The value of money does not remain the same at all points of time. The money available at the present time is worth more than the same amount in the future. It has the potential to earn returns (or interest as the case may be).
Consider the following options, assuming there is no uncertainty associated with the cash flow: Receiving Rs.100 now & receiving Rs.100 after one month. All investors would prefer to receive the cash flow now. Rather than wait for a month, though the amount to be received has the same value. This preference is attributed to the following reasons:
Instinctive preference for current consumption over future consumption.
The Ability to invest the Rs.100 for a month like a bank account or deposit. It earn a return so that it grows in value to more than Rs. 100 after one month. Clearly, Rs.100 available now is not equivalent to Rs.100 received after a month. The value associated with the same sum of money received at various points on the timeline is called the time value of money (popularly known as TVM). The time value of money received in earlier periods as compared to that received in later time periods will be higher. Since most decisions in finance involve cash flows spread over more than one period (monthly, quarterly, yearly etc.). The time value of money is a key principle in financial decision-making.
Present value
Present value is the amount that you would pay today for a cash flow that comes in the future. It brings the future value down to today’s price. It is based on the basic principle of time value of money that value of money keeps reducing as time passes. There are two ways in which the present value can be calculated. If there is a future value that has been given then this can be brought to the present by discounting it by the rate of return. This will give an idea of what the value of the future amount is worth today.
PV = FV/(1+r)^n Where FV= Future Value
PV= Present Value
r = rate of return for each compounding period
n = number of compounding periods For a one time receipt,
PV is calculated as per the following formulae:
PV = C/(1+r)^n
In case of a regular cash flow the present value can be calculated by the following formula
PV = C * ((1-(1/(1+r)^n))/r) Where C is the regular cash flow
Calculation of math problem
For example, Rajesh is going to receive a sum of Rs 6,500 a year for the next 8 years at an interest rate of 7 percent. He would like to know whether he should take the cash flow or a lump sum now and what this would be worth?
In this case using the formula or the excel function the Present value would come to PV = 6500/(1.07)^1 + 6500 /(1.07)^2 + 6500 / (1.07)^3 + 6500 /(1.07)^4 + 6500 / (1.07)^5 + 6500/( 1.07)^6 + 6500 / (1.07) ^7 + 6500 / (1.07)^8=38813.44
The present value can also be arrived at using the formula for a regular receipt PV = 6500*((1-(1/(1.07)^8))/0.07)
On the other hand, there can be a future payment of Rs 50,000 that might be received after a period of 5 year earning 6 per cent but this need to be evaluated in light of how much an investor should have in hand today.
This can be calculated as PV = 50000/(1.06)^5 PV = 37362.91
Future value
Future value represents what something is worth at some point in the future. There can be various amounts for which such a future value might need to be calculated and this will give an idea of the erosion in value from the current period.
FV = PV (1+r)^n Where FV= Future Value
PV= Present Value
r = rate of return for each compounding period
n = number of compounding periods
Note that the rate of return for each compounding period has to be adjusted for the frequency of compounding.
For example, if an investment pays 8% interest p.a. compounded quarterly, then the applicable rate of return for each compounding period is 8%/4, or 2%. Or it can be different for different periods in future.
The number of compounding periods (n) refers to the periodicity with which interest is paid on the investment during the year. For example, the Post Office Monthly Income Scheme (MIS) pays interest every month, while the Senior Citizens Scheme pays every quarter.
The greater the frequency of compounding, the more often interest is paid on interest, and the greater are returns earned through compounding.
Math problems
Consider the following example. Rajesh invests Rs.5 lakhs in a 5 year bank deposit that pays 8% interest compounded annually. What is the interest he earns from the investment in the following three scenarios?
Situation 1- The interest is used to pay the college fees of his daughter and there is no compounding.
Situation 2- The cumulative option is chosen and the interest is paid at maturity i.e. interest is compounded yearly.
Situation 3- If the interest is instead compounded quarterly and he chooses the cumulative option.
Under Situation 1 The interest income earned is: Rs.5 lakhs x 8% x 5=Rs.200,000 There is no compounding benefit since the interest is taken out and used and not re-invested. This is also known the simple interest.
Under situation 2 The maturity value will be= 500,000 x (1+8%)^5= Rs.734,664 Interest income earned over 5 years
= Rs.734,664- Rs.500,000= Rs.234,664
interest income is higher because the interest earned each year is re-invested and earns interest too. This is the compounding benefit.
