Ryan Thompson Texas A&M University • Math 482 Instructor: Dr. David Larson May 8, 2013 Final Paper: An Introduction to Interest Theory I. Introduction At some point in your life, you will most likely be involved in a financial transaction. It does not matter what path you choose to take your life in, you will inevitably have to manage your finances and secure your financial future in a way that you see fit. There is an incredibly wide variety of different types of transactions, from loans and mortgages to pension and benefit plans, but one thing common to nearly every financial transaction is the concept of interest. Because of this fact, it is beneficial to gain a better understanding of interest and its applications, as well as to develop an understanding of some of the more basic concepts that could become useful in everyday life. II. Interest Definition 1. The function 𝑎(𝑡) is the accumulation function for an investment at time t. It represents the amount of money in a fund at time t with an initial investment of $1.00 at time 0. When it comes to interest itself, there are two basic investment growth patterns through interest that many investments will follow. Definition 2. When interest gains in direct proportion to the time in years of the investment (𝑎(𝑡) = 1 + 𝑖 ∗ 𝑡), this is called simple interest. Definition 3. When interest gains exponentially in proportion to the time in years of the investment (𝑎(𝑡) = (1 + 𝑖)� ), this is called compound interest. The following is a graphical comparison of simple and compound interest: *From this point forward, simple interest will be ignored and only compound interest will be considered, as it is much more commonly used. One thing that must be taken into account is that interest periods may vary. In other words, investments can receive the dividends from the interest they gain in different time intervals. Interest most commonly gains annually, semi-annually, quarterly or monthly, but could gain for any agreed upon interval imaginable. Because interest periods vary, there must be a standard convention in which all interest rates are given. This standard convention is called the Nominal Interest Rate, and is given as an annual rate in the form 𝑖 (�) = ⋯ % where m indicates the number of interest periods that occur over the course of one year. If you divide this nominal rate by m, you will get the periodic interest rate. But what if you want to compare two rates with differing interest periods? It is important to note that the nominal interest rate does not accurately represent the real amount of interest an investment will earn in a year’s time if the investment period is less than one year. This is because the investment will be earning compound interest, and each periodic interest payment will also begin to earn money. Because of this, another method must be used to differentiate between rates. The most common practice in comparing interest rates is to compute their Effective Annual Interest Rate. This is done in the following way: � 𝑖 (�) 1 + 𝑖 = �1 + � 𝑚 where i indicates the effective annual interest rate. The effective annual rate provides you with the percent yield on an investment over a one year period. By normalizing each rate over a one year period, we have a means of comparing all rates. III. Time Value of Money Now that we understand the basics of interests, we need to explore what it is useful for. One very useful thing to do is to compare an amount of money today with its growth potential sometime in the future. For example, if you have $100 today, what is that $100 worth to you one year from now? If the annual interest rate is 10%, then your $100 today could be invested and be worth $110 in 1 year. This is the concept of Time Value of Money, comparing money with its potential worth sometime down the line. Time value of money however is not exclusive to looking forward in time. For example, say we want to invest money today so that it will grow to $100 in one year. How much money should we invest? If the annual interest rate is 10%, you could set up the following equation of value: Solving for X, you get 𝑋 ∗ (1 + 𝑖) = 100 𝑋 = 100 ∗ 1 1+𝑖 1 ≈ 90.91 1.1 This $90.91 is what’s called the Present Value of $100 due one year from now. What we did to the 𝑋 = 100 ∗ $100 was discount it one interest period by a factor of known as the present value factor. � ��� . This factor, commonly denoted by ν, is But how is present value useful? To