Fiscal mathematics trades with the job of puting money or capital. If a company ( or an single investor ) puts some capital into an investing, he/she will desire a fiscal return for it.
E.g. : IF A puts Rs 1000 into history with a bank, he will anticipate a return in the signifier of involvement, which will be added to the original investing in his history ; If An endeavor invests Rs 10,000 in an point of equipment, the company will anticipate to do some kind of net income out of the undertaking over the on the job life of equipment.
Investors may wish to cognize:
How much return will be obtained by puting money now for n old ages?
How much return will be obtained in n old ages ‘ clip by puting some money every twelvemonth for n old ages?
Time is an of import component in investing determinations. The longer an investing continues, the greater will be the return required by the investor. For illustration, if a bank lends Rs 20,000 to a company, it would anticipate a bigger involvement payment if the loan last for two old ages than if it lasts for merely one twelvemonth.
This clip factor in investing determination has a direct relation to rising prices every bit good as the worsening value of money over a period of clip. The cost of involvement would increase for a longer period of investing even if there is no rising prices. The consequence of rising prices is merely to increase the size of the return required by the investor over any period of clip.
The two major techniques of fiscal mathematics are intensifying and discounting. These techniques are really closely related to each other, and are discussed hereunder.
Sn= P + nrP1. Simple Interest: Simple involvement is the sum if involvement, which is earned in an equal sum every twelvemonth ( or month ) , and which is a given proportion of the original investing or ( chief ) . If a amount of money is invested for a period of clip so the sum of simple involvement, which accrues, is equal to the figure of accounting periods * the involvement rate * the sum invested, i.e. ,
Where P=Original amount invested ;
r= Interest rate ; n=Number of periods ; S=Sum after n periods, dwelling of the original capital plus involvement earned.
E.g. how much will an investor have after 5 old ages if he invests Rs 1000 at 10 % simple involvement per annum?
Sol: S5=Rs 1000 + ( 5 x 0.1 tens Rs 1000 ) = Rs 1500
2. Compound Interest:
Interest is usually calculated by the agencies of intensifying. If a amount of money, the principal, is invested at a fixed rate of involvement and the involvement is added to the principal and no backdowns are made, so the sum invested grows at an increasing rate as clip advancements, because involvement earned in earlier periods itself besides earns involvement subsequently on.
E.g. Rs 2,000 is invested to gain 10 % involvement. After 1 twelvemonth, the original chief plus involvement will amount to Rs 2,200.
Therefore involvement for 2nd twelvemonth will be on Rs 2200. So involvement will be earned on involvement.
Formula for combination:
Sn = P ( 1+r ) N
Where P – original amount invested
r= involvement rate, expressed as proportion ( 5 % =0.05 )
n=number of periods
Sn=sum after n periods.
Rs 2000 invested at 10 % per annum for 3 old ages
S3 = 2000 ( 1+0.10 ) 3
= 2000 ( 1.10 ) 3 ( See the compound value tabular array for 3years and 10 % )
=2000 x 1.331 = Rs 2,662.
The involvement earned over 3 old ages is Rs 662.
IF the money were invested for 6 old ages, it would be deserving
Rs 2,000 ( 1.10 ) 6 = Rs 2000 x 1.772 = Rs 3544.
The involvement earned over 6 old ages is Rs 1544.
E.g. what would be the entire value of Rs 5,000 invested now?
A. after 3 old ages, if the involvement rate is 20 % per annum.
B. after 4 old ages, if the involvement rate is 15 % per annum.
C. after 3 old ages if the involvement rate is 6 % per annum.
Sol: S3 = 5000 ( 1+0.20 ) 3 = 5000 x 1.728 = Rs 8,640.
S4 = 5000 ( 1+0.15 ) 4 = 5000 x 1.749 = Rs 8,745.
S3 = 5000 ( 1+0.06 ) 3 = 5000 x 1.191 = Rs 5,955.
Inflation: The same intensifying expression can be used to foretell hereafter monetary values after leting for monetary value rising prices. For e.g. if we wish to cognize how much a skilled employee will gain per annum in 5 old ages clip, given that he earns Rs 8,000 p.a. now and rewards rising prices is expected to be 10 % p.a.
