On some difference-delay equations arising in a problem of capital deposits

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria III: MATEMATYKA STOSOWANA XXXIX (1996)

Z b ig n ie w B a r t o s z e ws k i and M a r ia n K wa pis z

Gdansk

On some difference-delay equations arising in a problem of capital deposits

(Praca wplynqla do Redakcji 19.12.1995)

1. Introduction. We consider a real life problem: a. person has made a deposit of D

dollars in bank B, which calculates interest on this deposit at in = 100 • in% after each n+l-st quarter and the interest is compounded at the end of each consecutive year since the deposit date, which means that the interest is capitalised yearly. In the case discussed the basic time unit is a quarter but the conversion period - the time interval at the end of which the interest is compounded - is four quarters (for the terminology see [3]). We ask the following question: what is the balance of the person after n time units, i.e. after n quarters? Such a question is important to people planning various sorts of investments or making arrangements with life in- surance institutions. We cannot find an answer to this question in available literature. In particular, it is not to be found in recent books devoted to the subject [see 1-6]. We want to find general formulas that would allow us to express this balance by the other given quantities. Such formulas allow us to solve inverse problems consisting in finding the initial deposit Z)

, the in- terest rate i or the length of the period after which a capital reaches a given level. In this paper we give explicit formulas for the above-mentioned bal- ance. They can be applied to the problems with any number of payments.

At the end of this paper we give some examples of application of these

formulas to solving the mentioned inverse problems. The obtained formulas

make solving such problems easy. A straightforward application of backward

recurrence formulas derived from formula (2), although possible, is quite

troublesome.

2. Let Fn, n = 0 ,1 ,... , be the balance at the end of the n-th quarter.

We easily observe that F0 = Do,

Fi = F0 + ioFo = Fo(l F ?’o), Fi = F\ F /iFo = Fo(l + io F *i),

F% = Fo 12 F 0 = Fo(l + *o F 2*1 F *2)5

F 4 — F3 -f- 2*3 Fo = F0(l F ?’o F i\ F i ‘2 F *3)?

then

F

= F4 F 14 F 4 = F4(l + 2*4) = F0(l + io + H + *2 + *3)(1 + H)i F

= F 5 15 F 4 = F4(l + 14 + i.5) =

= Fo( 1 + io + ii + h + h )i 1 + H + is )5 Fj = F

+ ieF4 = F4( 1 + i4 + i.5 + ie) =

— Fo(l + io + 21 + io + 2*3 )(1 + i4 + is + ie)?

Fg = F7 + 2*7 F4 = F4(l + i4 + i.5 + 2-6 + 2*7) =

— F0( 1 + io + il + 2*2 + 2-3)(1 F 24 + 2*5 F ie F 27),

F 9 = Fg F igFg = Fg(1 + ig ) =

= Fo( 1 F io F 2*1 F 2*2 F 23)(1 F i4 + 2*5 + ie F i7)(l F is), and so on.

From this observation we conclude that the sequence Fn of the balances is defined by the following difference-delay equation

(1) Fn+i = Fn + inFln , 72 = 0,1, . .. ,

with 7n = 4 on, where a n = [ j ] . Here, the symbol [ar] stands for the entire part of the number x. The equation (1) can be seen as the fourth order difference equation with variable coefficients. However, this point of view does not help us to find an explicit formula for the solution of this equation with the initial condition Fb = Do.

3. We can discuss more general situation when a different than a quarter time unit is used and when the conversion period is equal to k, k = 1, 2, . . . , basic time units. Now, in = 100 • in%, 22. = 0,1, . . . , is the interest calculated by bank B at the end of n+ 1-st basic time unit and Fn, n = 0 , 1, . . . is the balance at the end of the n-th basic unit. Under these conditions we have the same difference-delay equation, however with

I11 order to find the explicit solution of (T) we first consider the case

when the sequence in is constant, i.e. in = 2 for all n = 0, 1, . . . . In this

On some difference-delay equations arising in a problem of capital deposits

77

case equation ( 1) can be rewritten as

(3) Fn+1 = Fn + iFln, n = 0 , 1 , . . . . We have the following

T h e o r e m 1. Under the conditions of the present section the solution of equation (3) with the initial condition F

= D

is expressed by the formula (4) Fn = D

( 1 + ki)an(l /3ni), ra = 0, 1, . . . ,

where (3n — n — kan.

P r o o f . From the definition of (3n it follows that n — kcxn + (3n. We will consider two cases:

1) n = kcxn + (3n, 0 < (3n < k - 1, 2) n — hcxn T (3n^ j3n — h 1.

