any specificgood are likely to behave completely different from the RPI. When buying a house, youmay wish to consult the House Price Index (current inflation rates are around 20% andbuyers fear that house prices start sinking - deflation, negative inflation rates). As apensioner you have different needs and there is an inflation index that takes this intoaccount (no salaries, no mortgage rates, more weight on medical expenses etc.).
We use the RPI in the context of investments where the primary aim is to accumulatemoney, and it is important to take account of a general decrease in value due to inflation.Here the RPI is a suitable index, since it reflects all expenses of the whole society.
Example 31 In January 2000 an investor put £1000 in a savings account at a (net)effective rate of 2% interest. His balance in January 2001 was £1020. The RPI tellsus that the basket that costed £2000166.60 in 2000 costed £2001171.10 in 2001. So the£20011020 are effectively only
£20001020 ∗ 166.60/171.10 = £200010201
1 + e2000
= £2000993.17
and we say that the real effective interest rate was −0.7%.
Like in physics, it helps to state explicitly the units. Therefore we introduced £2000
as the appropriate unit for value in 2000.The inflation rate e2000 had an informal character in this section. We shall properly
define and motivate it in the next section.For practical reasons, inflation indices only give monthly index values Q(m/12), and if
ever this time scale is not fine enough, one uses interpolation. Clearly, a constant rate ofinflation in the vague sense above corresponds to an exponentially growing index functionQ. Therefore, it is natural, even for non-constant rates, to use exponential interpolation,i.e. linear interpolation on ln(Q(·)).
Only the ratio of an index function at two different times enters inflation calculations.Index functions Q can therefore be standardized so as to be 100 on a specified day.
38 Lecture 8: Taxation and inflation
Example 32 If capital gains tax allows for inflation, then buying a financial product(equities, say) for £1998100 in January 1998 (RPI=159.5) and selling it for £2002180 inJanuary 2002 (RPI=173.3) entails tax deductions of
t2(180 − 108.65) since £1998100 = £1998100£2002173.3
£1998159.5= £2002108.65.
Lecture 9
Inflation models and real interest
In the last lecture we introduced inflation indices and showed how they can be used totake account of varying purchasing power of money when calculating future values ofinvestments. We shall here present a more systematic study of inflation and describe ageneral model.
9.1 Modelling inflation
Definition 17 Given an interest rate model δ(·) and an inflation index Q : [0, T ] →(0,∞), we define the real value of an amount C ∈ IR at time t ∈ [0, T ] as RV alt((t, C)) :=CQ(0)/Q(t). Given a cash flow c, we further define RV alt(c) := RV alt((t, V alt(c))).
The first definition is the formalisation of Example 31. The extension to cash flowscan only be done in this way if we want to keep the equivalence of cash flows, here c and(t, V alt(c)), under RV al. You may prefer writing it as the extension by linearity of
RV alt((r, C)) = RV alt((t, A(r, t)C)) = A(r, t)CQ(0)/Q(t).
Note that we chose 0 as our reference time. Everything is expressed in time 0 moneyunits £0. We could have chosen any other time. In fact, the passage to time s moneyunits £s can be made via the accumulation factors A(0, s). However, the primary purposeof inflation modelling and real value of money is to remove the coexistence of differentunits £s. So, we focus on valuing cash flows in £0 units. We even suppress this fact inour notation: RV alt always evaluates in £0 units for all t.
As an inflation index Q(t) merely represents the accumulated value of a certain basketof goods, it has the same structure as A(0, t) in any interest model. It is natural to imposesimilar regularity conditions and define a force of inflation from Q(t) in the same way asthe notion of the force of interest is related to A(0, t).
Definition 18 Let an inflation index function Q be of the form
Q(t) = exp
{∫ t
0
γ(s)ds
}
for a locally Riemann integrable function γ : [0,∞) → IR. Then we call γ the force ofinflation.
39
40 Lecture 9: Inflation models and real interest
Whereas A(0, s) based only on δ provides the transition to different money units £s
for values taken at all times t ≥ 0, we will now see that A(t, u) based on δ − γ realisesthe transition between values at times t and u when keeping the money unit fixed.
Proposition 18 Given a force of interest δ(·) and a force of inflation γ(·), we have
RV al0(c) = V al0(c), RV alu(c) = RV alt(c)A(t, u)
for all cash flows c and all t ≤ u, where A are the accumulation factors in the δ − γmodel. V al0 is associated with the δ model.
Proof: The first statement follows immediately from Definition 17. For the secondstatement we check
RV alu(c)A(u, t) = V alu(c) exp
{
−∫ u
0
γ(s)ds
}
exp
{∫ t
u
(δ(s) − γ(s)ds
}
= V alt(c) exp
{
−∫ t
0
γ(s)ds
}
= RV alt(c).
2
In the sequel we will call a model given by a force of interest δ and a force of inflationγ a (δ, γ) model. Valuation of cash flows in this model is done via RV alt(c). We have thenatural extensions of all concepts introduced for the δ-model, e.g. equivalence of cashflows: cA ∼ cB if RV alt(cA) = RV alt(cB) for all t ≥ 0.
Corollary 7 Two cash flows are equivalent for (δ, γ) if and only if they are equivalentfor δ. A cash flow is profitable for (δ, γ) if and only if it is profitable for δ.
In particular, you cannot beat inflation by choosing interest in advance rather thanin arrears. The whole story is just about different units: time t currencies with exchangerates decreasing as t evolves.
Although the profitability of a cash flow c is equivalent for models (δ, γ) for any forceof inflation γ and fixed force of interest δ, this does not mean that given a cash flowc, yield and real yield have the same sign. We discuss this at some length in the nextsection.
On the other hand, the (δ, γ) model is not equivalent to the (δ − γ) model since a(time-0) valuation of any cash flow has to be made in δ, only the evolution of real valuesfollows (δ − γ). The (δ, γ) model is equivalent to a (δ − γ) model with first all in- andoutflows re-expressed in time 0 money units.
Definition 19 For any force of interest γ and any t ∈ [1,∞), we define the inflationrate between t − 1 and t as
et = exp
{∫ t
t−1
γ(s)ds
}
− 1 =Q(t)
Q(t − 1)− 1.
Often, instead of Q(m/12), inflation statistics show em/12. We calculated these fromthe Q values of the RPI in the last lecture.
o13 Lecture Notes: Introduction to Actuarial Science 41
9.2 Constant inflation rate
A constant inflation rate e means (average) prices of goods increased by a rate e withinany one year, i.e. Q(t+1) = (1+e)Q(t) for all t, hence Q(t) = (1+e)t after normalisation.Clearly e is constant in particular if γ is constant and then e = exp{γ} − 1.
Now if γ and e, and δ and i are indeed constant, the accumulation factors A take theform
A(s, t) = exp
{∫ t
s
(δ(r) − γ(r))dr
}
=
(
1 + i
1 + e
)t−s
.
Definition 20 Given a constant (i, e) model, j = (1 + i)/(1 + e)− 1 = (i− e)/(1 + e) iscalled the real interest rate.
