Time value of money
The time value of money is a fundamental concept in finance that recognizes the potential change in the value of money over time. It is based on the premise that a dollar today is worth more than the same dollar in the future. The concept takes into account the earning potential of money when it is invested or used to generate returns over a given period.
The underlying principle of the time value of money is that money has the potential to grow through investment or earn interest over time. Therefore, a certain amount of money received or invested today is considered more valuable than an identical amount received or invested in the future.
The time value of money is influenced by several factors:
1. Opportunity cost:
By holding money today, you have the opportunity to invest it or earn interest,which means you are sacrificing potential returns by not doing so.
2. Inflation:
Inflation reduces the purchasing power of money over time. Therefore, the same amount of money in the future will generally buy fewer goods and services compared to today.
3.Risk and uncertainty:
Future cash flows are uncertain, and there is a risk associated with receiving money in the future. The time value of money accounts for this risk by considering the certainty of receiving money today.
04. Annuity: A series of periodic payments or receipts.
05. Ordinary Annuity: An annuity where payments are made at the end of each period.
06. Annuity Due: An annuity where payments are made at the beginning of each period.
To evaluate the time value of money, various financial calculations are used, such as present value (PV), future value (FV), and interest rates. These calculations help determine the current worth of future cash flows or the future value of an investment based on factors like interest rates and time periods.
The Usual Calculations in Time Value of Money (TVM)
1. Future value of a single sum
2. Present value of a single sum
3. Number of years or compounding periods
4. The interest rate earned per year or per compounding period
5. Present value of an Ordinary Annuity
6. Present value of an Annuity Due
7. Future value of an Ordinary Annuity
8. Future value of an Annuity Due
9. Periodic payments
"Abbreviation”
1. TVM =Time value of money
2. PV = present value
3. FV = Future value
4. I = Interest rate of return(interest rate)
5. N = number of compounding periods each year
6. T = number of years
7. R = return or interest rate per period(Typically 1 year)
8. PMT = payment Amount
9. CFn = Cash flow steam number
10. M =
Multi period compounding
11.EIR = Effective interest rate
12. A = Amount of installment
13.It will be followed by the payment of TK:
=This means amount payment consider as down payment & must be added to the total PV value determination.
Let’s starts with some formulas of Time value of money
10. According to Rule -69, N = 0.35+ 69/ r, r = 0.35+ 69/ n,
11.PV for a Perpetuity:
PV = C/R [ Here, C= Perpetual Cash Flow, r = Rate of Interest ]
12. PV for a Growing Perpetuity :
PV = C/r-g
Let’s starts with some practical example of Time value of money
Practical Example :(01)
MR, Karim, deposits Tk.5,00,000 today in bank. The bank promise to give 12% interest per annul. What will be the future value after 7 years in compounded annually?
Ans:
Here,
PV =Tk.5,00,000, r = 12%,or, 0.12 n =7 years ,FV =?
We know that,
FV = PV (1 + r) n
= 5, 00,000(1 + 0.12) 7
= 5,00,000*2.21068
= Tk.11, 05,341.00
Practical Example :(02)
Mr.X rahman . Wishes to deposit Tk.15,00,000 in a bank account at the rate of 10% compunded interest.
Rewuirement:
(I).How much money would be deposited at the end of 03 years?
Ans:-
FV = PV (1 + r) n
= 15,00,000(1 + 0.10)3
= 15,00,000 *1.331
= Tk.19,96,500(Ans)Practical Example :(03)
South East Bank LTD. Pays 12% interest on deposit which is compounded quarterly .If Tk.50, 000 is deposited today, fined the amount it will stand at the end of 8 years.
Ans:-
FV = PV (1 + r) n
= 50,000(1 + 0.03) (4*8)
= 50,000*2.5751
= Tk.1,28,754.00
Practical Example :(04)
You have a single amount of Tk.20,000 for 3 years at 8% p.a. At the end of 3 years you take the proceeds and invest them for 10,Years at 10% p.a. How much will you have after 13 years?
Here,
Pv=20,000,r =8% or,0.08 n=3 years ,FV=? Again, PV=25194 r=10% or,0.10 N=10 FV= ?
