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Given a chance, any rationally thinking individual would prefer to have an amount of money now, as compared to having the same at a later date. Why do you think we do this? This is because we give due weightage for ‘Time Preference for Money’ since we may otherwise lose the opportunity to earn additional income / interest.
Hence ‘Time Preference for money’ is considered very valuable due to the following reasons:
- Investment opportunities / opportunities cost and the element of cost
- Preference for current consumption to future consumption Inflationary trend
- Uncertainty
Due to the above reasons money loses value with time and hence a rupee on hand presently has a more value compared to a rupee receivable at a later date. Considering the interest and futuristic risk, the preference will be to have the even future value now itself, discounting the same at the rate of return. For instance, one may prefer Rs.100/- instead of Rs.105/- after one year. This means that the value of the higher amount of future is equivalent to a lesser present amount. In the example future Rs.105/- after one year is treated as equal to present Rs.100/-.
The principle of time value of money is the notion that a given sum of money is more valuable the sooner it is received, due to its capacity to earn interest. Central to the time value principle is the concept of interest rates. A borrower who receives money today for consumption must pay back the principal plus an interest rate that compensates the lender.
Interest rates are fixed in the marketplace and allow for equivalent relationships to be determined by forces of supply and demand. In an environment where the market-determined rate is 10%, it would be said that borrowing (or lending) Rs. 1,000 today is equivalent to paying back (or receiving) Rs. 1,100 a year from now.
The ‘stated annual rate’ (quoted rate) is the interest rate on an investment if an institution were to pay interest only once a year. In practice, institutions compound interest more frequently, either quarterly, monthly, daily and even continuously. However, stating a rate for those small periods would involve quoting in small fractions and would not be meaningful or allow easy comparisons to other investment vehicles. As a result, there is a need for a standard convention for quoting rates on an annual basis.
The ‘effective annual yield’ represents the actual rate of return, reflecting all of the compounding periods during the year. The effective annual yield (or EAR) can be computed given the stated rate and the frequency of compounding.
Financial decisions are to be made comparing the cash outlays / outflows and the benefits / cash inflows. A financial decision – financing or investment – taken today has implications for number of years in terms of cash flows. For a meaningful comparison, the variables – cash outflows and cash inflows at various points of time periods – shall be converted to a common period of time.
Effective annual rate (EAR) = (1 + Periodic interest rate) m – 1
Where,
m = number of compounding periods in one year
Periodic interest rate = (stated interest rate) / m
Example: Given is a stated interest rate of 9%, compounded monthly. Hence, EAR would be.
EAR = (1 + (0.09/12))12 – 1 = (1.0075) 12 – 1 = (1.093807) – 1 = 0.093807 or 9.38%
The effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in the formula), and the more frequent the compounding, the higher the EAR.
When solving Time Value of Money problems, one should first convert both the rate ‘r’ and the time period ‘N’ to the same units as the compounding frequency. If the problem specifies quarterly compounding (i.e. four compounding periods in a year), with time given in years and interest rate is an annual figure, one should start by dividing the rate by 4, and multiplying the time N by 4. Then, use the resulting r and N in the standard Present Value (PV) and Future Value (FV) formulas.
Example: (Compounding periods)
The future value of Rs. 10,000 five years from now is at 8%, but assuming quarterly compounding, there is quarterly r = 8%/4 = 0.02, and periods N = 4*5 = 20 quarters.
FV = PV * (1 + r)N = (10,000)*(1.02)20 =
(10,000)*(1.485947) = $14,859.47
Assuming monthly compounding, where r = 8%/12 = 0.0066667, and N = 12*5 = 60.
FV = PV * (1 + r)N = (10,000)*(1.0066667)60 = (10,000)*(1.489846)
= Rs. 14,898.46
FV and PV of a Single Sum of Money
If annual compounding of interest is assumed, these problems can be solved as
FV = PV * (1 + r)N
PV = FV * { 1 }
(1 + r)N
Where: FV = future value of a single sum of money,
PV = present value of a single sum of money, R = annual interest rate,
and N = number of years
Example: Present Value
At an interest rate of 8%, we calculate that Rs. 10,000 five years from now will be:
FV = PV * (1 + r)N = (Rs. 10,000)*(1.08)5 = (Rs. 10,000)*(1.469328)
FV = Rs. 14,693.28
At an interest rate of 8%, we calculate today’s value that will grow to Rs. 10,000 in five years:
PV = FV * (1/(1 + r)N) = (Rs. 10,000)*(1/(1.08)5) = (Rs. 10,000)*(1/(1.469328))
PV = (Rs. 10,000)*(0.680583) = Rs. 6805.83
Example: Future Value
An investor wants to have Rs. 1 million when she retires in 20 years. If she can earn a 10% annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to
Answer: The problem asks for a value today (PV). It provides the future sum of money (FV) = Rs. 1,000,000; an interest rate (r) = 10% or 0.1; yearly time periods (N) = 20, and it indicates annual compounding. Using the PV formula listed above, we get the following:
PV = FV *[1/(1 + r) N] = [(Rs. 1,000,000)* (1/(1.10)20)] = Rs. 1,000,000 * (1/6.7275) = Rs. 1,000,000*0.148644 = Rs. 148,644
Simple interest is a function of three variables: Original amount borrowed or Principal, Interest Rate, Number of time period.
SI = PV (r) (n)
Where, SI = Simple Interest
PV = Principal or Original Value
r = rate of interest per time period
[1 + (r) (n)]