Once an inflation rate (or indeed a possibly time-dependent force of inflation, e.g.given by interpolating a historical inflation index) has been fixed, there is a naturalnotion of real yields of cash flows as the real interest rate under which the cash flow haszero value:
Definition 21 Given a (mixed) cash flow c and a constant inflation rate e ∈ IR, the realyield ye(c), if it exists, is the unique root j of
0 =∞∑
k=1
(1 + j)−tk(1 + e)−tkCk +
∫ ∞
0
(1 + j)−t(1 + e)−tc(t)dt,
and under an inflation index Q rather than a constant rate e, of
0 =
∞∑
k=1
(1 + j)−tkQ(0)
Q(tk)Ck +
∫ ∞
0
(1 + j)−t Q(0)
Q(t)c(t)dt.
Under a constant inflation rate e, real yields and yields satisfy the same relationshipas real interest rates and interest rates:
Proposition 19 Let c be a cash flow with yield y(c) and a constant inflation rate e.Then the real yield of c exists and is given by
ye(c) =y(c) − e
1 + e.
Proof: Under interest rate i = y(c), NPV (i) = 0. By Proposition 18 also (i, e)-RV al0(c) = 0. In the (i, e) model the real interest rate is (i − e)/(1 + e) as required.
Conversely, for any real interest rate j ∈ IR we have a unique interest rate i =j(1 + e) + e. If the real value (i, e)-RV al0(c) = 0, then also NPV (i) = 0 and hencei = y(c) and this shows that the given root is unique, hence the real yield indeed exists.
2
42 Lecture 9: Inflation models and real interest
9.3 Inflation adjustments
We return to a time-dependent force of inflation γ, or even a general inflation index Q.If you can achieve a yield y(c) on cash flow c, you can, in principle, make this yield a realyield by adjusting the cash flow to inflation:
Definition 22 Given a discrete cash flow c = ((t1, C1), . . . , (tn, Cn)) and an inflationindex Q, we define the inflation adjustment cQ of c as
cQ = ((t1, CQ1 ), . . . , (tn, CQ
n )) =
((
t1,Q(t1)
Q(0)C1
)
, . . . ,
(
tn,Q(tn)
Q(0)Cn
))
.
Given a continuous cash flow c(t), t ∈ [0,∞), we define its inflation adjustment cQ(t),t ∈ [0,∞) as
cQ(t) =Q(t)
Q(0)c(t).
Proposition 20 If a cash flow c has yield y(c), then the inflation adjusted cash flow cQ
has real yield yQ(cQ) = y(c) under inflation Q.
Proof: We apply Definition 21 of the real yield equation for the constant e case) toexpress yQ(cQ) as the unique root j of
0 =
∞∑
k=1
(1 + j)−tkQ(0)
Q(tk)CQ
k +
∫ ∞
0
(1 + j)−t Q(0)
Q(t)cQ(t)dt
and we see the inflation rate cancel out leaving 0 = NPV (j) whose root is the yield y(c).2
Example 33 An index-linked security is the inflation adjusted cash flow cQ derived froma fixed-interest security c, i.e.
c =
((
1
2,D
2
)
, . . . ,
(
2n − 1
2,D
2
)
,
(
n,D
2+ R
))
⇒ cQ =
(
(
1
2,D
2
Q(1/2)
Q(0)
)
, . . . ,
(
2n − 1
2,D
2
Q(
2n−12
)
Q(0)
)
,
(
n,
(
D
2+ R
)
Q(n)
Q(0)
)
)
.
Typically, Q(t) = RPI(t− 8/12), since at the time of the dividend payment, the currentvalue of the RPI is not known. A delay of 8 months may seem long, but the advantageis, that both parties know the amount well in advance and can plan accordingly.
If after the term of the security, the holder wants to know his real yield on the security,he would use RPI rather than Q, so the inflation adjustment does not guarantee a realyield known in advance, but is usually very close to it.
Lecture 10
Uncertain payment and probabilisticmodels
So far, we have assumed all in- and outflows as well as interest rates were known. Thisis rarely the case in practice, and probabilistic models can help to deal with this.
10.1 An example
Example 34 Suppose, you are offered a zero-coupon bond of £100 nominal redeemableat par at time 1. Current market interest rates are 4%, but there is also a 10% risk ofdefault, in which case no redemption payment takes place. What is the fair price?
The present value of the bond is 0 with probability 0.1 and 100(1.04)−1 with prob-ability 0.9. The weighted average A = 0.1 ∗ 0 + 0.9 ∗ 100(1.04)−1 ≈ 86.54 is a sensiblecandidate for the fair price.
10.2 Notation and introduction to probability
We cannot give a full development of required probability theory here, but we shall discusssome of the main concepts by introducing our notation.
In the preceding example, the redemption payment R is to be modelled as a randomvariable that can take the values 100 and 0. The important information about R is itsdistribution, that is given by probabilities pR(0) + pR(100) = 1. This specification allowsto derive the distribution of related random variables like S = (1 + i)−nR.
Definition 23 The distribution of a discrete random variable R is represented by itsset of possible outcomes {rj : j ∈ IN} ⊂ IR (e.g. rj = j) and its probability massfunction (p.m.f.) pR : IR → [0, 1] satisfying pR(r) = 0 for all r 6∈ {rj : j ∈ IN} and∑
j∈IN pR(rj) = 1. We write for A ⊂ IR
P (R ∈ A) =∑
j∈IN:rj∈A
pR(rj).
To define a multivariate discrete random variable (R1, . . . , Rn), we replace IR by IRn inthe above phrases and denote the p.m.f. by p(R1,...,Rn).
43
44 Lecture 10: Uncertain payment and probabilistic models
Finally, we define for functions g : {rj : j ∈ IN} → IR
E(g(R)) =∑
j∈IN
g(rj)pR(rj)
provided the series converges.
Definition 24 The distribution of a continuous random variable R is represented byits Riemann integrable probability density function (p.d.f.) fR : IR → [0,∞) satisfying∫∞
−∞fR(r)dr = 1. We write for A ⊂ IR
P (R ∈ A) =
∫
A
fR(r)dr =
∫ ∞
−∞
1{r∈A}fR(r)dr.
and for functions g : IRn → IR
E(g(R)) =
∫ ∞
−∞
g(r)fR(r)dr
provided the Riemann integrals exist.To define multivariate continuous random variables (R1, . . . , Rn), replace IR by IRn
and integrals by multiple integrals, denote the p.d.f. by f(R1,...,Rn).
Definition 25 For a (discrete or continuous) random variable R we define the distribu-tion function FR and the survival function FR by
FR(t) = P (R ≤ t) = P (R ∈ (−∞, t])
FR(t) = P (R > t) = P (R ∈ (t,∞)) = 1 − FR(t), t ∈ IR.
If they exist, we define the mean µR and the variance σ2R of R as
µR = E(R) and σ2R = V ar(R) = E((R − E(R))2).
σR =√
σ2R is called the standard deviation of R.
Definition 26 (Discrete or continuous) Random variables R1, . . . , Rn are said to beindependent if
P ((R1, . . . , Rn) ∈ A1 × . . . × An) = P (R1 ∈ A1) . . . P (Rn ∈ An) (1)
for all A1, . . . , An ⊂ IR (such that the integrals exist in the continuous case). HereP (Rj ∈ Aj) = P ((R1, . . . , Rj, . . . , Rn) ∈ IR × . . . × Aj × . . . × IR) and we call the p.m.f.pRj
or p.d.f. fRjof Rj the marginal p.m.f. or p.d.f.