For first 03 years:
FV = PV (1 + r) n
= 20,000(1+0.08)3
= Tk.25,194 (Ans) Again for first 10 years:
FV = PV (1 + r) n
= 25,194 (1+0.10)10
= Tk.65,346 (Ans)Practical Example :(05)
Your father will need Tk.3,00,000 after 5 years from now. How much money should he deposit today to get the money at 12% compounded interest quarterly?
Ans:
PV = FV/ (1+r) n
PV = 300000/ (1+0.03) 4*5
PV = 300000/ (1+0.03) 20
PV = Tk.1,66,102.72
Practical Example :(06)
What will be the present value of Tk.6,000 to be received at the end of 6 years if the discount rate is 12%
Ans:
PV = FV/ (1+r) n
= 6000 / (1+0.12) 6
= Tk.3039.78
Practical Example :(07)
You are planning to buy a car next year that will cost Tk.1,00,000 at that time. If the interest rate is 18% how much should you set aside now and to de posit in a bank in order to pay for the car purchase.
Ans:
PV = FV/ (1+r) n
= 100000/ (1+0.18)*1
= Tk.84,746
Practical Example :(08)
You have opened a DPS at the Sonali bank. Your monthly deposit isTk. 500. If bank pays 12% compounding interest monthly. What will be the future value of you deposit after 10 years.
Ans:
FV=A {(1+r)n-1/ r } [ Assume Installment at the end of the month]
= 500 { (1+0.01)120-1/ 0.01}
= 500*230.04
=Tk.1,15,019
Practical Example :(09)
Kalpana opens a DPS account at sonali bank at Dhaka branch on 1st january2003 she deposit Tk100 at the beginning of every week into his DPS account . Sonali bank offers 12.50% interest compounded weekly .How much kalpana received on maturity. Assume that DPS will matured on 31st December 2017)
Ans:
Here,
A=Tk.100, m =52,r =12.50% or, 0.125/52 =0.0024, n=15*12=780 , FV=?
FV=A {(1+r)n-1/ r }(1+r) [When Installment at the beginning of the year]
FV=100 { (1+0.0024)780-1/ 0.0024}(1+0.0024)
=100*2291.62
=TK.2,29,162
Practical Example :(10)
You have a plan to buy a motorcycle after 6 years from now. The price of the motorcycle will be Tk.6,00,000 at that time. you want to accumulate this money in a bank account by monthly installment, which gives you 12% interest. What is your monthly installment now?
FV=Tk.6,00,000, m =12 , r =12.% or, 0.12/12 =0.01, n=6*12=72 , A (Amount of installment)=?
Ans:
FV=A {(1+r)n-1/ r }[When Installment at the end of the year]
Or, 600000=A {(1+0.01)6*12-1/ 0.01 }
Or, 600000=A {(1+0.01)72-1/ 0.01 }
Or, 600000=A (104.71)
Or, A= 600000/104.71
Or, A= TK.5730
Practical Example :(11)
You have a plan to accumulate Tk.12,00,000 ten (10) years from now for buying a car .you are deciding to deposit a certain amount of money in special account that will provide you 9.50% interest per Annam.
1. What will be your annual deposit at the beginning of year?
2. What will be your annual deposit at the end of year?
3. What will be your half –yearly deposit?
Ans:01
FV=A {(1+r)n-1/ r }(1+r) [When Installment at the beginning of the year]
Or, 12,00,000=A {(1+0.095)10-1/ 0.095 }(1+0.095)
Or, 12,00,000=A *17.0385
Or, A= 12,00,000/17.0385
Or, A= TK.70,431
Ans.02
FV=A {(1+r)n-1/ r } [When Installment at the end of the year]
Or, 12,00,000=A {(1+0.095)10-1} / r
Or, 12,00,000=A *15.5603
Or, A= 12,00,000/15.5603
Or, A= TK.77,120
Ans.03
FV=A {(1+r)n-1/ r } [When your half –yearly deposit?]
Or, 12,00,000=A{(1+0.0475)20-1/ 0.0475
Or, 12,00,000=A *32.2056
Or, A= 12,00,000/32.2056
Or, A= TK.37260
Practical Example :(12)
The PV=A {1-1/(1+r)n/r }(1+r) [When Installment at the beginning of the year]
PV=3000 { 1-1/(1+0.02)20/0.02 }}(1+0.02)
Or, PV=3000*16.6784
Or, PV= TK.50035
Practical Example :(13)
Wimax housing ltd. granted you Tk.40,00,000 as loan to buy a flat at Dhanmondi. The interest rate 12%. The loan should be paid in equal monthly installment for next 20 years. How much should be paid in each installment.