In the case 1) we have » n+i = a n, f3n+1 = j3n + 1 and we find that F

+

= D o(l -j- ki)an+1{ 1 + f3n+ii) =

= Fo(l + ki)a*(l + (/0n + l)i) =

= F0(l + ki)a” ( 1 + f3ni) + F0i(l + fci)a» =

= Fn -f- iF

(1 + ki)an = Fn iF\in, because from formula (2) we have

Fln = = F0(l + = F0(l +

Thus, formula (4) is satisfied.

In the case 2) we have n + 1 = kan+1 + fln+i = kan+i = + 1)?

which means that o:n+i = a n -f 1, (3n + i = 0. From this and formula (3) it follows that

Fn+1 = Fo(l + fc0“ *+,(l + Pn+1>) = Fo(l + k i r - +l =

— Foil -f- ki)an{ 1 + {k — 1)0 F *Fo( 1 + ki)a" = Fn + iF-yn.

This proves that in this case equation (4) is also satisfied. ■

4. Now we will consider the case when in = i, n = 0,1,... and the person makes an additional deposit of b dollars at the end of each n4-1-st basic time unit. In this case the balance Fn+1 at the end of n+l-st basic time unit satisfies the following difference-delay equation

Fn+i = Fn 4- iFln 6, n = 0, 1, . .. . ( 5 )

We can claim

T h e o r e m 2. The solution of equation (5) for i / 0 with the initial condition F

= D0 is expressed by the formula

(6) Fn = (1 + ki)an{ 1 + jdni) —

, n — 0 ,1, . . . .

P r o o f . We observe first that the constant sequence — | is a special solution of equation (5). (For / = 0, b / 0 there is no constant solution of equation (5)). So, the linearity of equation (5) implies that the general solution of equation (5) takes the form

(7) Fn = D{ 1 -f k i)“ n(l + fini ) — r,

where D is an arbitrary constant. Now to find that the solution of equation (5) with the initial condition

=

has the form (6 ) it is enough to specify the arbitrary constant D in such a wav that the initial condition is satisfied, i.e.

D0 = D - ~ . b

? This implies the formula (6). ■

5. Finally, we are also able to wrrite a general formula for the solution of equation ( 1) with an arbitrary sequence in. Namely, we have

T h e o r e m 3. Under the conditions of section 4 with an arbitrary sequence in the solution of equation ( 1) with the initial condition F

=

is expressed by the formula

a n — 1 Ar(r+1)-1 n — 1

(8) Fn = Fb| J"J ( l + is'j ( l + — 0, 1, .. .

r=0 s=kr j = k a n

Here, as usual, we mean that

- l - l

n ^ 1, X / ; = 0-

s=0 j =0

P r o o f . Observe first that cv7n = a n, so

a „ - l k(r-j-l) — 1

- f 0 I ] ( l + Yl ll

r

—0

Nowr wre consider two cases:

1) n = a nk T j3n with 0 < /3n < k — 1 and consequently

On some difference-delay equations arising in a problem of capital deposits

79

^n+l — ^ni 3n+1 — 3n T 1. Then

&n —1 /c ( r + l) -l

Fn+l = To [ ( l + ^ h

r= 0 s=kr

oen— 1 kr+k — 1

1 + ^

3’ k ot n

n — 1

~ [ PI + X^ ?'s) ^ ^F X/ ijj F *nFln — Fn + ^inFlm

r=0 s—kr j — kan

2) n = a nk + (3n with /3n = k - 1. Then a n+1 = a n + 1, /3n+l = 0 and

a„ + i - l fc(r+l)-l

Fn+l = F0 [ n ( l + E

r=0

k r+ k—l

1 + 1 ] ii ) ~

3 — + l

r=0 a „ - l

s=kr kr+k —l

- F° i n (i+12 '*) ( i +

12

^{) -}

k ex n -f- k — 1

=ir° [ n ( i + x^ ^ ) . ( i + x i

j — kot n

r= 0 s=kr

because

Hence,

ij = 0 for /?< = a nk + k — 1.

j = ^(a„ + l)

a „ - 1 A;(r+1)-1

Tn+i = ^0 [ j j ( l + ^ *

r=0 s=kr

ctn— 1 kr+k — l + h a n+ k - i - F 0 J J ( l + ^ 2 f ) =

kcxn k— 2

! + T *j) +

j = k a r

r= 0 a „ - l kr+k— 1

s=kr

n — 1

- F 0 [ n ( i +

12

Y.

a n—1 k r+ k —l

+ in ' F

^1 + =

r—0 s=kr

— Fn T inF'yn . In this way the Theorem is proven. ■

6. Let us make some comments about the equation (1). We assume that

(9) 0 < i' < in < i.