Proposition 21 R and S are independent if and only if (1) holds for all Aj = (−∞, tj],tj ∈ IR, j = 1, . . . , n, if and only if
p(R,S)(r, s) = pR(r)pS(s) or f(R,S)(r, s) = fR(r)fS(s)
for all r, s ∈ IR in the respective discrete and continuous cases (for the continuous casewe actually need some continuity assumptions or exceptional sets, that are irrelevant inpractice. In fact p.d.f.’s are only essentially unique).
o13 Lecture Notes: Introduction to Actuarial Science 45
In practice, the dependence structure can always be derived from a small set of inde-pendent random variables, and dependencies only arise when these are transformed.
Example 35 Assume, that the life time T of a light bulb has a geometric distributionwith parameter q ∈ (0, 1), i.e. P (T = n) = (1 − q)qn, n = 0, 1, . . .. Then the randomvariables Bn = 1{T=n}, i.e. Bn = 1 if T = n and Bn = 0 if T 6= n, are Bernoulli variableswith parameter (1 − q)qn. Of course, the Bn are not independent, since e.g.
P (B0 = 1, B1 = 1) = P (T = 0, T = 1) = 0 6= P (B0 = 1)P (B1 = 1).
We recall that µR is the average value of R, also in the sense that
Proposition 22 (Weak law of large numbers, Tchebychev’s inequality)Given asequence of independent, identically distributed random variables (Rj)j≥1 with existingmean µ = µRj
, we have
P
(∣
∣
∣
∣
∣
1
n
n∑
j=1
Rj − µ
∣
∣
∣
∣
∣
> ε
)
→ 0 as n → ∞
for all ε > 0, i.e. P − limn→∞1n
∑nj=1 Rj = µ, where P − lim denotes the limit in
probability.
More precisely, if the variance σ2 = σ2Rj
exists, P
(∣
∣
∣
∣
∣
1
n
n∑
j=1
Rj − µ
∣
∣
∣
∣
∣
> ε
)
≤ σ2
nε2.
σ2R is a measure for the spread of the distribution of R. Its definition σ2
R = E((R −µR)2) can be read as the expected (squared) deviation from the mean. Also the precedingproposition indicates that a high σ2
R means more deviation from the mean.
10.3 Fair premiums and risk under uncertainty
Definition 27 In an interest model δ, the fair premium for a random cash flow (of fixedlength)
C = ((T1, C1), . . . , (Tn, Cn))
(typically of benefits Cj ≥ 0) is the mean value
A = E (DV al0(C)) =
n∑
j=1
E (Cjv(Tj))
where v(t) = exp{−∫ t
0δ(s)ds} is the discount factor at time t.
Proposition 23 If the times of a random cash flow are fixed Tj = tj and only the
amounts Cj are random, the fair premium is A =
n∑
j=1
E(Cj)v(tj) and depends only on
the mean amounts.
46 Lecture 10: Uncertain payment and probabilistic models
Proof: The deterministic vtj can be taken out of the expectation. 2
Proposition 24 If the amounts of a random cash flow are fixed Cj = cj and only the
times Tj are random, the fair premium is A =
n∑
j=1
cjE(v(Tj)) and in the case of constant
δ we have E(v(Tj)) = E(exp{−δTj}) = E(vTj) which is the so-called Laplace transformor (moment) generating function.
Proof: Again, the deterministic cj can be taken out of the expectation. 2
The fair premium is just an average of possible values, i.e. the actual random valueof the cash flow C is higher or lower with positive probability each as soon as DV al0(C)is truly random. In a typical insurance framework when Cj ≥ 0 represents benefits thatthe insurer has to pay us under the policy, we will be charged a premium that is higherthan the fair premium, since the insurer has (expenses that we neglect and) the risk tobear that we want to get rid of by buying the policy, and that we assess now.
Call A+ the higher premium that is to be determined. Clearly, the insurer is concernedabout his loss (DV al0(C) − A+)+, or expected loss E (DV al0(C) − A+)+. In some specialcases this can be evaluated, sometimes the quantity under the expectation is squared (so-called squared loss). A simpler quantity is the probability of loss P (DV al0(C) > A+). Inthe next lecture we shall use Tchebychev’s inequality to indicate that the insurer’s riskof loss gets smaller with an increasing number of policies.
Lecture 11
Corporate bonds and uncertainpayment
Every company has a (usually small) risk of default that should be taken into accountwhen assessing any corporate investments. This can be dealt with in the probabilisticterms of the preceding lecture.
11.1 Uncertain payment
Before discussing pricing issues specific to corporate investments, we introduce someterminology.
Definition 28 An uncertain cash flow C is a sequence (tj, cj, Bj)j≥1 with tj ∈ [0,∞),cj ∈ IR and Bj ∼ B(1, pj) Bernoulli random variables with parameters pj ∈ [0, 1] and/ora random function (c, B) : [0,∞) → IR × {0, 1} such that B(t) ∼ B(1, p(t)). pj and p(t)are interpreted as the probabilities that the corresponding in- or outflow takes place. Wemake no restrictions on the dependence structure of the Bj and B(t).
Restrictions on the dependence structure of the Bernoulli random variables are notnecessary since we shall here only take expectations of linear functionals of these randomvariables, and these do not depend on the dependence structure.
Uncertain cash flows are special cases of generalised cash flows introduced in Lecture1. The randomness only enters via payment/no-payment possibilities for any individualin- and outflow.
Example 36 (Gambling) A stake of £100 on ’Impair’ in Roulette with no repaymentin case of loss and payment of £200 in case of success can be modelled as
((0,−100, 1), (1, 200, B))
with B ∼ B(1, 18/37).
47
48 Lecture 11: Corporate bonds and uncertain payment
Proposition 25 Given a mixed uncertain bounded value cash flow C and an interestmodel δ(·), its mean value (fair premium in the sense of Definition 27) is
E(DV al0(C)) =
∞∑
j=1
pjcjv(tj) +
∫ ∞
0
p(t)c(t)v(t)dt.
Proof: Just note that E(Bj) = pj and E(B(t)) = p(t). Technically, we need somethinglike absolute summability of cj and absolute Riemann integrability of c to exchangeexpectation and summation/integration. This is the case if DV al0(C) is bounded. 2
Definition 29 The expected yield of a mixed uncertain cash flow C is the solution i of
E(NPV (i)) = 0
provided it exists and is unique.
This is the yield of the expected cash flow, not the mean of the random yield, exceptin the following example where the yield equation is linear in i.
Example 37 In Example 36, the mean yield is i such that
−100 +18
37200(1 + i)−1 = 0 ⇒ i = − 1
37≈ −2.7%.
The mean yield is negative: you should not gamble. Note furthermore, that the timeunit is very small (one minute, say), so the effective hourly rate is, for instance
(1 − 0.027)60 − 1 = −80.6%.
It really isn’t worth it.As there are only two possible scenarios (gain or loss), we can also calculate the
(random) yield. With probability 18/37 (gain)
−100 + 200(1 + ig)−1 = 0 ⇒ ig = 1 = 100%
and with probability 19/37 (loss)
−100(1 + il) + 0 = 0 ⇒ il = −1 = −100%.
Strictly speaking, the yield equation (equation of value at time 0) has no solution. Sinceequations of value at all times are equivalent, we chose the equation at time 1, wheni = −1 is the solution. Actually, looking at yield equations at all in this simple exampleis just to give an example on the usage of yield equations. It is obvious that the gain is100% if you win, −100% if you lose.
