Time Value of Money
If one is to receive € 1000 today, he or she can use it to either satisfy his needs (e.g. take a memorable vacation) or invest it (lend it out) for a definite period of time. If the individual decides to invest the € 1000, he or she will be foregoing the benefit of the immediate use of that money for the satisfaction of his or her needs (e.g. vacation) for the whole of the investment duration. The individual deserves compensation for the postponement of satisfaction (opportunity cost) over the agreed period of time. This compensation is normally done hrough the payment of interest.
A debtor can pay you € 1000 today or you can wait for one year and the debtor will pay you € 1100.
Interest Rate
Interest: Interest on the principal over a year (day or month). This is the price for borrowing a unit of currency and keeping it over a given period of time.
Simple Interest = Principal x Time x Rate/100 (ptr/100). € 1000 deposited in a fixed deposit account over one year at 5 % interest: S I = 1000 x 1 x 5/100 = € 50. The account will have € 1050 at the end of the year.
Time is often given in years, rate is in percentage. The rate (in percentage) can be given in decimal notation i.e. 0.05 in place of 5% (5/100).
Deriving the compounding formular from the simple interest formula: Assuming that a loan is taken for the duration of 1 year (any number multiplied by 1 remains unchanged), let the principle be P and the interest rate be r (decimal notation), the interest shall be given by rP. At the end of the year, the borrower should pay back the loan plus the interest i.e. a total of P+rP = P(1+r).
If the borrower makes no repayment, his or her debt will be P(1+r), which must be paid back at the end of the second year with interest of rP(1+r). This means that he or she shall have to pay a total of P(1+r)+rP(1+r) = P(1+r)2.
After three years, he or she will owe P(1+r)3, after four years, he or she will owe P(1+r)4 and after n years he or she will owe P(1+r)n and this is the compounding formula.
Compounding: the process of finding the future value of a given principal sum (present value) invested over a period of time with interest being paid not only on the principle but also on interest earned.
Compound interest: Interest accumulated in such a way that interest is being paid not only on the principal but also on interest already earned. This addition of interest on principal and interest on interest to the principal is called compounding. As seen in the compounding formula derived above, the principal sums used in interest calculation for later years includes the interest paid in ealier years.
Future Value , (FV) is the total amount at the end of the investment period, Present Value (PV) is the principal sum (the money you have today; that which you intend to invest) r is the nominal rate of interest, n is the total number of years (periods).
FV = PV(1+r)n that is, A = P(1+r)n
This formula gives the future value of an investment whose present value (principal) is accruing interest at a fixed interest rate (r) for n periods. More often, annual compound interest is calculated using the formula: A = P(1+(r/n))nt
Where A = value after t periods (FV), P = principal amount (initial investment), r = annual nominal interest rate (not reflecting the compounding), n = number of times the interest is compounded per year, t = number of years the money is borrowed for.
In the previous example (€ 1000 deposited for one year in a fixed deposit account paying interest at 5% per annum with the bank paying you a total of € 1050 at the end of one year): If instead of depositing the money for only one year, you decide to keep it in the bank fixed deposit account for two years, the amount at the end of the two years will be made up of the interest in year 1 + the interest in year 2 + the principal.
Interest in year 1 is: € 1000 x 1 x 5/100 = € 50
Interest in year 2 is: € 1050 x 1 x 5/100 = € 52.50
The original amount deposited was: € 1000
Total amount (Future Value) at the end of 2 years: € 50 + € 52.50 + € 1000 = € 1102.50.
Applying the compounding formula presented above: A = € 1000(1+(5/1))1×2 = € 1102.50
The compound interest earned is € 102.50. As can be seen the principal in year 2 was made up of the original € 1000 and the interest earned in year 1. The interest in year two was therefore being paid on the original amount (Present Value) as well as on the interest that had been earned in year 1.
To know the compound interest rate that would be achieved if an initial investment of PV returns a value FV after n accrual period, the formula would be: r = (FV/PV)1/n – 1.
To know the number of periods (years) required to get FV given the PV and the interest rate (r), the relevant formula is n = [log(FV) – log(PV)]/log(1+r).
Doubling your money: How long will it take for your money to double when it is invested in a savings account at 5%? And how long will it take to double at 7% interest rate? A simple rule of thumb provides that at x% interest rate, money will double in {70}/{x} years. So at 5%, a deposit doubles in 14 years, and at 7% it doubles in 10 years. This is the “Rule of 70”.
How is the rule derived? If an amount A is invested at an interest rate of r, the problem is to determine n such that [ 2A=A(1+r)n] : algebra gives [ n={log 2}/{log(1+r)}].
Discounting: the process of finding the present value of future cash flows. Given the compounding formula, one can find the NPV formula by solving for PV:
If FV = PV(1+r)n, then: FV/(1+r)n = {[PV(1+r)n]/(1+r)n} and so FV/(1+r)n = PV.
Net Present Value (NPV)
PV = FV/(1+r)n that is P = A/(1+r)n
This formula gives the present value (principal) which would be needed to produce a specific future value (amount) if interest accrues at the rate of r for n periods. The question is, how much should I invest today in order to receive a given amount of money after a given period of time in view of the prevailing rate of interest?
NPV compares the present value of money today to the present value of money in the future. It is the sum of all discounted future cashflows
NPV = ∑ FVn /(1+r)n
In our earlier example where a debtor who owes you € 1000 today is willing to pay you € 1100 in a year’s time, you can compare this proposal to that which the best fixed deposit account at a commercial bank offers.
If you take the € 1000 today and deposit it for one year in a fixed deposit account paying interest at 5% per annum, the bank will pay you a total of € 1050 at the end of one year. This implies that the bank that pays 5% interest on fixed deposit account will require you to deposit € 1047.62 with them on for one year if you wanted to have € 1100 at the end of the 12 months (1 year). €1100/(1+0.05)1 = € 1047.62 (NPV). So, all other factors held constant, the debtor is making you a better offer since you commit a smaller amount to gain an extra € 100.