Ans:
PV=A {1-1/(1+r)n/r } [When Installment at the end of the year]
Or, 40,00,000= A {1-1/(1+0.01)240/0.01 }
Or, 40,00,000 =A*90.8194
Or, A= 40,00,000/90.8194
Or, A= TK.44,043
Practical Example :(14)
You have taken a loan amount of Tk. 30,000 from IFIC bank. You will have to pay the loan with 5% interest by 5 equal annual installments. Prepare loan amortization schedule. (Assume 1st payment is made at the end of the year.)
Ans:
PV=A { 1-1/(1+r)n/r}[When Installment at the end of the year]
Or, 30,000= A { 1-1/(1+0.05)5/0.05}
Or, 30,000=A*4.3294
Or, A= 30,000/4.3294
or, A= TK.6,929
Loan amortization schedule
|
Year (01) |
Beginning loan (2) |
Installment (03) |
Interest (04) 5%*(2) |
Principal(5)(3-4) |
Loan (liability)(2-5) |
|
01 |
30,000 |
6929 |
1500 |
5429 |
24571 |
|
02 |
24571 |
6929 |
1229 |
5700 |
18871 |
|
03 |
18871 |
6929 |
944 |
5985 |
12886 |
|
04 |
12886 |
6929 |
644 |
6285 |
6601 |
|
05 |
6601 |
6929 |
328*(6929-6601) |
6601
|
0 |
Practical Example :(15)
You have taken a loan amount of Tk. 30,0,000 from Dutch bangle Bank bank. You will have to pay the loan with 10% interest by 6 equal annual installments. Prepare loan amortization schedule. (Assume 1st payment is made at the Beginning of the year.)
Ans:
PV=A { 1-1/(1+r)n/r}(1+r) [When Installment at the beginning of the year]
Or, 300000 = A { 1-1/(1+0.10)6/0.10}(1+0.10)
Or, 300000=A*4.7908
Or, A= 300000/4.7908
Or, A= TK.62,620
Loan amortization schedule
|
Year (01) |
Beginning loan (02) |
Installment (03) |
Interest (04) |
Principal(5) |
Loan (liability) |
|
0 |
3,00,000 |
62,620 |
0 |
62,620 |
2,37,380 |
|
01 |
2,37,380 |
62,620 |
23,738 |
38,882 |
1,98,498 |
|
02 |
1,98,498 |
62,620 |
19,850 |
42,770 |
1,55,728 |
|
03 |
1,55,728 |
62,620 |
15,573 |
47,047 |
1,08,681 |
|
04 |
1,08,681 |
62,620 |
10,868 |
51,752 |
56,929 |
|
05 |
56,929 |
62,620 |
5,691* (62,620-56,929) |
56,929 |
0 |
Practical Example :(16) (CMA -Exam-May-2023)
RD Foods LTD borrowing Tk.5,00,000 to finance a project involving an expansion of its existing factory. It has obtained an offer from City Bank. The terms of the loan facilities are as follows:
Annual interest rate :22%
Duration =2 years
Interest method=quarterly compounded interest
payment plan=equal installments at the end of each quarter
Requirement:
(i) Compute the quarterly installment
(ii)Prepare a loan amortization schedule to show the period interest charges ,installment payments,principle payments,and balances of the loan at the end of each quarter.