Proposition 26 If T ∼ Exp(µ) and Bj = 1{T>tj}, B(t) = 1{T>t}, then the mean yieldof (c, p) is e−µ(1 + y(c)) − 1.
o13 Lecture Notes: Introduction to Actuarial Science 49
Proof: The equation for the mean yield i = eδ − 1 is
∞∑
j=1
e−µtjcje−δtj +
∫ ∞
0
e−µtc(t)e−δtdt = 0
and clearly y(c) = eµ+δ − 1 as required. 2
Exponential times are very useful in the theory of stochastic processes. This is becauseof their lack of memory property : given T > t, the conditional law of T − t is stillexponential with the same parameter. This can be interpreted as follows if T is seen asa killing time: at every time t the likelihood of being killed is the same. The parametercan be seen as the killing rate or killing force, or as the arrival rate of a Poisson process.Exponential random variables are the continuous pendant to geometric random variableswhich in turn can be viewed as the time of first success in a sequence of independentidentically distributed Bernoulli trials. This shows the constant rate idea: at any timebefore success, there is the same probability that the trial is successful.
11.2 Pricing of corporate bonds
Let
c = ((1, DN), . . . , (n − 1, DN), (n, DN + N))
be a simple fixed-interest security, but issued by a company. The risk of insolvency ofthe company needs to be added to c: model the insolvency time T as a random variableand define Bk = 1{T>k}. A corporate bond is then the uncertain cash flow C given by
((1, DN, B1), . . . , (n − 1, DN, Bn−1), (n, DN + N, Bn)).
Here we assume implicitly that insolvency entails a complete loss of coupon paymentsand initial capital. In certain situation a refined model may be more adequate.
The main result of this section applies for general cash flows c that are “killed” at acontinuous insolvency time T . We denote the killed cash flow by c[0,T ].
Proposition 27 Let δ(·) be an interest rate model, c a cash flow and T a continuousinsolvency rate, then
E(
δ-DV al0(
c[0,T ]
))
= δ-DV al0(c)
where δ = δ + µ and µ(t) = fT (t)/FT (t).
Proof: First note that
FT (t) = P (T > t) = exp
{
−∫ t
0
µ(s)ds
}
50 Lecture 11: Corporate bonds and uncertain payment
as is easily verified by taking logarithms and differentiation. p(t) = P (T > t) is theparameter of B(t), and pj = P (T > tj) of Bj. Therefore, the formula in Proposition 29takes the form
E(δ-DV al0(C)) =
∞∑
j=1
cje−∫ tj0 µ(s)+δ(s)ds +
∫ ∞
0
c(t)e−∫ t
0 µ(s)+δ(s)dsdt
as required. 2
The important special case is when δ and µ are constant and c is a corporate bondinfluenced by the insolvency time T .
Corollary 8 Given a constant δ model, the fair price for a corporate bond C with insol-vency time T ∼ Exp(µ) is
A = E(δ − DV al0(C)) = δ − DV al0(c)
where δ = δ + µ.
Often, problems arise in a discretised way. Remember that a geometric randomvariable with parameter p can be thought of as the first 0 in a series of Bernoulli 0-1trials with success (1) probability p.
Proposition 28 Let c be a discrete cash flow with tk ∈ IN for all k = 1, . . . , n, T ∼geom(p), i.e. P (T = k) = pk−1(1 − p), k = 1, 2, . . .. Then in the constant i model,
E(
i-V al0(c[0,T )))
= j-V al0(c)
where j = (1 + i − p)/p.
Proof: P (T > k) = pk. Therefore
E(
i-V al0(c[0,T )))
=
n∑
k=1
ptkck(1 + i)−tk .
2
Lecture 12
Uncertain investment projects andrisk
12.1 Pricing of equity shares
The evolution of share prices largely depends on expectations in the future profitability ofthe company’s business. One method of explaining share prices is by discounting futuredividends:
Definition 30 Given a probabilistic (or deterministic) model for the future dividendpayments (Dk)k≥1 of a share and an interest rate model δ(·), the discounted dividendprice of the share is given by
A =∑
k≥1
v(k)E(Dk).
The definition assumes that dividends are paid once per time unit. Generalisationsare straightforward. It is implicitly assumed that the random variables Dk include theno-payment possibility due to insolvency (or other financial restrictions).
If a share is held in perpetuity, this is the fair price of this share.
Example 38 An equity share is expected to pay constant dividend D forever with norisk of insolvency. Interest rates are expected to be constant i. Then the discounteddividend price for the share is
A =∑
k≥1
(1 + i)−kD =D
1 + i
1
1 − (1 + i)−1=
D
i.
51
52 Lecture 12: Uncertain investment projects and risk
12.2 Examples: Comparison of investment projects
Example 39 Suppose you can buy a corporate bond of nominal N = £100, 000, termn = 10 years, coupon rate of 8% payable annually. Suppose both purchase and redemp-tion are at par. Denote the bond by c. If the company has a 1% probability of insolvencyevery year, what is the expected yield of the bond?
The insolvency time T is geometric with parameter p = 99% since it is determinedby independent Bernoulli trials.
The yield of the bond when ignoring the possibility of insolvency is 8%. So, forj = 8%, we have
j-DV al0(c) = 100, 000.
By Proposition 28,
i = p(1 + j) − 1 = 6.92%
gives the interest rate under which the corporate bond has an expected i-V al0(c[0,T ]) of100, 000. This is the yield, since E(NPV (i)) = 0 for c[0,T ].
Example 40 You buy 10 years leasehold of a small office building for £157,000. Therent is at £2,000 per month, but in any month there is a risk of vacancy of 10%. Whatis the yield of the investment? The expected equation of value is
157, 000 = 0.9 × 24, 000 × a(12)
10|i
and the solution is i = 6.93%.
Example 41 Compare the investment projects from the preceding two examples assum-ing a market interest rate of i0 = 6%.
The second example discounted at a market rate of i0 = 6% gives an expected profit(time 0 value) of
−157000 + 0.9 × 24, 000 × a(12)
10|i0= 6303.78
The expected profit of the first example under i0 = 6% is the profit at the ratej0 = (1 + i0 − p)/p of the cash flow without risk of default, hence
−100000 + 8000a10|j0 + 100000(1 + j0)−10 = 6505.69
Hence, although the yield of the second investment was slightly higher, the firstinvestment is more profitable in a market of 6% interest.
o13 Lecture Notes: Introduction to Actuarial Science 53
12.3 Individual risk models
Definition 31 The individual risk model
(Nj, Xj)1≤j≤n
describes a portfolio of n ∈ IN insurance policies (over a given fixed short time period,ignore any effects of interest). For each policy j = 1, . . . , n the number of claims Nj ∈{0, 1} and (if Nj = 1) the amount of the claim Xj ∈ (0,∞) are random variables,independent but not necessarily identically distributed for different j.
We denote by Yj = NjXj the payment under the jth policy and by S = Y1 + . . . + Yn
the aggregate total claim amount of all policies.
12.4 Pooling reduces risk
Assume an individual risk model with n policies. The random total claim amount S mustbe ensured by premium payments A, say, to be determined. Usually, the fair premiumE(S) leaves too much risk to the insurer. E.g. the loss probabilities P (S > A) is usuallytoo high. The following result suggests to set a higher premium to ensure a low lossprobability.
