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Time Value of Money
Very Important topic for RBI Phase 2 as minimum 1 or 2 questions are always
asked in RBI PHASE 2 FM paper
Annuity: An annuity is any series of equal payments that are made at regular intervals.
Though the word "annuity" comes from the Latin for "yearly," the periods between
payments in an annuity can be just about anything -- years, months, weeks; it doesn't
matter as long as the interval is consistent. An annuity can also last for a short period --
say a few months -- or for decades. An annuity will be either an ordinary annuity or an
annuity due. The difference lies in the timing of each payment relative to the period the
payment covers.(RBI Phase 2 exam 2017)
Ordinary Annuity: With an ordinary annuity, the payment comes at the end of the
covered term. The typical home mortgage is an example of an ordinary annuity. When
you pay your mortgage on Sept. 1, for example, you're actually paying for the use of
your home (and the use of the lender's money) for August. You'll pay for September on
Oct. 1, and so on. Most annuities are ordinary annuities, which is why they're called
"ordinary." Other common examples include interest payments from bonds and
payments on installment loan.
“A mortgage is a loan in which property or real estate is used as collateral. The borrower
enters into an agreement with the lender (usually a bank) wherein the borrower receives cash
upfront then makes payments over a set time span until he pays back the lender in full.”
An Ordinary Annuity has the following characteristics:
• The payments are always made at the end of each interval
• The interest rate compounds at the same interval as the payment interval.
Annuity Due: In an annuity due, the payment comes at the beginning of the term. The
most familiar application of the annuity due is rent. When you pay apartment rent on
Sept. 1, you're paying for the use of the apartment in September. Unlike with a
mortgage, when your first payment typically isn't due until after your first full month in
the home, your first rent payment is due when you move in. Insurance premiums are
another common example of an annuity due; you pay today for coverage in the future.
In an annuity due, the payments occur at the beginning of the payment period.
Generally in RBI Examination questions are asked on ordinary annuity related only.
If question is asked on annuity due then we can solve the same also using this formula:
Annuity Due = Ordinary Annuity x (1 + i)
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Present Value of an Ordinary Annuity:
F.V. = P.V (1+ i/m)m*n
This formula is similar to compound interest formula that we have studied in
Mathematics in 8th or 10th class.
P.V. = Present Value
F.V. = Future value
i = the rate of return or interest rate etc. (expressed as fraction, e.g. 6 per cent = 0.06)
m = number of times per year that interest is compounded
n = number of years invested
Suppose annuity C is paid every year for n years:
Then P.V. of C in 1st year will be :
F.V = P.V. ( 1+ i/m)m*n
P.V. = C/(1 + i) :consider m =1 assuming interest is paid annually. n= number of years
invested
Then P.V. of C in 2nd year will be :
P.V. = C/(1 + i)2
Then P.V. of C in 3rd year will be :
P.V. = C/(1 + i)3
….
….
P.V. = C/(1 + i)n
The present value of an n-period annuity with payment C and interest i is given by:
C/(1 + i) + C/(1 + i)2 + C/(1 + i)3+ …………………. + C/(1 + i)n
This is similar to G.P.
a +ar+ar2+…………….+ arn
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Sum of G.P.= a ( 1 - rn )
(1-r)
For the above a = C/(1 + i)
C * { 1 – 1/(1+i)n} = C*{ 1 – 1/ (1+i)n} = C{ 1 – 1/ (1+i)n}
1+ i 1 - 1/(1+i) 1+ i * (1+i- 1) i
1+i
The present value of an N-period annuity with payment C and interest i is given
by: P.V. of Annuity = C*{ 1 – 1/ (1+i)n} / i also called PVAF (present value
annuity/interest factor).PVAF/PVIF
For calculating Compound Value Annuity Factor (CVAF) or Future value of an
annuity for payment/investment C for n periods and interest i is given by:
F.V. of an annuity = C{ (1+i)n – 1 } / i
Present value factor or Future value of an investment which will be paid after n
n
years: F.V. = P.V (1+ i) (Question in RBI Phase 2 exam 2017)
Present value factor
Q.1. Rs. 2,000 is invested at annual rate of interest of 10%. What is the amount
after 2 years if the compounding is done?
(a) Annually? (b) Semi annually? (c) Monthly? (d) Daily?
Solution:
(a) The annual compounding is given by:
F.V. = P.V (1+ i)n
N = 1, i= 0.10 and Present Value = 2000
= 2,000 (1.1)2 = 2,000 × 1.21 = 2,420
(b) For Semiannual compounding, n = 2x2 = 4, i = 0.1/2 = 0.05
FV= 2000 (1+ 0.05)4 = 2,000x1.2155 = 2,431
(c) For monthly compounding, n = 12 ⋅ 2 = 24, i = 0.1/12 = 0.00833
FV= 2000 (1+ 0.00833)12 = 2,000x1.22029 = 2440.58
(d) For daily compounding, n = 365 ⋅ 2 = 730, i = 0.1/(365) = 0.00027
FV= 2,000 (1.00027) = 2,000x1.22135 = 2,442.70
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Q.2.Similarly if we have to find out that how much amount is to be invested at
present at annual rate of interest of 10% to get rupees 3630 after 2 years if the
compounding is done annually?
Then
n
F.V. = P.V (1+ i)
P.V. = F.V. / (1+ i)n
P.V. = 3630/ (1 + 0.10)2 = 3000
Rs. 3000 have to be invested to get 3630 after 2 years @ 10%.
Future value of an annuity for payment/investment C for n periods and interest i
is given by:
F.V. of an annuity = C{ (1+i)n – 1 } / i
Q.3.Find the amount of an annuity if payment of Rs .500 is invested annually for 7 years
at interest rate of 14% compounded annually.
Here C = 500, n = 7, i = 0.14
F.V. of an annuity = 500 { (1+0.14)7-1} / 0.14 = 5,365.25
Q.4.If we have to find out that how much amount to be invested per year to get Rs
10730.50 after 7 years at a rate of 14% compounded annually.
Then
F.V. of an annuity or CVAF = C{ (1+i)n – 1 } / i
Here F.V. = 10730.50, n = 7, i = 0.14
10730.50= C*{ (1+0.14)7-1} / 0.14
C= 10730.50/10.7305 = 1000
Rs. 1000 are to be invested annually.
Q.5. Determine the present value of an annuity of Rs.700 each paid at the end of each
of the next six years. Assume an 8 per cent of interest per annum.
Present value of an annuity = C{ 1 – 1/ (1+i)n} / i
Here C = 700, n = 6, i = 0.08
Present value = 700{ 1 – 1/ (1+0.08)6} / 0.08 = 700X4.623 = 3236.10
Using these formulas we can solve questions asked in RBI related to bonds.
Bonds related numerical which are also being asked in RBI will be released soon.