Formula: Sn = P ( 1+r ) N
= Rs 8,000 ( 1.10 ) 5
= Rs 12,888 p.a.
Withdrawal of capital or involvement
IF an investor takes money out of an investing, it will discontinue to gain involvement. Therefore, if an investor, A puts Rs 3,000 into a bank sedimentation history, which pays involvement at 8 % p.a. and makes no backdowns except at the terminal of twelvemonth 2, when he takes out Rs 1,000. What would be the balance in his history at the terminal of 4 old ages?
Original investing Rs. 3,000.00
Interest in twelvemonth 1 ( 8 % ) 240.00
Investing at the terminal of twelvemonth 1 3,240.00
Interest in twelvemonth 2 ( 8 % on Rs 3,240 ) 259.20
Investing at the terminal of twelvemonth 2 3,499.20
Less: Withdrawal 1,000.00
Investing at the start of twelvemonth 3 2,499.20
Interest in twelvemonth 3 ( 8 % on Rs 2,499.20 ) 199.94
Investing at the terminal of twelvemonth 3 2,699.14
Interest in twelvemonth 4 ( 8 % on Rs 2,699.14 ) 215.93
Balance at the terminal of 4 old ages 2,915.07
A better attack to the solution would be as follows:
Rs 3,000 invested for 2 old ages at 8 % would increase
In value to Rs 3,000 ( 1.08 ) 2 Rs. 3,498
Less: backdown 1,000
Rs 2,498 invested farther 2 old ages at 8 % would
Increase in value to
Rs 2,498 ( 1.08 ) 2 Rs 2,913
Changes in the rate of involvement
If the rate of involvement alterations during the period of an investing, the combination expression would be amended somewhat as follows –
Sn = P ( 1+r1 ) ten ( 1+r2 ) ( n-x )
Where, r1 = initial rate of involvement
ten = figure of old ages in which the involvement rate r1 applies
r2 = Next rate of involvement
n-x = equilibrating figure of old ages in which the involvement rate r2 applies
Illus: If Rs 8,000 is invested now, to gain 10 % involvement for 3 old ages and so 8 % p.a. in the subsequent old ages, what would be the size of entire investing at the terminal of 5 old ages?
Sol: Sn = P ( 1+r1 ) ten ( 1+r2 ) ( n-x )
= 8000 ( 1.10 ) 3 ( 1.08 ) 2
= 8000 ( 1.331 ) ( 1.166 )
= Rs 12415.568
Illus: An investor puts Rs 10,000 into an investing for 10 old ages. The rate of involvement earned is 15 % p.a. for the first 4 old ages, 12 % p.a. for the following 4 old ages and 9 % p.a. for the last 2 old ages. How much will the investing be worth at the terminal of 10 old ages?
Sol: Sn = P ( 1+r1 ) ten ( 1+r2 ) Y ( 1+r2 ) ( n-x-y )
= 10000 ( 1.15 ) 4 ( 1.12 ) 4 ( 1.09 ) 2
= 10000 ( 1.749 ) ( 1.574 ) ( 1.188 )
Illus: An point of equipment costs Rs 61,000 now. The rate of rising prices over the following 4 old ages is expected to be 16 % , 20 % , 15 % and 10 % . How much will the equipment cost after 4 old ages?
Sol: Rs 61,000 ( 1.16 ) ( 1.20 ) ( 1.15 ) ( 1.10 ) = Rs 107413.68
Semi-annual and other intensifying periods:
Semi one-year combination means that there are two intensifying periods within a twelvemonth. Very frequently the involvement rates are compounded more than one time in a twelvemonth. Savings establishments, peculiarly give compound involvements semi-annually, quarterly and even monthly.
Assume Mr Y places his nest eggs of Rs 1,000 in a biennial clip sedimentation strategy of a bank with 6 % involvement compounded semi – yearly. He will be paid 3 % involvement compounded over 4 periods – each of 6 months ‘ continuance.
How 3 % ?