Here,
PV =Tk.5,00,000
Frequency ,M=4
Term (in years) = 2
No.periods N=(2*4) = 8 years
Interest rate = 22% or 0.22 ,Quarterly = (0.22/4) =0.055
Ans:
Req-(i)
We know that ,
PV=A { 1-1/ (1+r)n/r } [ When Installment at the end of the year]
Or, 5,00,000 = A { 1-1/(1+0.055)8 /0.055}
Or, 5,00,000 =A*6.33456590 [by using calculator]
Or, A= 5,00,000 /6.33456590
or, A= TK. 78932.01(Ans)
Now we calculate Loan amortization schedule,
Req-(ii)
|
Period (01) |
Beginning loan (2) |
Installment (quarter)(03) |
Interest (04) (quarter) 5.50%*(2) |
Principal (5)(3-4) |
Out standing balance(2-5) |
|
01 |
5,00,000 |
78932.01 |
27500 |
51432.01 |
448567.99 |
|
02 |
448567.99 |
78932.01 |
24671.24 |
54260.77 |
394307.23 |
|
03 |
394307.23 |
78932.01 |
21686.90 |
57245.11 |
337062.12 |
|
04 |
337062.12 |
78932.01 |
18538.42 |
60393.59 |
276668.53 |
|
05 |
276668.53 |
78932.01 |
15216.77 |
63715.24
|
212953.29 |
|
06 |
212953.29 | 78932.01 |
11712.43 |
67219.57
|
145733.72 |
|
07 |
145733.72 | 78932.01 |
8015.35 |
70916.65 |
74817.07 |
|
08 |
74817.07 | 78932.01 |
4114.94 |
74817.07
|
- |
Practical Example :(17)
Compute the Total present value of the following
cash inflows(uneven cash flow but non sequential period)
Tk.10,000 Today, Tk.12,000 after one year, Tk. 15,000 after two years,Tk.10,000 after five years and Tk. 20,000 after 9 years. Assuming the discount rate is 10%.
Here,
FV1=10,000, FV2=12,000, FV3=15,000, FV4=10,000, FV5 =20,000, r=10% or 0.10 n =0,1,2,5,9
We know that,
PV= FV1/(1+r)n + FV2/(1+r)n+FV3/(1+r)n+FV4/(1+r)n+FV5/(1+r)n
PV= 10000/(1+0.10)0 + 12000/(1+0.10)1+15000/(1+0.10)2+10000/(1+0.10)5+20,000/(1+0.10)9
PV= 10000+10909+12397+6209+8482
PV=TK .47997
Practical Example :(18)
Determined the number of years to be required for initial deposit of Tk.15,000 grow at Tk.25,000 at annual rate 12.50%.
Ans:
We know that,
FV = PV (1 + r) n[PV= principle amount]
Or, 25000=15000(1+0.125)n
Or, 25000/15000= (1.125)n
Or, 1.667 = (1.125)n
Or, log1.667= nlog(1.125) (Taking log in both side)
Or, n= log1.667/log 1.125
Or,n=4.34 years
Practical Example :(19)
At what time will a sum of TK.5,000 will be doubled at 5% compound interest.
We know that,
FV = PV (1 + r) n [PV= principle amount]
Or,10,000 = 5,000(1+0.05)
Or, 10000/5000 = (1.05)n
Or, 2 = (1.05)n
Or log 2 = n log(1.05)
Or, n = log2/log 1.05
Or, n=14.21 years.
Practical Example :(20)
Taking a loan of Tk.9,000 from a bank, a man was unable to pay it till the end of 4 years. Thus the bank demanded Tk. 11,250 from him. How much % of interest compounded yearly on the demanded money he was paid?
Ans:
We know that,
FV = PV (1 + r) n [PV= principle amount]
Or,11,250 =9,000(1+r)4
Or, 11,250/9,000 =(1+r)4
Or, 1.25=(1+r)4
Or, (1+r)4=1.25
Or,1+r =4√1.25 [4 Then shift , then ^1.25 & then Enter)
Or r= 1.0573-1
Or, r=0.0573
Or, r=5.73%(Ans)
Practical Example :(21)
You want to double your deposit at 10% interest. What will be the period according to the Rules of 72?
ANS:
We know that, According to Rule-72
N =72/r
N =7/10
=7.2 years
Practical Example :(22)
According to Rule-69
You want to double your deposit at 12% interest rate. How much time it will take according to the Rules of 69?
ANS:
We know that, according to Rule-69
N =0.35+69/r
N =0.35+69/12
=6.1 years
Practical Example :(23)
Proof: Compute the required interest rate to double your amount by 8 years according to rule of 69. And prove your calculation assuming the principle amount of Tk.20, 000.
