Ef 5370 Lecture 1

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EF5370 Mathematics and Statistics for Financial

Service

Lecture 1

Xuan S. Tam

Sept. 3, 2013

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Outline

Cashflows

Discount Functions

Calculating the Discount Function Constant Interest

Values and Actuarial Equivalence

Regular Pattern Cashflows

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Cashflows

A basic application of actuarial mathematics is to model thetransfer of money

Insurance companies, banks and other financial institutions

engage in transactions that involve accepting sums of moneyat certain times, and paying out sums of money at other times

Let time 0 refer to the present time, and time k will thendenote k time units in the future

Assume that all funds are paid out or received at integer timepoints, that is, at time 0, 1, 2, ....

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Cashflows

The amount of money received or paid out at time k will becalled the net cashflow at time k and denoted by ck

A positive value of ck denotes that a sum is to be received,

while a negative value indicates that a sum is paid out The entire transaction is then described by listing the

sequence of cashflows, which will refer to this as a cashflowvector, c = (c0, c1, ...,cN)

N is the final duration for which a payment is made

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Cashflows

Suppose I lend you 10 units of capital now and a further 5units a year from now. You repay the loan by making threeyearly payments of 7 units each, beginning 3 years from now.The resulting cashflow vector from my point of view isc = (10,5, 0, 7, 7, 7)

From your point of view, the transaction is represented byc = (10, 5, 0,7,7,7)

This lecture is to provide methods for analyzing transactions

in terms of their cashflow vectors

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Cashflows

There are several basic questions that could be asked: When is a transaction worthwhile undertaking? How much should one pay in order to receive a certain

sequence of cashflows? How much should one charge in order to provide a certain

sequence of cashflows? How does one compare two transactions to decide which one is

preferable?

we could answer all of them if we could find a method to put

a value on a sequence of future cashflows

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Cashflows

We pay interest for the privilege of borrowing money today,which lets us consume now, or we advance money to others,giving up our present consumption and expecting to becompensated with interest earnings

In addition, there is the effect of risk. If we are given a unit ofmoney today, we have it. If we forego it now in return forfuture payments, there could be a chance that the party whois supposed to make remittance to us may be unable or

unwilling to do, and we expect to be compensated for thepossible loss

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An Analogy with Currencies

Suppose that I give you 300 Canadian dollars, 200 US dollars,and 100 Australian dollars. How much money did I give you?

Let v(c, u) denote the value in Canadian dollars of 1 US

dollar. Assume that v(c, u) = 1.20, which means that a USdollar is worth 1.20 Canadian dollars

Similarly, letting a stand for Australian, we will assume thatv(c, a) equals 0.90, which means 90 Canadian cents will buy 1

Australian dollar

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v function returns the value of one unit of the second

coordinate currency in terms of the first coordinate currency

Let v(c, u) denote the value in Canadian dollars of 1 USdollar. Assume that v(c, u) = 1.20, which means that a USdollar is worth 1.20 Canadian dollars

If it takes 1.20 Canadian dollars to buy 1 US dollar, then asingle Canadian dollar is worth 1/1.2 = 0.833 US dollars. Thatis, v(u, c) = v(c, u)1 = 0.833, v(a, c) = v(c, a)1 = 1.111

Consider v(u, a): the amount of US dollars needed to buy oneAustralian dollar

v(u, a) = v(u, c)v(c, a) = 0.750

The real-life currency relationships do not hold exactly due tocommissions and other charges

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Lets return to the original problem, we could say that thetotal was equivalent to 300 + 200v(c, u) + 100v(c, a) = 630

Canadian dollars or 630v(u, c) = 525 US dollars Similarly, the total in Australian dollars can be computed

immediately as 630v(a, c) or alternatively as 525v(a, u), bothof which are equal to 700

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Discount Functions

We want to value a sequence of cashflows, which are all in thesame currency, but which are paid at different times

Conversion factors are needed to convert the value of money

paid at one time to that paid at another Let v(s, t) denote the value at time s, of 1 unit paid at time t

[Note again that our convention is that the 1 unit goes withthe second coordinate. In other words, 1 unit paid at time t isequivalent to v(s, t) paid at time s.]

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Discount Functions

In the case where s< t, you can interpret v(s, t) as theamount you must invest at time s in order to accumulate 1 attime t

In the case where s> t you can interpret v(s, t) as theamount that you will have accumulated at time s from an

investment of 1 at time t

The fundamental relationship: v(s, t)v(t, u) = v(s, u), for alls, t, u. (2.1)

1 unit at time u is equivalent to v(t, u) at time t, and this

v(t, u) at time t is equivalent to v(s, t)v(t, u) at time s,showing that 1 unit at time u is indeed equivalent tov(s, t)v(t, u) at time s

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Discount Functions

Definition 2.1: A discount function is a positive valuedfunction v, of two nonnegative variables, satisfying (2.1) forall values of s, t, u.