6 % twice a twelvemonth for 2 old ages, i.e. 4 times wholly
So, 6 % divided by ( m ) 4 times = 1.5 % ten ( n ) 2 old ages = 3 % for each 6 months.
Formula: – Tin = P 1 + R manganese
= 1000 ( 1 + 0.06/2 ) 2 x 2
= 1000 ( 1 + 0.03 ) 4
= 1000 ( 1.03 ) 4
In the compound value table see 3 % in 4 twelvemonth column
= 1000 ten 1.125
= Rs 1125
( Difference between the two methods is due to rounding off done in intensifying tabular array )
Interest is paid in 4 equal installments after every 3 months.
E.g. : – How much does a sedimentation of Rs 5,000 grow to at the terminal of 6 old ages, if the nominal rate of involvement is 12 % and the frequence of combination is 4 times a twelvemonth? The sum after 6 old ages will be:
Sn = P 1 + R manganese
= Rs 5,000 ( 1 + 0.12/4 ) 4 x 6
= Rs 5,000 ( 1.03 ) 24
In the combination tabular array we will see in 3 % in 24th twelvemonth
= Rs 5000 x 2.033= Rs 10,165
Effective versus nominal rate
We have seen above that Rs 1,000 grows to Rs 1,060.90 at the terminal of a twelvemonth of nominal rate of involvement is 6 % and combination is done semi-annually. This means that Rs 1,000 grows at the 6.09 % .
A company may offer to investors 10 % p.a. involvement, pay 5 % every 6 months at a compound rate of involvement. The effectual one-year rate of involvement would be:
R = 1+ K m _ 1
Where R = effectual rate of involvement ; k = nominal rate of involvement ; m = frequence of intensifying per twelvemonth.
R = ( 1+ 0.1/2 ) 2 – 1
= ( 1.05 ) 2 – 1
= 0.1025 or 10.25 %
Similarly, if a bank offers to its depositors a nominal 12 % public address system, with the involvement collectible quarterly, the effectual rate of involvement would be 3 % compound every 3 months i.e.
Formula = ( 1+0.12/4 ) 4 – 1
= ( 1.03 ) 4 – 1
= 0.126 or 12.6 %
E.g. cipher effectual one-year rate of involvement:
a. 15 % compound quarterly
B. 18 % compound monthly
( Using reckoners )
R = ( 1+0.15/4 ) 4 – 1
= ( 1.0375 ) 4 – 1
= 0.1587 or 15.87 %
R = ( 1+0.18/12 ) 12 – 1
= ( 1.015 ) 12 – 1
= 0.1956 or 19.56 %
Future value of an Annuity
A general definition of an rente would be a changeless amount of money each twelvemonth for a given figure of old ages.
It is a watercourse of equal one-year hard currency flows.
Ordinary rente: Payments or grosss occur at the terminal of each period.
Suppose you deposit Rs 1,000 every twelvemonth in a bank for 5 old ages, involvement rate is 10 % compound, so value of this series of sedimentations at the terminal of 5 old ages will be: –
Rs 1,000 ( 1.10 ) 4 + 1,000 ( 1.10 ) 3 + 1,000 ( 1.10 ) 2 + 1,000 ( 1.10 ) 1 + 1000 ( 1.10 ) 0
= Rs 1,000 ( 1.4641 ) + 1,000 ( 1.331 ) + 1,000 ( 1.21 ) + 1,000 ( 1.10 ) + 1,000
= Rs 6,105
Formula = FVAn = A ( 1+k ) n-1 + A ( 1+k ) n-2 A ( 1+k ) n-3+…..+ Angstrom
= A ( 1+k ) n-1
Where FVAn = future value of an rente which has a continuance of n periods
A = changeless periodic flow
K = involvement rate per period
n = continuance of the rente
the term ( 1+k ) n-1 is referred to as the hereafter value involvement factor for an
rente ( FVIFAk, n ) . The value for this factor for combinations Ks and N is given in Table A.2 ( future value involvement factor for rente )
Illus: 4 equal one-year payments of Rs 2,000 are made into a sedimentation history that pays 8 % involvement per twelvemonth. What is the future value of this rente at the terminal of 4 old ages?