Proposition 29 Given a random variable Y1 with existing mean µ and variance σ2,representing the benefits from an insurance policy, we have
P
(
Y1 ≥ µ +σ√δ
)
≤ δ,
and A1(δ) = µ + σ/δ is the premium to be charged to achieve a loss probability below δ.
Given n independent and identically distributed Yj from n independent policies, weobtain
P
(
n∑
j=1
Yj ≥ n
(
µ +σ√nδ
)
)
≤ δ.
i.e. An(δ) = µ + σ/√
nδ suffices if the risk of n policies is pooled.
Proof: The statements follow as consequences of Tchebychev’s inequality:
P
(
n∑
j=1
Yj ≥ n
(
µ +σ√nδ
)
)
≤ P
(∣
∣
∣
∣
∣
1
n
n∑
j=1
Yj − µ
∣
∣
∣
∣
∣
≥ σ√nδ
)
≤ σ2
nσ2
nδ
= δ.
2
54 Lecture 12: Uncertain investment projects and risk
The estimates used in this proposition are very weak, and the premiums suggestedrequire some modifications in practice, but adding a multiple of the variance is oneimportant method, also since often this variance can be easily calculated.
The second and certainly not less important observation in this result is that thepremiums An(δ) decrease with n. This means, that the more policies an insurer can sellthe smaller gets the risk, allowing him to reduce the premium. The proposition indicatesthis for identical policies, but in fact, this is a general rule about risk. You can also testit on a personal level: often insurance policies are sold in packages. You can insure yourhouse, its contents, personal liability, travelling etc. in one policy. This is much cheaperthan insuring every single item or liability separately.
Lecture 13
Life insurance: the single decrementmodel
’Single decrement’ means that there is only one change of state, from ’alive’ to ’dead’.More general ’multiple decrement models’ include e.g. illness.
13.1 Uncertain cash flows in life insurance
Assume that a certain cash flow c is restricted to some (residual) lifetime T . This wasour situation for corporate bonds and related cash flows stopping at the insolvency timeT of a company. Here we think of T as the time to death of a human being (life).
Example 42 (Pure endowment) A pure endowment provides just one unit payment(n, 1) if and when the person insured lives at time n, i.e. if T > n. Clearly, the fairpremium in an interest model δ(·) is A = v(n)P (T > n).
More typically, the cash flow c might be a pension payable from now or from somefuture date. To price these, we can reformulate Proposition 27 in Lecture 11.
Proposition 30 Let δ(·) be an interest rate model, c a cash flow and T the time of deathof a life, modelled by a continuous random variable, c[0,T ] the cash flow c restricted to[0, T ], then the fair price for c[0,T ] is
E(
δ-DV al0(
c[0,T ]
))
= δ-DV al0(c)
where δ = δ + µT and µT (t) = fT (t)/FT (t).
In fact, µT (t), t ≥ 0, is of importance in the sequel. It has a name:
Definition 32 For any continuous random variable T in [0,∞) modelling a lifetime, thefunction µ(t) = fT (t)/FT (t) is called the force of mortality.
We will give a motivation for this name in the next section. We conclude this sectionintroducing an important class of examples, life annuities.
55
56 Lecture 13: Life insurance: the single decrement model
Example 43 (Life annuities) An elementary life annuity for a life with residual lifetime T , is the cash flow C = c[0,T ] obtained by restricting the perpetuity c = ((n, 1))n∈IN
to [0, T ], i.e.
c[0,T ] = ((1, 1), (2, 1), . . . , ([T ], 1))
where [T ] denotes the integer part of T , i.e. [T ] ≤ T < [T ] + 1.Analogously, pthly and continuously payable life annuities are the restricted pthly and
continuously payable perpetuities.
Example 44 Given a (residual) life time T ∼ Exp(µ), the fair price of an elementarylife annuity c[0,T ] in a constant δ interest rate model is
A = (δ + µ)-DV al0(c) =1
eδ+µ − 1=
p
1 + i − p
where i = eδ − 1 and p = e−µ = P (T > 1).
13.2 Conditional probabilities and the force of mor-
tality
Definition 33 Given a random variable T and a set B ⊂ IR such that P (T ∈ B) > 0,then the conditional distribution of T given {T ∈ B} is defined for A ⊂ IR by
P (T ∈ A|T ∈ B) =P (T ∈ A, T ∈ B)
P (T ∈ B)=
P (T ∈ A ∩ B)
P (T ∈ B).
We still think of T as a life time. Conditional probabilities arise naturally in questionslike the following.
Example 45 Given that somebody reaches his 65th birthday, what is the probabilitythat he survives his 80th birthday:
P (T > 80|T > 65) = P (T > 80)/P (T > 65).
Note that the conditional probability exceeds the unconditional P (T > 80). Intuitivelythis is because the older you get, the more likely it is that you survive your 80th birthday.
Example 46 You would expect that the older you get, the more likely it is that you diewithin the following year, month, day etc. This is not reflected in the density fT of T .fT is usually even assumed decreasing. In fact, the phrase ”the older you get” indicatesconditioning on survival, and the probabilities that we expected to increase are
P (T ≤ t + ε|T > t), t ≥ 0,
for ε = 1, ε = 1/12, ε = 1/365 etc., respectively.
Dealing with several values of ε is a bit inconvenient, but
Proposition 31 Given a continuous random variable T ∈ [0,∞), we have
1
εP (T ≤ t + ε|T > t) → µT (t), as ε → 0
o13 Lecture Notes: Introduction to Actuarial Science 57
Proof: Just note that
P (T ≤ t + ε|T > t) =P (t < T ≤ t + ε)
εP (T > t)=
FT (t + ε) − FT (t)
εFT (t)→ fT (t)
FT (t)= µT (t).
2
The force of mortality expresses your infinitesimal likelihood to die provided you arestill alive. It therefore is a measure of your risk to die now.
13.3 The curtate future lifetime
As many cash flows involve payments only on a discrete lattice, integer times, say, it isuseful to discretise lifetime distributions.
Definition 34 Given a life time random variable T , we call K = [T ] the associatedcurtate lifetime.
K is the number of complete time units (years) lived.
Example 47 If T ∼ Exp(µ), we have K = [T ] ∼ geom(e−µ). To see this, denotep = e−µ and note that
P (K = k) = P (k ≤ T < k + 1) =
∫ k+1
k
µe−µtdt = e−kµ − e−(k+1)µ = pk(1 − p).
This motivates the expression in terms of p in Example 44.
More generally, we can have µ and p depend on time:
Proposition 32 The distribution of K is given by P (K = k) = P (k ≤ T < k + 1). Kcan be seen as the first 0 in a series of independent Bernoulli trials with varying success(1) probabilities
pk = P (T ≥ k + 1|T ≥ k) = exp
{
−∫ k+1
k
µT (s)ds
}
,
the latter provided T is a continuous random variable.
Proof: The first statement is clear from the definition of [·]. We further express theprobabilities as
P (K = k) = P (T ≥ k) − P (T ≥ k + 1) = P (T ≥ k)(1 − P (T ≥ k + 1|T ≥ k))
= P (T ≥ k)(1 − pk) = p0 . . . pk−1(1 − pk)
by induction. Clearly, this is also the probability that the first 0 in a series of independentBernoulli trials with varying success probabilities pj occurs at k.