Then the sequence Fn defined by equation ( 1) is nondecreasing and

< Fln < Fn,

because 0 < 7n < n. Using this inequality we find that

q q v

Fn + i'F

< Fn + inFo < Fn + inFln = Fn+1 =

— Fn + inFln < (1 + in)Fn < (1 + i)Fn From this inequality it follows that

n —1 n — 1

F

^1 + is'j < Fn < F

(1 + is).

s=0 s—0

The upper bound for Fn in the last inequality is just the amount which the man would obtain if the interest was compounded at the end of each quarter (or if the conversion period was one quarter instead of one year).

The lower bound for Fn in the same inequality represents the result for the conversion period equal to + 00. We also have the following inequality (11) F0(l + m') < Fn < ^0( 1 + *')"■

We see that for ir, in and i satisfying inequality (9) the last inequality gives us other (less sharp) estimates of the balance Fn. The upper and knver bounds for Fn in (11) represent the amounts the person would obtain if the interest was constant and equal to i' or i respectively and the conversion period was equal to +00 or one quarter respectively.

Finally, observe that equation (3) joins the two patterns according to which the interest is earned. Namely, if 0 < n < k, then formula (4) gives us

Fn = D0(l + n i),

which is the standard formula for calculating simple interest, because for such n a n = 0 and 0 n = n.

If k = 1, then for all natural n we have a n = n, (3n = 0 and from (4) we get

Fn = D 0(l + i)n,

which is the standard formula for calculating compound interest. The same holds for equation ( 1).

7. The results established above will help us to solve practical problems.

For instance, under the conditions of Theorem 1, when the basic time unit is a quarter and the interest 100i% is paid quarterly and compounded every four quarters, one can ask the following standard questions:

1) What will be the balance after two and a half years if 6000 US dollars

is deposited in bank B which offers the interest i = 4%?

On some difference-delay equations arising in a problem of capital deposits

81

To answer this question we fix n = 10 and apply formula (3) to get Fio = 6000(1 + 4 • 0.04)2( 1 + 2 • 0.04) = 8719.49 .

If the interest was compounded quarterly then, we would get Fio < F[o = 6000(1 + 0.04)10 = 8881.47.

But for the case when the conversion period is greater than 10 we get Fj'o = 6000( 1 + 10 • 0.04) = 6000 • 1.4 = 8400 < F10 = 8719.49

< F10 = 8881.47 .

2) What is the present value of 10000 US dollars that we will receive in three years if the interest offered by bank B is equal to 5%?

To answer this question we use formula (3) with n = 12, F\i — 10000.

We have

(1 + 4-0.05)‘^ (

+ /312.

.05)' Since 012 = 3, (312 = 0, we obtain

_ 10000

~ ( W ^5787.04.

3) What is the present value of 10000 US dollars that we will receive in three and a half years if the interest offered is as above, i.e. / = 0.05?

Now, n — 14, F14 = 10000. Thus

(T + 4 • 0.05)ai4 (1 + j3\4 ■ 0.05) and as « i 4 = 3, /3

= 2 we have

10000 10000

(1 -2)3( 1 + 0.1) ” ( 1.2)3( 1-1) 5260.94 .

4) What interest should bank B offer so that a deposit of 8000 US dollars will amount to 11000 US dollars in two and a half years?

To answer this question we first fix n =■ 10, D

— 8000, F\0 = 11000.

Then we solve the equation

11000 = 8000(1 + 4*)2(1 + 2/')

with respect to i. Using, for example, the computer program Mathematica and executing the operation: Roots [ 128/3 + 128r -fi 40/ — 1.5 = = 0, /] we find i = 0.0337353.

We would like to stress that we are able to answer these standard ques-

tions due to formula (4) established in Theorem 1. The other theorems can

be used in a similar way.

References

[1] M.V. B u tch er, C.J. Nesbitt, Mathematics of Compound Interest, Edwards Brothers, Inc., Ann Arbor, Michigan, 1971.

[2] P.C. C a rtleg e, A Handbook of Financial Mathematics, Euromoney Publication PLC, London, 1991.

[3] H.U. G erber, Life Insurance Mathematics, Springer-Verlag, Zurich, 1990.

[4] G. G ilb e rt, D. K o e h le r, Applied Finite Mathematics, McGraw-Hill, 1984.

[5] D. Maki, M. T h o m s o n , Finite Mathematics, McGraw-Hill, 1983.

[6] S. O s ta s ie w ic z ,W . R o n k a -C h m ie lo w ie c , Elements of Financial and Insurrance Calculations, Akademia Ekonomiczna Wroclaw, 1993 (in Polish).

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