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What is Growing Annuity and Deferred Annuity?

Growing Annuity:
 It is a fixed series of cash flow for fixed time, where initial cash flow is growing at a constant rate.

Deferred Annuity: It is a fixed series of payments which is received in some future date. For example in retirement plan amount invested into annuity a/c and after completion of fixed period, payments are received by plan holder.

Formulas:

Growing Annuity
Deferred Annuity
Present Value (PV0)
P/r-g[1-(1+g) n/ (1+r) n]
PVGA/(1+r) t
Future Value
P[(1+r) n-(1+g) n/r-g]
FV GA*(1+r) t
Present Value (PVAD)
P/r-g[1-(1+g) n/ (1+r) n] (1+r)
PV G AD /(1+r) t
Future Value  (FV AD)
P[(1+r) n-(1+g) n/r-g](1+r)
FV G AD *(1+r) t
Where,
P = cash flows
R = rate per period
G = growth rate
N= time period
T= deferred time period
PVG AD= Present Value of Growing Annuity Due
PVGA= Present Value of Growing Annuity
FVGA= Future Value of Growing Annuity
FVG AD=Future Value of Growing Annuity Due

Growing Annuity Problems:

Example 1: In pension plan employer provides fixed percentage of contribution to the employee‘s retirement. Employees must contribute Rs.6000 each year, 3% per year growing annually due to inflation and discount rate is 6% per annum. What is the pension cost of an employee’s who will retire after 40 years?
PVGA= C1/r-g [1-(1+g) n/ (1+r)n]
= 6000/0.06-0.03 [1-(1+0.03)40/(1+0.06) 40]
=136571.51

Example 2: Calculate future value of growing annuity whose initial cash flow is Rs.8000 each year and interest rate and growth rate is 6% and 3% respectively for 30 years.
FVGA= P [(1+r) n-(1+g) n/r-g]
= 8000[(1+0.06) 30-(1+0.03)30/0.06-0.03
=884327.65

Example 3: Find out the present value of annuity due if a person paid Rs.1, 000 annually for 5 years. The rate of interest is 5% annually and the growth rate is 2%.

Solution:
PVGAD = P/r-g [1-(1+g) n/ (1+r) n] (1+r)
= 1, 000 / 0.08 – 0.02 [1-(1+0.02) 5/ (1+0.08) 5] (1+0.08)
= 1, 000 / 0.06 [1-(1.10/ 1.47)] 1.08
= 16, 667 * 0.27
= 4, 500

Example 4: Suppose Mr. X has decided to invest in Company ABC shares and get dividend of Rs. 50 per share at the beginning of the year and he bought 100 shares. It is estimated that the dividend will increases in future 0.5% for 8 years. The discount rate of interest is 7.2%. Find out the future value of annuity for 8 years.

Solution:
FVGAD = P [(1+r) n-(1+g) n/r-g] (1+r)
= 5, 000 [(1+0.072) 8-(1+0.005) 8/0.072-0.005] (1+0.072)
= 5, 000 [(1.74 – 1.04) / 0.067] * 1.072
= 56, 012

Deferred Annuity Problems:

Example 1: Find out present value of deferred annuity that will pay Rs. 20000 per month for 5 years after being deferred for 4 years and rate of interest is 7% compounded monthly?
PVDA= P*[1-(1+r)-n/ r] / (1+r) n
= 20000 [1-(1+0.005833) -60 /0.005833] / (1+0.005833)48
= 76427.138

In case of Present Deferred Annuity Due:
PV Deferred Annuity Due =PVA*(1+r)/ (1+r) t
                       OR
= P * (1- (1 + r)-n / r) *(1+r)/ (1+r) t
= 1015931.77/ (1+1.005833) 48
= 768462.31  

Example 2: Suppose you deposit Rs. 6000 for 7 years starting 3rd year from now and interest is 6% compounded annually, what is the future value of deferred annuity in 11 year?
FVDA= P [(1+r) n-1/r]*(1+r) t
=6000 [(1+0.06) 7-1/0.06]*(1+0.06) 2
=56782.28

In case Future Deferred Annuity Due:
FV Deferred Annuity Due= FVAD* (1+r) t
= P*[((1+r) n – 1) / r] * (1+r) * (1+r) t
= 53384.81*(1+0.06)2

=59983.17

Example 3: Suppose Mr. Sharma has decided to invest Rs. 8, 000 each year payments starts from 2nd year from now @ 7.5% per annum and the annuity period is 15 years. The withdrawal of cash is at the beginning of the year. Find out how much amount he will get in future if there is 2% growth rate?
Solution:
FV G AD *(1+r) t
= P [(1+r) n-(1+g) n/r-g] (1+r) * (1+ r) t
= 8, 000 [(1+0.075) 15-(1+0.02) 15/0.075 - 0.02] (1+0.075) 2
= 8, 000 [(2.96 – 1.35) / 0.055] (1.16)
= 2, 71,625

Example 4: Find out the deferred growing annuity due with the help of following information:
·        Cash flow = Rs. 2, 500
·        Time = 14 years
·        Payment starts 5th year from now.
·        Growth rate = 0.16%
·        Discount rate = 5.9%

Solution:
PV G AD / (1+r) t
OR
PVGDA = P/r-g [1-(1+g) n/ (1+r) n] (1+r) / (1+r) t
= 2, 500/0.059-0.0016 [1-(1+0.0016) 14/ (1+0.059) 14] (1+0.059) 5
= 2, 500 / 0.0574 [1- (1.02 / 2.23)] (1.33)
= 43, 554 *0.72
= 31, 359


Comments

  1. UnknownNovember 16, 2017 11:43 am

    Wow! Your writing is really to the point, Keep your good work up. India business solution ( http://www.consultibs.in/ )provides all kind of solution to Corporate Consultant Companies for analysis market. Business strategy is our focus.

    ReplyDelete
  2. RobMarch 27, 2023 10:14 am

    This comment has been removed by the author.

    ReplyDelete
  3. RobMarch 27, 2023 10:17 am

    There are errors in the equations for Future Value. The equations are written as P[(1+r)^n-(1+g)^n /r-g] and the final bracket is misplaced. In addition the last /r-g needs parentheses around the r-g so that the subtraction will be performed before the division.
    It should be [(1+r)^n-(1+g)^n] / (r-g).

    The same issue is present in each of the examples that use these FV equations. In at least one instance there are unmatched parenthesis.
    I did not check the calculations to see if they were done using the incorrect order of operations as shown in the equations or - maybe the calculations are right and it is just a typesetting issue in the writing.

    ReplyDelete

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