FVA4 = Rs 2,000 ( FVIFA8 % , 4 ) = Rs 2,000 ( 4.507 ) = Rs 9,014.
= Rs 2,000 ( 1.08 ) 3 + 2,000 ( 1.08 ) 2 + 2,000 ( 1.08 ) 1 + 2,000
= Rs 2,000 x 1.260 + 2,000 ten 1.166 + 2,000 ten 1.080 + 2,000
= Rs 2520 + 2332 + 2160 + 2000
= Rs 9,012.
Annuity DUE: When the one-year payments are at the beginning of each twelvemonth so a alteration in expression.
FVAn = A ( 1+k ) ( 1+k ) n-1
so merely excess is multiplying ( 1+k ) one more clip as payments are made at the beginning of each twelvemonth.
Present ( discounted ) value of a individual sum
A rupee today is worth more than a rupee to be received one, two or three old ages from now. Calculating the present value of future hard currency flows allows us to put all hard currency flows on a current terms so that comparings can be made in footings of today ‘s rupees.
For e.g. Which should you prefer – Rs 1,000 today or Rs 2,000 10 old ages from today? Assume that both amounts are wholly certain and your rate of involvement is 8 % p.a. ( i.e. you could borrow or impart at 8 % ) . Present worth of Rs 1,000 is easy – it is deserving Rs 1,000. However, what is Rs 2,000 received at the terminal of 10 old ages deserving to you today? Or we can inquire the inquiry what sum ( today ) would turn to be Rs 2,000 at the terminal of 10 old ages at 8 % compound involvement. This sum is called present value of Rs 2,000 collectible in 10 old ages, discounted at 8 % .
Finding the present value is merely the contrary of combination.
Sn = P ( 1+r ) N
P = Sn 1 N
The factor ( 1/ ( 1+r ) N ) is called the discounting factor or present value involvement factor ( PVIFk, N )
So we can rearrange the above equation as
P = Sn ( PVIFk, N )
= 2,000 ( PVIF8 % , 10 )
= 2,000 ( 0.463 )
= Rs 926
So eventually if we compare this present value sum of Rs 926 to the promise of Rs 1,000 to be received today, we should prefer to take Rs 1,000. In present value footings we would be better off by Rs 74 ( Rs ( 1,000-926 ) .
In the above job every rupee is handicapped to such an extent that each was deserving merely about 46 paisa. The greater the disadvantage assigned to a future hard currency flow, the smaller the corresponding present value involvement factor.
Present value of uneven series
A company has an option, whether to pass Rs 22,500 on an point of equipment, in order to obtain hard currency net income of:
Year 1 Rs 7,500
Year 2 Rs 10,000
Year 3 Rs 6,250
Year 4 Rs 1,250
If the company requires a return of 10 % public address system, is the undertaking worth while? Use Net nowadays value method.
The net present value is negative ; that means cost of undertaking is more than its return, so it will be cheaper to put elsewhere at 10 % than to put in the undertaking.
Present ( discounted ) value of an rente
To cipher the present value of a changeless one-year flow or rente, multiply the one-year hard currency flows by the amount of the price reduction factors for the relevant old ages.
For e.g. there is changeless one-year hard currency flow of Rs 8,000 p.a. for 5 old ages, twelvemonth 1 to 5 at 11 % involvement.
8,000 x 0.9001
Plus 8,000 tens 0.8116
Plus 8,000 tens 0.7312
Plus 8,000 tens 0.6587
Plus 8,000 tens 0.5935
= 8,000 ten 3.6951 = Rs 29,559
In the Present value involvement factor for an rente tabular array see the present value factor of Re 1 p.a. for 5 old ages at 11 % , which is 3.6951.
Multiply the one-year hard currency flow with the PVIFA.
So, Rs 8,000 x 3.6951 = Rs 29,559.
If we go by expression method
( 1+r ) n – 1
PVA = a R ( 1+r ) N
( 1+0.11 ) 5 – 1
= 8000 0.11 ( 1.11 ) 5
= 8000 ten 1.68506 – 1
= 8000 ten 3.6958
= Rs 29,566.