ANS:
We know that, according to Rule-69
r =0.35+69/n
r=0.35+69/8
=8.98%
Proof
We know,
FV = PV (1 + r) n
FV = 20000 (1 + 0.0898) 8
FV= TK.39,792 (Approx)
(Proof)…
Practical Example :(24)
The nominal interest rate on a loan is 12% per year. Compute the EIR (Effective interest rate)if
a) The interest is compounded annually
b) The interest is compounded semi-annually
c) The interest is compounded quarterly
d) The interest is compounded monthly
ANS:
a) The interest is compounded annually
we know that,
EIR= { (1 +r/m)m-1} (100)
= { (1 +0.12/1)1-1} (100)
= { (1 +0.12)1-1} (100)
= { (1.12)1-1} (100)
=12%
b) The interest is compounded semi- annually
EIR= { (1 +r/m)m-1} (100)
= { (1 +0.12/2)2-1} (100)
= { (1 +0.06)2-1} (100)
= { (1.06)2-1} (100)
=12.36%
c) The interest is compounded quarterly
EIR= { (1 +r/m)m-1} (100)
= { (1 +0.12/4)4-1} (100)
= { (1 +0.03)4-1} (100)
= { (1.03)4-1} (100)
=12.55%
d) The interest is compounded monthly
EIR= { (1 +r/m)m-1} (100)
= { (1 +0.12/12)12-1} (100)
= { (1 +0.01)12-1} (100)
= { (1.0.01)12-1} (100)
=12.68%
Practical Example :(25)(CMA Exam-September-2023)
Mr.karim has decided to start saving for his retirement. Beginning on his twenty first birthday, he plans to invest Tk.2,000 each birthday into a savings investment earning a 7 percent compounded annual rate of interest. He will continue this savings program for a total of 10 years and then stop making payments. But his savings will continue to compound at 7 percent for 35 more years, until karims retires at age 65.
On the other hand,Mr. Rahim also plans to invest Tk.2,000 a year on each birthday, at 7 percent, and will do so for a total of 35 years. However, he will not begin his contributions until his thirty first birthday. How much will Karims and Rahims savings programs be worth at the retirement age of 65? Who is better off financially at retirement, and by how much?
Answer: (Part-Mr. Karim)
Here.
Interest (r) = 7%, or, 0.07
A = 2000
N = 10 years
1st step:
At first we should calculate the FV value of 2,000 each installment over the period 10 years
We know that,
FV = A {(1+r) n-1/ r} [Assume Installment at the end of each year]
= 2,000 {(1+0.07)10-1} / 0.07
= 2,000 {(1.07)10-1} / 0.07
= 2,000*13.816
= Tk.27632
2st steps:
Since after 10 years Mr.Karim stop the installment at the end of each the year , Now Tk.27,632(calculate above) is the base value will re -consideration for continue to compound interest at 7 percent for 35 more years, until Mr.Karims retires at age of 65.
Here.
Interest (r) = 7%, or, 0.07
A = 27,632 (consider a single deposited for next 35 years)
N = 35 years
Then, recently Future value(FV) under re-consideration over the next 35 years is -
We know that,
FV = PV (1 + r) n
= 27,632 {(1+0.07)35} (consider a single deposited for next 35 years)
= 27,632 {(1.07)35}
= 27,632*10.6770049
= Tk.2,95,027(round).
Answer: (Part-Mr. Rahim)
We know that,
FV = A {(1+r) n-1/ r} [Assume Installment at the end of each year]
= 2,000 {(1+0.07)35-1} / 0.07
= 2,000 {(1.07)35-1} / 0.07
= 2,000*138.237
= Tk.2,76,474.
Comments section: -
Mr karims investment program is worth(Tk2,95,027-Tk.2,76,474)=Tk.18,553 more at retirement than MR.Rahims program.
Practical Example :(26) (CMA Exam-January-2023)
The director of “corporate practice bd Ltd,” are considering two payments option for the purchase of a new cotton processing plant to produce Yarn.
Option (1) - (Cash purchase option)
This option requires dimidiate payment of the full price of the plan. If the corporate practice bd Ltd chooses this option, it will pay the cash price of Tk.8,00,379 today. The bd Ltd. Plans to raise the required amount by borrowing from City bank, the bank has offer to lend to meet cash price to Bd Ltd at an annual interest rate of15% with monthly compounding. The loan, interest & other charges are to be amortized by continuing even installments of TK.27,952.26 each made at the end of each month over the next three years.
Option (2) - (Credit purchase plan)
Under this option vendor requires an immediate down payment followed by a series of even payments. If the corporate practice bd Ltd chooses this option, it will be required to pay of Tk.50,000 today. This will be followed by the payment of Tk.1,16,100 at the end of each quarter over the next two years. The annual interest rate implicit in this credit purchase plan is 20% per annum.