58 Lecture 13: Life insurance: the single decrement model
In the case of a continuous T we use
P (T > t) = exp
{
−∫ k
0
µT (s)ds
}
from the proof of Proposition 27, and we further calculate
P (T ≥ k + 1|T ≥ k) =P (T ≥ k + 1)
P (T ≥ k)= exp
{
−∫ k+1
k
µT (s)ds
}
.
2
13.4 Insurance types and examples
In the sequel we assume given a life with time to death T , and K = [T ], and a constantδ interest model.
We have seen pure endowments in Example 42 and life annuities in Examples 43 and44. Three more types are important.
Example 48 (Whole life insurance) Pay one unit at the end of the year of death, i.e.at time K+1. The random discounted value at time 0 is Z = e−δ(K+1) and A(δ) = E(Z) =E(e−δ(K+1)) is the fair premium. We can also calculate E(Zm) = E(e−mδ(K+1)) = A(mδ)and V ar(Z) = A(2δ) − A2(δ).
Example 49 (Term insurance) Pay one unit at the end of the year of death if deathoccurs within n years. No payment is made if the life survives n years.
Example 50 (Endowments) Formally, this is the sum of a term insurance and a pureendowment: pay one unit at the end of the year of death if death occurs within n years,otherwise pay one unit after n years.
There has been a lot of effort to find suitable parametric families of lifetime distribu-tions that have simple survival functions. The most elementary and most notable onesare
Example 51 (Gompertz-Makeham) µT (t) = A+Bct for A > 0, B > 0, c > 0, whichimplies
FT (t) = exp{−At − m(ct − 1)}
where m = B/ log(c)
Example 52 (Weibull) µT (t) = ktn for k > 0, n > 0, which implies
FT (t) = exp
{
− k
n + 1tn+1
}
.
In practice, one does not restrict to two or three parameters, but estimates laws ofK = [T ] essentially among all distributions on IN or {0, 1, . . . , 109} from past experience.This law of K is the main information given on lifetables.
Lecture 14
Life insurance: premium calculation
14.1 Residual lifetime distributions
Actuaries often work with large populations of different ages. It is therefore convenientto introduce notation taking account of age.
Definition 35 (Family of residual lifetime distributions) For a given (full) lifetimeT we define a family of random variables (actually probability distributions)
P (Tx ∈ A) = P (T − x ∈ A|T > x) or FTx(t) =
FT (t + x)
FT (x)
for A ⊂ IR+, t ∈ IR+, x ∈ [0, ω) where ω = inf{t ≥ 0 : FT (t) = 0} is the maximal agepossible under T . Tx is called the residual lifetime of a life aged x.
Proposition 33 Any family of residual lifetime distributions (Tx)x∈[0,ω) in the sense ofDefinition 35 has the consistency property
P (Tx+y ∈ A) = P (Tx − y ∈ A|Tx > y)
for all x, y ∈ IR+ such that x + y < ω. Vice versa, any consistent family of residuallifetime distributions is uniquely determined by a full lifetime distribution of T = T0.
Proof: The distribution function, and hence the survival function uniquely determinesthe distribution of a random variable, it therefore suffices to note that
P (Tx − y > t|Tx > y) =FTx
(y + t)
FTx(y)
=
FT (x+y+t)
FT (x)
FT (x+y)
FT (x)
=FT (x + y + t)
FT (x + y)= FTx+y
(t).
The second statement is obvious as the consistency condition contains the definition ofthe law of Ty for x = 0. 2
The law of Tx can be used both as the residual lifetime distribution of a life aged xnow and as the conditional residual lifetime distribution beyond x of a life that is youngernow given it survives to age x. This is often useful in applications.
59
60 Lecture 14: Life insurance: premium calculation
14.2 Actuarial notation for life products
We recall that K = [T ] denotes the total number of completed future years of a life.Analogously, we denote Kx = [Tx] for a life aged x now. Actuaries have extensivenotation related to lifetimes. We only introduce some key notation here.
Definition 36 Actuaries use the following shorthand for lifetime distributions
tpx = FTx(t) = P (Tx > t) and tqx = 1 − tpx = FTx
(t) = P (Tx ≤ t)
where a pre-index t = 1 is usually suppressed: px = 1px and qx = 1qx. The force ofmortality is also denoted
µt+x = µTx(t).
Note that for t = k ∈ IN, the symbols also denote the survival and distributionfunction of the curtate lifetime K. qx is particularly important being the probability ofdying within a year for a life aged x.
Example 53 In Proposition 32 we used notation pk = P (T ≥ k + 1|T ≥ k) = P (Tk >1) = P (Kk > 1) consistent with the definition here to interpret the (curtate) lifetime asthe first failure in a series of Bernoulli experiments. The pk (or the qk = 1 − pk), k ∈ INdetermine the law of K. This observation is crucial when reading lifetables that provideestimates of just qk, k ∈ IN.
Further notation is particularly useful to represent and relate the fair prices of maininsurance products.
Example 54 (Whole life assurance) Given a constant interest rate model, the fairpremium of a whole life insurance is denoted
Ax = Ax(δ) = E(exp{−δ(Kx + 1)}) =
∞∑
k=0
vk+1kpxqx+k
where the underlying force of interest δ is often suppressed.
Example 55 (Term assurance, pure endowment and endowment) The fair pre-mium of a term insurance is denoted by
A1x:n|
=n−1∑
k=0
vk+1kpxqx+k.
The superscript 1 above the x indicates that 1 is only paid in case of death within theperiod of n years.
The fair premium of a pure endowment is denoted by A 1x:n| = vn
npx. Here the super-script 1 indicates that 1 is only paid in case of survival of the period of n years.
The fair premium of an endowment is denoted by Ax:n| = A1x:n|
+A 1x:n|
, where we couldhave put a 1 above both x and n, but this is omitted being the default, like in previoussymbols.
o13 Lecture Notes: Introduction to Actuarial Science 61
14.3 Lifetables
Assume that there is a (curtate) lifetime distribution K that we want to estimate (andassociated Kx). We know how to express it in terms of one-year death probabilities qx,x = 0, 1, 2, . . .. The naive way to estimate qx is the following.
Proposition 34 Given a sample of n people observed between age x and x + 1, recordBj = 1 (Bj = 0) if person j died (survived), j = 1, . . . , n. Then Bj ∼ B(1, qx) can beassumed independent random variables and
qx = qx(n) =1
n
n∑
j=1
Bj → qx
in probability, as n tends to infinity.
Proof: Bj ∼ B(1, qx) is clear since qx is the one year death probability of any personaged x. The convergence in probability follows from the weak law of large numbers. 2
For given data, qx is an estimate of qx. Life tables show such estimates rather thanany true values behind, although they write qx rather than qx, for historical reasons, sincesuch ’estimates’ have been used much longer than the stochastic approach behind. Anextract from a life table is reproduced in Appendix A.
