Financial Mathematics can vary from elementary equations, like calculating simple interest on a principal value to as complicated as utilising partial differential equations when it comes to risk analysis. However, in this introductory post to financial mathematics - let’s cover a few of the basic fundamentals.
The Role of Mathematics in Finance.
In What is Corporate Finance, the importance of cash-flows were discussed. In simple terms, it can even be taken a step further back and stated that cash flows are simply the funds that are transferred or ‘flow’ between two entities either immediately or some time in the future as a result of a financial contract. Our particular concern is over the cash-flows that are transferred at some time in the future. This introduces three important concept; Rate of Return, Interest Rate and Time Value of Money.
Rate of Return (R.O.R.).
Rate of Return or ‘ROR’ is a simple arithmetic that creates a relationship between the cash inflows and cash outflows. It is also referred to as ‘ROI’ (Return on Investment) or simply as return. It is the profit(amount gained – amounted invested) as a percentage of the amount invested.
Interest Rate is simply the Rate of Return (as seen above) on debt. In this context, the cash-flow IN is the amount lent to the borrowing party and the cash-flow OUT is the amount returned back to the creditor after one period. Thus, effectively making the Rate of Return the interest rate.
Time Value of Money.
Cash-flows that are transferred at different points in time cannot simply be added or subtracted from each other. This is due to the ‘time-value of money’ theorem that states;
“A dollar today is worth more than a dollar tomorrow.”
The idea is simple; money received today can be reinvested to earn more cash (interest) tomorrow. Hence a lump sum of money today is not the same as that amount tomorrow because you can invest today’s amount – and by tomorrow – have more than what you had initially. Since this theory holds for any amount of money and for any amount of time – businesses’ must account for cash-flows that occur in multiple time frames. This theory is crucial in Finance, as significant amount of time may elapse between the in-flows and out-flows of cash.
If you are interested, my post over at StudySaurus on “Why isn’t $1 today the same as a $1 tomorrow” touches on the theory of Time-Value of Money.
Present Value vs. Future Value
Present Value is the value of a lump-sum amount today.
Future Value is the value of that same lump-sum amount in some time in the future, compounded appropriately.
Simple Interest vs. Compound Interest.
Simple Interest is used when a lump sum amount is paid after a single time period (t) on a principal amount (P), at an agreed interest rate (r).
$ Interest = Principal (P) x Time Periods (t) x Interest Rate (r)
Therefore, the total amount payable at the end of the time period is the Present Value + Interest Gained, which is the Future Value (FV):
Compound Interest is used when interest accumulates on to the principal amount (PV) over several time periods (n), thus earning interest on interest. This notion of interest earned from one period accumulating to earn more interest on a later period is the main difference between compound and simple interest. The compounding interest rate is denoted by (i).
This can be transformed to make the Present Value (PV) the subject;
The fundamental law governing time-value calculations of Present Value (PV) and Future Value (FV) are based on Compound Interest.
Nominal Interest Rates vs. Effective Interest Rates vs. Real Interest Rate.
The Nominal Interest Rate is the interest rate that is quoted when interest is compounded more or less frequently than the time period specified in the interest rate.
Example: Interest rate of 8% per annum, compounded semi-annually.
The Effective Interest Rate is a converted nominal interest rate, so that the compounding frequency is the same as the interest rate quoted. The equation used to convert a nominal interest rate to an effective interest rate is:
Nominal interest rates are converted into effective interest rates so they can be more easily compared.
So, using our Interest rate of 8% per annum (j = 0.08), compounded semi-annually (m = 2) example – let’s find the effective interest rate.
The effective interest rate of the nominal 8% per annum, compounded semi-annually is therefore 8.16 % per annum. In other words, 8% per year compounded 2 times a year is the same as 8.16% per year compounded just once a year.
The Real Interest Rate is the effective or nominal interest rate after taking into account the effects of inflation.
Continuous Interest Rates & Geometric Rates of Return.
Some investment opportunities offer “continuous interest”, whereby interest is calculated all the time – or continuously - effectively making the time period between each compound approach zero. In such an investment, the ‘continuously compounding’ model must be used;
The geometric rate of return or the ‘average compound rate of return’ is an average interest rate over a fixed time period – when the interest rate per year varies. For example, an investment can grow in some years and fall in others – but at the end of the investment period – an ‘average interest rate’ can be calculated using the following;
Annuities – a steady stream of money.
An annuity is a stream of equal cash-flows, equally spaced in time. With annuities, it’s important to differentiate between the 3 major types – ordinary annuities, annuity due and deferred annuities and realise which type you are working with. The main difference between ordinary annuities and an annuity due is only a shift in cash-flows. See the differentiation explained below.
Ordinary annuities are annuities in which the time period between the valuation date of the annuity and the first cash-flow is equal to the time period between any two subsequent cash-flows. In other words, if the valuation of the annuity is today and you are promised to receive an equal payment every month – the first payment must be one month from today.
Calculating the Present Value (PV) or Future Value (FV) or an Ordinary Annuity.
Annuities Due are annuities in which the first cash-flow occurs immediately. An annuity due of n cash flows is in essence the same as an ordinary annuity of (n – 1) cash flows, plus an immediate cash flow of CF. In other words, if the valuation of the annuity is today and you are promised to receive an equal payment every month – the first payment must be from today onwards.
The present value of an annuity due has a similar but slightly varied formula;
A deferred annuity is similar to a annuity due but instead of an immediate payment – there is a prolonged delay between the valuation date and the arrival of the first cash-flow. The critical difference here is that in order to be classified as a deferred annuity the first cash-flow must be delayed by a time period that is greater than the time period between each subsequent cash-flow.
For illustrative purposes, let’s call the delay between valuation (t = 0) and the first cash-flow is the ‘gap’. If the ‘gap’ was zero (0) then it would be an annuity due and if the gap was exactly the same amount of time as the time between each subsequent cash-flow; it would be an ordinary annuity. If the gap was > 1 time period, then it is a deferred annuity.
Perpetuities – a never ending incoming steam.
Quite similar to an ordinary annuity, an ordinary perpetuity assumes that the cash-flows are never ending. In practicality, this is like depositing a certain amount of money in a government backed bank – the cash-flows received by the investor is assumed to go on forever.
Calculating the Present Value (PV) or an ordinary perpetuity is quite simple. It is simply the promised future CF divided by the accepted rate of return, or interest rate;
Since money has a value that is dependent in which time-frame it is transferred from one entity to another – Financial mathematics is mainly used to discount or compound sums of money to a comparable time period.
Simple Interest and Compound Interest are the two main types of growing a capital through interest – but the advertised interest rate or the nominal interest rate is usually not the same as the real or effective interest rate.
Furthermore, annuities and perpetuities require a slightly more advance use of mathematics. These equations are utilized in order to calculate the present value (PV) of multiple cash-flows over either a fixed or an unlimited amount of time.