Requirement:
(i). Find the Present value of all the payments under the cash purchase option.
(ii). Find the Present value of all the payments under the credit purchase option.
(iii). Which of the two options do you recommended to the company? Explain.
Answers:
Req-(i) Part –Cash purchase option
Here.
Annual Interest (r) = 15%, or, 0.15, or by monthly= (0.15/12) = 0.0125
A = Tk.27,952.26
Frequency(m) = 12,
Term (years)= 3 years, So, Number of period(n)=3*12 =36
Let’s start to apply formula,
We know that,
PV= A {1-1/(1+r) n/r} [When Installment at the end of the year]
= 27,952.26{1-1/ (1+0.0125)36/0.0125}
= 27,952.26 { 1-1/(1.0125)36/0.0125}
=27,952.26 *28.84726737
=8,06,346.318
Answers:
Req-(ii) Part –Credit purchase option
Here.
Annual Interest (r) = 20%, or, 0.20, or by quarterly= (0.20/4) = 0.05
A = Tk.1,16,100
Frequency(m) = 4,
Term (years)= 2 years, So, Number of period(n)=4*2 =8
We know that,
PV = A {1-1/(1+r) n/r} [When Installment at the end of the year]
= 1,16,100{1-1/ (1+0.05)8 /0.05}
= 1,16,100 {1-1/ (1.05) 8/0.05}
= 1,16,100*6.463212759
= 7,50,379
PV of payments = 50,000+7,50,379
=TK.8,00,379
Req-(iii) Part –recommendation section
We are, recommended option is the credit purchase plan, since its PV is lower than the cash purchase option.
Practical Example :( 27 ) (CMA Exam-September-2022) (FIM)
You may choice of two annuities, Annuity X & Annuity Y. where X is an annuity due with cash inflow of Tk.9,000 for each of 6 years.
On the other hands,” Y” is an ordinary annuity with a cash inflow of TK.10,000 for each of 6 years.
Assume that you can earn 15% on your investments.
Requirement:
(i). On purely subjective basis, which annuity do you thinks is more attractive? Why?
(ii). Find the future value at the end of 6 years for both annuities
(iii). Use your finding in part(ii) to indicate which annuity is more attractive? Why? Compare your finding to your subjective response in part(i)
Answers:
Req-(i)
On the surface, annuity Y looks more attractive than X because it provides Tk.1,000 more each year than does annuity X.
Req-(ii) Part – for “X”
Here,
Annual Interest (r) = 15%, or, 0.15,
A = Tk.9,000
Frequency(m) = 1,
Term (years)= 6 years, So, Number of period(n)=6*1= 6
Let’s start to apply formula,
We know that,
FV = A {(1+r) n-1/ r} (1+r) [When Installment at the beginning of the year]
FV= 9,000 {(1+0.15)6 -1/ 0.15} (1+0.15)
FV= 9,000{(1.15)6 -1/ 0.15} (1.15)
FV= 9,000 *10.0667992
FV =Tk.90,601
Req-(ii) Part – for “Y”
We know that,
FV = A {(1+r) n-1/ r} [When Installment at the end of each year]
FV= 10,000{(1+0.15)6 -1/ 0.15}
FV= 10,000{(1.15)6 -1/ 0.15}
FV= 10,000*8.753738438
FV =Tk.87,537.3843
Req-(iii)
Annuity X is more attractive because its future value at the end of 6 years TK.90601 is greater than annuity Ys end of future value Tk.87537. 3843.The subjective assessment in part(i) was incorrect. The benefit of receiving annuity X, s cash inflow at the beginning of each year appears to have outweighed the fact that annuity Y, s annual cash inflow which occurs at the end of each year is Tk.1000 larger (Tk.10000 vs Tk.9000) than annuity X, s.
Practical Example :( 28 ) (CMA Exam-May-2022) (FIM)
The rule of “72” suggests that an amount will be double in 12 years at a 6% compound annual interest rate or double in 6 years at a 12% annual interest rate. Is this a useful rule and is it an accurate one?
Requirement:
Is this a useful rule, and is it an accurate one?
Answer:
For interest rates likely to be encountered in normal business situations the Rule of "72" is a pretty accurate money doubling rule. Since it is easy to remember and involves a calculation that can be done in your head , it has proven useful