Internal Rate of Return
The rate of return (ROR) is the speed at which the money that you invested comes back to you. It is normally provided in percentage per annum. If you invest € 1000 and you will be getting € 100 back every year, then the rate of return would be (100/1000)x100 = 10 % per year. In cases where the amount received back is not constant over a predetermined time, the rate of return may not be so obvious and in such cases, the internal rate of return often helps in decision making.
The Internal Rate of Return (IRR) is that particular rate of return that equates the present value of a project’s or an investment’s cash inflows with the present value of its cash outflows i.e. it sets out the net present value equal to zero (it is the rate at which an investment breaks even). Internal rate of return is the generally required low cost of capital to accept the project. IRR is also sometimes referred to as the discounted cash flow rate of return (DCFROR).
Calculation: Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows from the net present value as a function of the rate of return. A rate of return for which this function is zero is an internal rate of return. The period is usually given in years (but other periods e.g. months can be used instead).
P0/(1+r)0+ P1/(1+r)1+ P2/(1+r)2+……..+ Pn/(1+r)n = 0
This can also be written as: P0(1+r)-0 + P1(1+r)-1 + P2(1+r)-2 +……..+Pn(1+r)-n= 0
Quite often, the value of r (IRR) in the NPV equation cannot be found analytically. In this case, numerical methods (iteration) or graphical methods must be used. As an example, if you invest € 1000 and you will be getting back € 600 in year 1 and € 600 in year 2, then the internal rate of return can be obtained from the present value formula either via iteration or through graphical methods. Iteration: The NPV expression in this case would be: -1000(1+r)-0 + 600(1+r)-1 + 600(1+r)-2
The initial investment (cash outflow) has a minus (negative) sign before it because money is flowing out of your pocket while the cash inflows have plus (positive) sign because money is coming back to your pocket. In the expression, we will be trying to find out r (which we still do not know). The r which equates this NPV expression to 0 is the IRR.
-1000(1+r)-0 + 600(1+r)-1 + 600(1+r)-2 = 0
Here, r is not to be solved for by using the usual algebraic methods but rather by trial and error (iteration) as demonstrated below:
Lets assume r were 14% or 0.14 and plug it to the equation.
Is: -1000(1+0.14)-0 + 600(1+0.14)-1 + 600(1+0.14)-2 = 0 ? Is: -1000(1.14)-0 + 600(1.14)-1 + 600(1.14)-2 = 0 ? No because -1.2 is not equal to 0, r is not 14%.
Now lets assume and test if r might be 12% or 0.12
Is: -1000(1+0.12)-0 + 600(1+0.12)-1 + 600(1+0.12)-2 = 0 ? Is: -1000(1.12)-0 + 600(1.12)-1 + 600(1.12)-2 = 0 ? No because 1.4 is not equal to 0, r is not 12%.
Now let us try something in between 12% and 14 % by assuming r is 13% or 0.13.
Is: -1000(1+0.13)-0 + 600(1+0.13)-1 + 600(1+0.13)-2 = 0 ? Is: -1000(1.13)-0 + 600(1.13)-1 + 600(1.13)-2 = 0 ? The equation boils down to 0.09 (almost 0) = 0, so we conclude that the r which equates the NPV to 0 is 13%, so IRR = 13%.
Exercise: Assume a coffee machine is to be bought at € 100 to help reduce waiting time and improve business over two years with returns on years 1 and 2 being € 60 each.
-100/(1+r)0+60/(1+r)1 +60/(1+r)2 = 0
Cash inflows are + values while cash outflows are – values. Percentages (growth rates) are entered in decimal notation such that 13 % will be written as 0.13. You can now find the IRR for this small investment.
Use: It is used in capital budgeting to measure and compare the profitability of investments i.e. it is basically used to measure the efficiency of capital investments.
In the context of savings and loans the IRR is also called the effective interest rate. This rate of return is referred to as being internal because its calculation does not incorporate external factors (e.g., market interest rate or inflation etc).
Decision criterion: An investment is considered acceptable if its internal rate of return is greater than an established minimum acceptable rate of return or cost of capital. If the IRR is greater than the cost of capital, accept the project. If the IRR is less than the cost of capital, reject the project. The cost of capital should include the opportunity cost of tying the money in the investment before hand, rate of inflation and interest rate on loan if the funds are borrowed.
If comparing two projects with different IRRs, the higher a project’s IRR, the more desirable it is to undertake the project. A firm (or individual) should, in theory, undertake all projects or investments available with IRRs that exceed the cost of capital. Investment may however be limited by availability of funds to the firm and/or by the firm’s capacity or ability to manage numerous projects.
Because the internal rate of return is a rate quantity, it is an indicator of the efficiency, quality, or yield of an investment. This is in contrast with the net present value, which is an indicator of the value or magnitude of an investment.
Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV. Apparently, managers find it easier to compare investments of different sizes in terms of percentage rates of return than by dollars of NPV. However, NPV remains the “more accurate” reflection of value to the business. IRR, as a measure of investment efficiency may give better insights in capital constrained situations. However, when comparing mutually exclusive projects, NPV is the more appropriate measure.
Annual Percentage Rate: The Annual Percentage Rate (APR) is the unit cost of money. It is a unit rental cost indicating the cost of borrowing €1 and keeping it for one year. An APR of 25% means that it will cost you 25 cents to borrow € 1 and keep it for one whole year.The APR helps one compare the true cost of different loans.
Other relevant topics covered by us include:
CAGR (Compounded Annual Growth Rate)
Payback period
Break-even analysis