Taking s= t= u, we deduce that v(s, s)v(s, s) = v(s, s) and,since v(s, s) is nonzero, we verify that v(s, s) = 1, for all s.(2.2)

From this we deduce that v(s, t)v(t, s) = v(s, s) = 1 so thatwe recover the relationship that v(s, t) = v(t, s)1. (2.3)

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Discount Functions

Although we have called v a discount function, the commonEnglish usage of the word really applies to the case wheres< t

In that case, v(s, t) will be normally less than 1, and the

function is returning the discounted amount of 1 unit paid ata later date

Some authors would prefer to define accumulation functionwhere s> t

We will suppose that, given any financial transaction, there isa suitable discount function that governs the investment of allfunds

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Discount Functions

There are many factors governing this choice and it willdepend on the nature of the transaction

It may simply reflect the preferences of the parties for present

as opposed to future consumption It may reflect the desired return that an investor wishes to

achieve In many cases it is based on a prediction of market conditions

that will determine what returns can be expected on investedcapital

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Calculating the Discount Function

The currency example indicated that we can calculate all thevalues of a discount function just by knowing those at pointswith a common comparison point. In most applications it is

convenient to take this as time 0 To simplify notation, we drop the first coordinate in this case

and define v(t) = v(0, t) (2.4)

It follows from (2.1) that v(s, t) = v(s, 0)v(0, t) and then

from (2.3) that v(s, t) = v(t)v(s)

(2.4)

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From (2.4), it will be sufficient to know v(n) where n is anynonnegative integer

To calculate v(n), (2.1) can be extended from one involvingthree terms to an arbitrary number. That is, given timest1, t2, ..., tn, v(t1, t2)v(t2, t3)v(tn1, tn) = v(t1, tn) (2.5)

Take for example n = 4. v(t1, t2)v(t2, t3)v(t3, t4) is equal toV(t1, t3)V(t3, t4) by applying (2.1) to the first two terms. Byanother application of (2.1) it is equal to v(t1, t4) We have

extended (2.1) to a formula involving four terms

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It follows from (2.5) that v(n) = v(0, 1)v(1, 2)...v(n 1, n),(2.6)

We need only know v(n 1, n) for all positive integers n.

Given such values, we can use the recursion formulav(n) = v(n 1)v(n 1, n), v(0) = 1, (2.7)to calculate all values ofv(n)

The information we need is then summarized by the vectorv = [v(0), v(1), v(2), ..., v(N)] , where N is the final duration

at which a nonzero cashflow occurs

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Exercise: You are given a discount function v wherev(1, 3) = 0.9, v(3, 6) = 0.8, v(8, 6) = 1.2

How much must you invest at time 1, in order to accumulate10 at time 8? If you invest 100 at time 3, how much will have accumulated

by time 8?

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Interest and Discount Rates

In practice, rather than specifying v(k 1, k) directly, it ismore common to deduce this quantity from the correspondingrates of interest or discount

Given any discount function v and a nonnegative integer k,these are defined as follows

Definition 2.2 The rate of interest for the time interval k to k+ 1is the quantity

ik = v(k+ 1, k) 1

.

I d Di R

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Definition 2.3 The rate of discount for the time interval k tok+ 1 is the quantity

dk = 1 v(k, k+ 1)

Note that an investment of 1 unit at time k will producev(k+ 1, k) = 1 + ik units at time k+ 1

Similarly, an investment of 1 dk = v(k, k+ 1) units at timek will accumulate to 1 unit at time k+ 1

Given any of the three quantities v(k, k+ 1), ik or dk we caneasily obtain the other two

I d Di R

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For example, using the definitions and (2.3), it isstraightforward to deduce that

dk = ikv(k, k+ 1) =

ik

1 + ik

ik = dkv(k+ 1, k) =dk

1 dk(2.8)

Remark: i0

is the interest rate for the first time interval

C t t I t t

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Constant Interest

Suppose we believe that accumulation of invested fundsdepends only on the length of time for which the capital isinvested, and not on the particular starting time. That is, wepostulate that for all nonnegative s, t, h,

v(s, s+ h) = v(t, t+ h) (2.9)

If this holds, then

v(s+t) = v(0, s+t) = v(0, s)v(s, s+t) = v(0, s)v(0, t) = v(s)v(t)

C t t I t t

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Constant Interest

If we assume that v be continuous, we must have v(t) = vtand therefore that v(s, t) = vts for some constant v

For such a discount function, the rate of interest ik is aconstant i = v1 1, and the rate of discount dk is a

constant d = 1 v The discount function is therefore conveniently given by i, the

constant rate of interest

If we want to know how much we will accumulate at time nfrom an investment of 1 at time 0, this is justv(n, 0) = (1 + i)n, the usual starting point for the compoundinterest

Values and actuarial equivalence

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Suppose we are given a cashflow vector c = (c0, c1, ..., cN)and a discount function v

We want to calculate the single payment at time zero that isequivalent to all the cashflows, assuming that the time valueof money is modeled by the given discount function v

This amount is commonly referred to as the present value ofthe sequence of cashflows, and sometimes abbreviated as PV

It is the amount we would pay at time zero in order to receive

all of the cashflows

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Example 2.1 Let v(k) = 2k for all k. This is a constantinterest rate per period of 100%. In other words, moneydoubles itself every period. Suppose we are to receive 12 unitsat time 2, but will be required to payout 8 units at time 3.