In practice, there are some complications and ways to address these that we onlyindicate here:
• incomplete observations (people entering an insurance contract between ages x andx + 1, or whose contract reaches the end of its term);
• small sample sizes for high x, e.g. x = 105; a procedure called ’graduation’ averageswith neighbouring values and thus also completes the picture up to a maximal age;
• the more data one considers, the older they are; since mortality changes with time,estimates are likely not to be up to date; a procedure called ’extrapolation’ projectsforward the development during past years;
• people can be classified into groups with significantly different mortality: male/female,smoker/nonsmoker, job groups, state of health, type of insurance chosen etc; thereare specific tables for the more important combinations of these;
• insurance contracts can often only be made when in a good state of health; thisdecreases mortality significantly, and it is customary to take account of this for twoinitial years, denoting decreased death probabilities by q[x] and q[x]+1 respectively.With these included, a life table has three columns and the relevant entries for anygiven situation are the age row and the qx+2+n, n ∈ IN, in the last column.
Example 56 The premium for a 4-year temporary life assurance of a life aged 55 as-suming an interest rate i = 4% is
A 155:4| = vq55 + v2p55q56 + v3p55p56q57 + v4p55p56p57q58,
where we recall that px = 1 − qx.
62 Lecture 14: Life insurance: premium calculation
Given a sum assured of N = £10, 000 and without taking account of a good initialstate of health we read off the life table given in Appendix A and calculate
NA 155:4| ≈ 356.26.
Taking account of a good initial state of health, i.e. using q[55] and q[55]+1 instead of q55
and q56, we obtain from the same table
NA 1[55]:4| ≈ 293.13.
14.4 Life annuities
Example 57 Given a constant i interest model, the fair premium of an ordinary (re-spectively temporary) life annuity for a life aged x is given by
ax =∞∑
k=1
vkkpx respectively ax:n| =
n∑
k=1
vkkpx.
For an ordinary (respectively temporary) life annuity-due, an additional certain paymentat time 0 is made:
ax = 1 + ax respectively ax:n| = 1 + ax:n|.
Life annuities occur, in particular when life product premiums are multiple ratherthan single premiums: in the simplest case level advance premiums are paid until death.The cash flow of premium payments is therefore a life annuity. We shall see in the nextlecture that this observation allows to calculate the amount of fair level premiums.
14.5 Multiple premiums
Definition 37 Given a constant i interest model, let C be the cash flow of insurancebenefits, the annual fair level premium of C is defined to be
Px =E(DV al0(C))
ax.
Proposition 35 In the setting of Definition 37, the expected dicounted benefits equal theexpected discounted premium payments.
Proof: The expected discounted value of premium payments ((0, Px), . . . , (Kx, Px)) isPxax = E(DV al0(C)). 2
Example 58 We calculate the fair annual premium in Exercise 56. Since there shouldnot be any premium payments beyond the term of the assurance, the premium paymentsare a term life annuity-due ((0, 1), (1, 1), (2, 1), (3, 1)) restricted to the lifetime K55. First
a[55]:4| = 1 + vp[55] + v2p[55]p[55]+1 + v3p[55]p[55]+1p57 ≈ 3.74143
and the annual premium is therefore calculated from the life table in Appendix A as
P =NA 1
[55]:4|
a[55]:4|
= 78.35.
Lecture 15
Some elements of General Insurance
This lecture is a brief digression to motor, health and property insurances. We essentiallydiscuss some concepts for premium calculation and give an important example that showsagain the effect of pooling.
15.1 Premium principles
In the sequel, we think of random variables S as the total claim amount from an insurancepolicy or a portfolio of policies.
Definition 38 A premium principle is any rule H that assigns with every random vari-able S a real number or ∞, denoted H(S). H(S) − E(S) is called safety loading. IfH(S) = ∞, the risk represented by S is called uninsurable.
This vague definition reflects the multitude of possibilities.
Example 59 (Net premium principle) The fair premium from Definition 27, morecommonly called the net premium is the special case H = E, i.e. H(S) = E(S).
Example 60 (Expected value principle) H(S) = (1+λ)E(S) suggests a safety load-ing proportional to E(S).
Example 61 (Variance principle) H(S) = E(S)+αV ar(S) suggests a safety loadingproportional to V ar(S) as in Proposition 29.
Example 62 (Standard deviation principle) H(S) = E(S) + β√
V ar(S) suggestsa safety loading proportional to the standard deviation of S.
Example 63 (Percentile principle) H(S) = min{A|P (S > A) ≤ δ} suggests theminimal premium that bounds the probability of loss by a given level. Proposition 29does this in an approximate way: the level is not exceeded but the premium is notminimal in this property.
Example 64 (Exponential principle) H(S) = log(E(exp{aS}))/a.
63
64 Lecture 15: Some elements of General Insurance
There are other and more general principles using utility functions that we do notdiscuss here. The exponential principle plays an important role. All principles haveadvantages and disadvantages. A more thorough discussion with practical relevancerequires an understanding of the market structure.
15.2 The Central Limit Theorem and an example
Proposition 36 (Central Limit Theorem) For independent and identically distributedXn, n ∈ IN, with finite variance σ2 < ∞ and mean µ we have
E
(
f
(
X1 + . . . + Xn − nµ√nσ2
))
→ E (f(Z)) as n tends to infinity
for all bounded continuous functions f : IR → IR, where Z ∼ N(0, 1) is standard normal.
This is usually applied as
X1 + . . . + Xn − nµ√nσ2
≈ Z
for ”large” n.
Example 65 Assume an individual risk model with n = 1000 policy holders who havea chance of q = 5% each to produce a claim (denote Nj = 1) of amount Xj ∼ Exp(λ),λ = 0.001, i.e. E(Xj) = 1/λ = 1000.
The net premium per policy is
E(NjXj) = E(Nj)E(Xj) = q/λ = 50.
For each policy, the probability of loss for the insurer is
P (NjXj > 50) = P (Nj = 1, Xj > 50)
= P (Nj = 1)P (Xj > 50)
= (1/20)e−50λ
≈ 4.76%.
For each policy, the expected loss for the insurer is
E((NjXj − 50)+) = E(
(Xj − 50)+ 1{Nj=1}
)
= E((Xj − 50)+)P (Nj = 1)
=1
20
∫ ∞
50
(x − 50)λe−λxdx
=
∫ ∞
0
yλe−λ(y+50)dy
= λe−50λ ≈ 951.23
o13 Lecture Notes: Introduction to Actuarial Science 65
If we double the premium, we get
P (NjXj > 100) ≈ 4.52% and E((NjXj − 100)+) ≈ 904.84.
For all policies together, S = N1X1 + . . . + NnXn, the probability of loss for the insureris
P (S > 50n) = P
(
S − nE(NjXj)√
nV ar(NjXj)> 0
)
≈ P (Z > 0) = 1/2
by the Central Limit Theorem. Note that we did not need the value of
V ar(NjXj) = E((NjXj − 50)2)
= E(
(Xj − 50)21{Nj=1} + 5021{Nj=0}
)
=1
20E(
(Xj − 50)2)
+19
20502
=1
20
(
E(
(Xj − E(Xj))2)
+ (E(Xj) − 50)2)
+19
20502
= 97500,
but we shall need it when we now double the premium to get
P (S > 100n) = P
(
S − nE(NjXj)√
nV ar(NjXj)>
50n√
nV ar(NjXj)
)
≈ P (Z > 5.064)
≈ 2.05 ∗ 10−7.