demonstrate that, let’s look at an example: Example 1. The parents of three children ages 1, 3, and 6 wish to set up a trust fund that will pay $25,000 to each child at age 18 and $100,000 to each child at age 21. If the trust fund will earn at an effective annual interest of 10%, what amount must the parents now invest in the fund? Solution First we must determine when each payment in the trust fund will be paid out. Call the present day time 0, and then determine the payouts for each child at their respective times by subtracting their current age from the age at which they receive the payment. Current Age 1 3 6 Turns 18 at What Time? 17 15 12 Turns 21 at What Time? 20 18 15 Using this chart, we can set up a timeline to demonstrate the payments: +$X -$25K 0 12 -$125K -$25K -$100K -$100K 15 17 18 20 Now to find what must be paid for the timeline, we can discount all the payments and find their combined present value using the given annual interest rate of 10%. What we want is for the present value of our deposit into the fund, which is simply X, minus the present value of the future payments to be equal to zero. Moving our deposit to its own side, this ends up being � 𝑋 = 25000 ∗ ν�� + 125000ν�� + 25000ν�� + 100000ν�� + 100000ν�� where ν = �.�. So it turns out that X = $75,686.14 must be paid to set up this trust fund. As you can see, present value can be used to determine the cost of some stream of payments made at some point in the future. The cost of something should be equal to the present value of its future payments, since their values are said to be equivalent. We will now look into a few different types of these payment streams. IV. Payment Streams Definition 4. An annuity is a stream of level, regularly spaced payments for a specific amount of time periods. A general case would look like the following: k 0 1 k ⋯ 2 ⋯ k k n-1 n Since this is a common payment stream, we would like to come up with a general formula for its present value that we could apply to all payment streams of this type. Derivation 1. 𝑃𝑉 = 𝑘𝑣 + 𝑘ν� + ⋯ + kν��� + kν� = 𝑘ν(1 + ν + ν� + ⋯ + ν��� + ν��� ) � = 𝑘ν ∑��� ��� ν = 𝑘ν ∗ ���� ��� � ���� = 𝑘 ∗ ��� ∗ � = 𝑘 ∗ ��� ∗ 𝑃𝑉 = 𝑘 ∗ ���� � ��� ���� �� � � ��� To find the formula for the accumulated value of this same stream of payments, we can just move the present value we found n interest periods into the future. Derivation 2. 𝐴𝑉 = (1 + 𝑖)� ∗ 𝑘 ∗ � ���� � � = 𝑘 ∗ (1 + 𝑖) ∗ � � − =𝑘∗� (���)� 𝐴𝑉 = 𝑘 ∗ � � − �� � (���)� �� �� � � Using these formulas, we can now find the present and accumulated values of an annuity of this general form. A level annuity isn’t the only kind of common payment stream though. An annuity could also contain payments that increase/decrease, and they could do so either arithmetically or geometrically. General formulas could be derived for each of these cases in a similar manner to those above. Definition 5. A perpetuity is an annuity that continues for an infinite number of interest periods. This can be thought of as simply being a general annuity as displayed above, but as 𝑛 → ∞. The accumulated value of any perpetuity is infinite, since payments never end. But there is a way for us to determine the present value of these payments, by taking their limit as such: � 1 − ν� lim 𝑘 ∗ �→� i Since ν = ��� and i > 0, 0 ≤ ν ≤ 1 which implies that ν� → 0 𝑎𝑠 𝑛 → ∞. ∴ 𝑃𝑉 = 𝑘 ∗ 1 𝑖 Just like annuities, perpetuities can have payments that are not always level, and it is also possible to determine a general formula for their present values in a similar way, by taking limits as n goes to infinity and simplifying the resulting equation. V. Conclusion As you have seen, even the most basic concepts in interest theory can be used to understand and solve real world problems. The concepts presented are very useful for financial transactions that nearly every household nationwide must complete, most notably when it comes to dealing with loan repayments of loans taken out, specifically for home and auto loans. The applications of these concepts reach much further than basic applications though, and these concepts are important to understand in order to be able to learn about the higher levels of financial mathematics. VI. References Broverman, Samuel A. Mathematics of Investment and Credit. 5th ed. Publications, 2010. Print. Winsted, CT: ACTEX Cherry, Harold, and Rick Gorvett ASM Study Manual for Exam FM/Exam 2. 11th ed. Greenland, NH: Actuarial Study Materials, 2012. Print.