Find the present value, and verify that it makes sense.


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Example 2.1 Let v(k) = 2k for all k. This is a constantinterest rate per period of 100%. In other words, moneydoubles itself every period. Suppose we are to receive 12 unitsat time 2, but will be required to payout 8 units at time 3.

Find the present value, and verify that it makes sense. Solution: From (2.10) the present value is

12v(2) 8v(3) = 12(1/4) 8(1/8) = 2. Note that the 12units received at time 2 will accumulate to 24 at time 3. Wethen have to payout 8, leaving an accumulation of 16 units bytime 3


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Note that it was convenient in the above example to comparethe amounts accumulated at the time of the last payment.

This is known as the accumulated value and in general isgiven by

N

k=0

ckv(N, k)

Definition 2.4 : For any time n = 0, 1, ...,N, the value attime n of the cashflow vector c with respect to the discountfunction v is given by

Valn(c; v) =

N

k=0

ckv(n, k)

A single amount that we would accept at time n in place of allthe other cashflows according to the discount function v


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The values at various times are related in a simple way. Sincev(m, k) = v(m, n)v(n, k), it follows immediately that

Valm(c; v) = Valn(c; v)v(m, n) (2.11)

For the particular case of values at time 0 we will use a specialsymbol:

a(c; v) = Val0(c, v)

The letter a is a standard actuarial symbol that is used to

stand for annuity, another name for a sequence of periodicpayments


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When there is only one discount function under consideration

we often suppress the v and just write Valn(c) or a

For the particular case of values at time 0 we will use a specialsymbol:

a(c; v) = Val0(c, v)

We can express and calculate a conveniently by expressing itin vector form:

a(c) = v c = vcT (2.12)

The second term is the (scalar) inner product of the twovectors. The third views the vectors v and c as 1xN matrices,with the superscript T denoting a matrix transpose


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Valk(c+ d) = Valk(c) + Valk(d),Valk(c) = Valk(c), (2.13)for any cashflow vectors c and d, scalar , and duration k.

We often wish to compare the values of two sequences of

cashflows Definition 2.5: Two cash flow vectors c and e are said to be

actuarially equivalent with respect to the discount function vif, for some nonnegative integer n,

Valn(c; v) = Valn(e; v)


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q

Example 2.2: A lends B 20 units now and another 10 unitsat time 1. B promises to repay the loan by two payments,

made at time 2 and time 3. The repayment at time 3 is to betwice as much as that at time 2. IfA wishes to earn interestof 25% per period, what should these repayments be?


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q

Solution: Let K be the unknown payment at time 2. Wewant to find K so that the vectors c = (20, 10, 0, 0) ande= (0, 0,K, 2K) are actuarially equivalent.

PV of advances = 20 + 10(0.8) = 28PV of repayments = K[(0.82 + 2(0.83)] = 1.664KEquating values to make the advances and repaymentsactuarially equivalent, K = 28/1.664 = 16.83. The borrowerpays 16.83 at time 2 and 33.66 at time 3


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q

Example 2.3: Let c = (2, 4,3,5). Assume a constantinterest rate of 0.25. Find the actuarially equivalent vector byapplying the replacement principle with time k = 2 and thesubset {1, 3}.

Hint: i.e. find a K such that e= (2, 0,K, 0) is actuariallyequivalent to c


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Solution: The value ofc1 and c3 at time 2 is4(1.25) 5(1.25)1 = 1. Making the replacement, we obtain

the vector (2, 0,2, 0) that is actuarially equivalent to c, ascan be verified by direct calculation


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We assume throughout the section that the discount functionis given by v(n) = vn for some constant v.

Consider the vectors (1n) and jn = (1, 2, ..., n 1, n)

a(1n) = 1 + v+ v2 + ... + vn1

Multiplying by v, va(1n) = v+ v2 + v3 + ... + vn

Subtracting the second equation from the first and dividing by(1 v) gives

a(1n) =

1 vn

1 v (2.14)


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A similar trick handles the vector jn by reducing to the levelpayment case:

a(jn) = 1 + 2v+ 3v2 + ... + nvn1

Multiplying by v, va(jn) = v+ 2v2 + 3v3 + ... + nvn

Subtracting the second equation from the first and dividing by(1 v) gives

a(jn

) =

a(1n) vn

1 v (2.15)


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Exercise 1: You are given interest rates i0 = i1 = 0.25,i2 = i3 = 1. You have entered into business transaction whereyou will receive 2 at time 0, 5 at time 3, and 10 at time 4, inreturn for a payment by you of 3 at time 2. In place of allthese cashflows you are offered a single payment made to youat time 1. What is the smallest payment you would accept?


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Exercise 2: A loan of 20, 000, made at an interest rate of6%, is to be repaid by level yearly payments for 10 years,beginning 1 year after the loan is advanced. Just beforemaking the seventh repayment, the borrower wishes to repay

the entire loan. If interest rates remain unchanged, what is the outstanding

balance? Suppose interest rates have dropped to 5%. How much will

the borrower have to pay if the lender uses the lower interest

rate to calculate the outstanding balance?

Ef 5370 Lecture 1

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