Under scenario 3 Here the interest is compounded quarterly so this requires the rate to be divided by 4 while the total quarterly period are 20 during the 5 years The maturity value will be= 500,000 x (1+(8%/4)^20) = 500,000 x (1+2%)^20= Rs.742,974 Interest income earned over 5 years = Rs.742,974- Rs.500,000= Rs.242,974
The interest income is higher than scenario 2 because the frequency of compounding is higher. The interest is paid each quarter and this earns interest for the remaining period
Rate of return
The rate of return is the percentage rate that is earned on a particular investment. There are times when the investor has just the amount that has been earned but this needs to be converted into a rate of return. This will enable proper comparison with other instrument and options that are present in the market and will aid in proper decision making too.
In financial markets, the time value of money is always taken into account. It is assumed that if an investment provides a series of cash inflows, they can be re-invested to earn a positive return.
CAGR
Alternatively, an investment that does not have intermediate cash flows, is assumed to grow at an annual rate each year, to be compounded every year to reach the final value.
The compounded annual growth rate (CAGR) of an investment is the underlying compound interest rate that equates the end value of the investment with its beginning value.
Consider the following formula for FV:PV (1+r)^n = FV. A sum of money at the current point in time (PV) grows at a rate of r over a period n to become a future value (FV). CAGR is the rate r, which can be solved as: CAGR is computed using the above formula, given a beginning and end value for an investment and the investment period in years.
Since FV and PV represent end and beginning values of the investment for which CAGR is to be computed, the formula for CAGR (in decimals, not %) can be written as: CAGR = ((End Value/Beginning Value) ^ (1/n)) – 1
Calculation
The resulting CAGR has to be multiplied by 100 to be expressed in percent terms. For example, consider an investment of Rs.100 that grows to Rs.120 in 2 years. In this case: End Value (or FV) = 120 Beginning Value (or PV) = 100 No. of years ‘n’ = 2
Substituting in the formula for CAGR we have: 120 = 100*(1+r/100) ^ 2 We consider that Rs.100 has grown to Rs.120 over a 2 year period at CAGR of r. Rearranging the terms and writing CAGR instead of r we get: 120/100 = (1+CAGR) ^ 2 CAGR = ((120/100) ^ (1/2)) – 1 CAGR = ((1.2) ^ (1/2)) – 1 = 1.095 – 1 = 0.095 = 9.5%.
CAGR is the accepted standard measure of return on investment in financial markets, except in case of returns that involve periods of less than one year. The following example shows how CAGR is computed for a mutual fund investment. Assume that Rs. 10.50 was invested in a mutual fund and redeemed for Rs. 12.25 at the end of 3 years. What is the compounded rate of return?
In this problem, Rs.10.50 grew at some compounded rate to become Rs.12.25 at the end of 3 years. To solve for the CAGR, we use the formula: CAGR = ((12.25/10.5) ^ (1/3)) – 1 = 5.27% The same formula may be applied for fractional compounding periods.
Consider this example: An investor purchased mutual fund units at an NAV of Rs.11. After 450 days, she redeemed it at Rs.13.50. What is her compounded rate of return, assume that it’s a non-leap year? In order to use the CAGR formula, period of 450 days has to be converted into years or 450/365 years. CAGR = ((13.5/11) ^ (365/450)) – 1 = 0.1807 = 18.07%
Periodic investments or pay-outs
There are a lot of areas where the investor will be making a regular or a periodic payment. The most common example is that of a loan, where there is a regular Equated Monthly Instalment (EMI) being paid out to the lender each month. It is essential to know the amount that would be paid, so that there can be a proper planning made of how the amount should be accumulated
Annuity
An annuity is a sum of money paid at regular periods, such as monthly, quarterly, annually. A common example of an annuity is pension.
Annuities can be of two types (1) Fixed annuity and (2) Flexible annuity.
Fixed Annuity means that fixed returns are received at regular periods. For instance, a fixed deposit with a bank paying 5.5 % p.a. on the investment for a predetermined term assured (for example, for the next 5 years).
Floating annuities are those in which the returns are benchmarked to inflation or index returns or any other return as specified in the indenture agreement at the time of buying. So, the annuities paid are not fixed, but change in line with the chosen benchmark.
Annuities are used extensively during retirement wherein there is the need for a regular cash flow and this is generated by several investment instruments. An annuity is referred to by various terms but the key point is that the feature of all these are that there is a regular sum of money being received.
Thanks for reading.