To get 4.52%, we need a safety loading of 16.72 per policy. The expected losses in theaggregate case can be approximated as
E(
(S − 50n)+)
=√
V ar(S)E
((
S − E(S)√
V ar(X)
)+)
≈√
V ar(S)E(Z+)
=√
V ar(S)
∫ ∞
0
x√2π
e−x2/2dx
≈ 3939.24
and
E(
(S − (50 + c)n)+)
≈√
V ar(S)E
((
Z − cn√
V ar(S)
)+)
.
which is
E(
(S − (50 + 50)n)+)
≈ 1.1841 ∗ 10−5 and E(
(S − (50 + 16.72)n)+)
≈ 5.8043.
66 Lecture 15: Some elements of General Insurance
In practice, insurance companies explicitly set aside some of their capital that they areprepared to spend in the unlikely event that premium income is not sufficient. In thiscase, there is another threshold that should be analysed. If this is exceeded, the insurancesuffers so-called technical ruin upon which yet other safety measures apply, either moreinternal funds or reinsurance policies that provide a network of essentially all insurersto maximise pooling effects on the highest possible level of all insurances. Premiums forreinsurance and related topics are beyond the scope of this course.
Lecture 16
Summary: it’s all about Equationsof Value
16.1 Summary
Roughly, one can say that the course so far has consisted of three parts. Some of thematerial mentioned below was introduced on the Assignment sheets as straightforwardextensions of lecture material.
16.1.1 Basic notions used throughout the course
The first third (lectures 1,2,3 and parts of 4,6) introduced into
1. Cash flows (discrete cash flows, continuous cash flows, mixed cash flows, generalisedrandom cash flows, modelling with cash flows)
2. Compound interest theory (simple interest, time-dependent compound interest,constant interest rates, simple discount)
3. Valuation of cash flows (accumulated values, discounted values, time values)
4. Annuities (ordinary, pthly and continuously payable, perpetuities, deferred annu-ities and annuities-due)
5. Yields of cash flows (definition and existence of yields)
Expressed differently, the following notation is the most important.
c = (tn, cn)n≥0, c = (c(t))t∈IR+, C = (Tn, Cn)n≥0
δ(t), v(t), i, v, d, δ, i(p), d(p)
DV alt(c) + AV alt(c) = V alt(c)
an|, sn|, a(p)n|
, an|, m|an|, an|, a∞|
y(c)
67
68 Lecture 16: Summary: it’s all about Equations of Value
16.1.2 Deterministic applications
The second third (lectures 5,7-9 and parts of 4,6) concerned applications without uncer-tainty. The following list regroups the main topics.
1. Mortgages and loans (repayment schemes, prospective/retrospective method foroutstanding capital, interest/capital repayment components, APR, flat rate)
2. Fixed interest securities (valuation, running yield and yield to redemption, incomeand capital gains tax)
3. Investment projects (profitability, yields, discounted and undiscounted payback pe-riod, comparison of investments, cross-over rates)
4. Funds (Money-weighted rate of return, Time-weighted rate of return, Linked inter-nal rate of return, funds as investment)
5. Inflation (RPI index, inflation rate models, real yields, inflation-adjusted payments,capital gains tax taking into account inflation)
16.1.3 Applications with uncertaincy
The last third (lectures 11-15 and parts of 10) concerned models with some randomingredients.
1. Risk (Variance of random values, loss probabilities, fair prices and other premiumprinciples, pooling of risk)
2. Corporate bonds (Insolvency time, general pricing formula, pricing under exponen-tial or geometric insolvency time)
3. Discounted dividend model
4. Single decrement model (lifetime distributions, curtate lifetime, lifetables, generalvaluation, temporary, whole life and endowment assurances, annuities, single andmultiple premiums)
5. Uncertain investment projects
Everything that has been done in the deterministic setting can be done under uncer-tainty with random values replaced by expected values. It then makes sense to ask forthe variability. Calculating variances is one indicator.
o13 Lecture Notes: Introduction to Actuarial Science 69
16.2 Equations of value
The most important method throughout the course is via equations of value
i-V alt(c) = X
where the interest rate (or yield) i, the time t, the value X or some unknown quantityin the specification of the cash flow c is to be determined. Virtually everything relies onthis.
Example 66 (Accumulated and discounted values) X unknown, t end of term ort = 0.
Example 67 (Yields, real yields, cross-over rates, APR, MWRR, TWRR) iunknown, X = 0; i-RV alt(c) = 0; ...
Example 68 (Payback periods) Here actually 0-AV alt(c) = 0 or i-AV alt(c) = 0, tunknown
Example 69 (Capital gains tax) Purchase price P of a security unknown given theyield i, say coupons c, redemption proceeds R:
P = V al0(c) + V al0((n, R − (R − P )t2))
Example 70 (Fair prices under uncertainty) Here actually E(V alt(C)) = X, Xunknown
Example 71 (Multiple premiums) Expected discounted benefits equal expected dis-counted level premium payments; level premium amount P unknown.
Always ask yourself: What is the cash flow? What is the equation of value?
16.3 Examination
The exam paper will consist of 6 questions (not 8 as in most other papers you are goingto sit). Questions will be marked out of 25 as usual, so you should aim for 4 questionsas usual, the usual marking scheme applies. This means, your amount choice is reduced.This is a concession to the Institute of Actuaries who do not allow any choice in theirexams.
You will require a calculator. Please check the University regulations in good time,because you may have to familiarise yourselves with more basic calculators than yourown.
The style of questions is not like assignment questions. There is usually a bookworkpart asking you to define certain notions or reproduce or reprove important formulas,and checking your understanding of these. A question may cover more than one lectureor more than one of the topics above. Three questions concern MT and three questionsHT, but as HT builds on MT they may include explicitly some MT material.
70 Lecture : Summary: it’s all about Equations of Value
16.4 Hilary Term
Peter Clark who taught a class this term, will take over teaching most lectures next term.The following topics from the Institute of Actuaries 102 paper remain:
1. Investments: risk characteristics
2. Investments: stability (reaction on small changes in parameters)
3. How to determine the market interest rate model: term structure of interest rates
4. Stochastic interest rate models
5. Arbitrage free pricing (cf. o10)
6. Forward contracts, futures, options (cf. o10)
Some of these quite naturally lead to some 109 (Financial economics) issues, but PeterClark chooses what to teach you.
16.5 Assignment 7
I prepared an Assignment 7 on Life insurance, since you have not yet applied lifetables,and next term may not get back to them. I try to organise a voluntary class in a mathslecture theatre for week 2 - assuming that normal classes start in week 3 again. Thereare some unknowns, but you will be given clear information in Peter Clark’s first lecture.
Appendix A
A 1967-70 Mortality table
The following table is an extract from mortality tables of assured lives based on datacollected between 1967 and 1970. These tables were used for examination purposes bythe Faculty and Institute of Actuaries and are provided on their website at
http://www.actuaries.org.uk/students/FT-A1967-70.pdf
The extract below can be found at the top of page 7 of the pdf file.
Age [x]
5354555657
5859606162
q[x]
.00376288
.00410654
.00447362
.00486517
.00528231
.00572620
.00619802
.00669904
.00723057
.00779397
q[x]+1
.00519413
.00570271
.00625190
.00684424
.00748245
.00816938
.00890805
.00970168
.01055365
.01146756
qx+2
.00844128
.00941902
.01049742
.01168566
.01299373
.01443246
.01601356
.01774972
.01965464
.02174310
Age x + 2
5556575859
6061626364
71
