Sequence and Series in Finance
Explore Applications of Sequence and Series in Finance
Sequence and Series are very basic concepts which we study during our early mathematics classes yet they have very powerful and beautiful applications in Finance. Some of the applications include valuation of financial contracts such as annuity and perpetuity contracts and bonds. Other applications include calculation of mortgage payments for fixed-rate mortgages. Sequence and Series are also used in converting functions to series which makes calculations easier for computers. This is done using advanced theorems in mathematics such as Taylor’s theorem.
In this article, we will first introduce sequence and series with a particular example of geometric sequence and geometric series. As a next step, we will re-visit annuity and perpetuity contracts, and explore the applications of geometric sequence and series in calculating the Present Value (PV) of these contracts.
Sequence
A sequence is an arrangement of numbers in a particular order. A sequence can be generated by taking its n-term. Consider the example of an geometric sequence which is given and below:
Here, a is the first term and r is the common ratio.
Series
A series is the sum of all the terms of a a sequence. For the geometric sequence given above, the series (sum) of first n terms is expressed as below:
The above sum can be calculated by multiplying the above equation by common ratio, r, and then subtracting above equation from the multiplied equation. However, let’s skip the derivation and directly look at the formula for the sum of first n terms of a geometric sequence, which is given as below.
Amazing! Isn’t it? We have derived a simple formula where we don’t care about the number of terms in our sequence to calculate the sum of the series. We don’t have to enter all the terms individually in our calculator. All we care about is first term (a), common ratio (r) and number of terms (n).
In the next sections, we will explore some amazing applications in Finance such as valuation of annuity and perpetuity contracts.
Applications in Finance
As mentioned in the beginning of the article, the concept of Sequence and Series, especially geometric series, has many applications in Finance. Here, we will discuss about the calculation of PV for annuity and perpetuity contracts.
Annuity
Recall that annuity contracts allows the investor to receive fixed amount of payment at the end of each year (until maturity) by investing a lump sum payment at the beginning of the contract.
Let A denotes the amount of annuity payment received at the end of each year, r denotes the spot-rate, and let n payments are made by the contract. Therefore, the sequence of present values of all the annuity payments is a geometric sequence which looks as below:
Here, first term is A and common ratio is 1/(1+r).
In the above example of annuity payments, the series is the sum of all the present values of annuity amounts. It is given as below:
The formula for the PV of an annuity contract is given as below.
Perpetuity
Let’s extend the concept to perpetuity contracts. Perpetuity contracts are similar to annuity contracts with the only difference that they pay a fixed annual amount till perpetuity, unlike an annuity where the payment of annual amounts stops after n payments. To calculate the PV of a perpetuity, we can simply use the formula for PV of an annuity and replace n with infinity. Doing so, we can calculate the PV of a perpetuity as below:
Fixed-Rate Mortgage
A fixed-rate mortgage is a mortgage loan where the interest rate is fixed for the entire term of the loan. Consider a person John who wants to buy a home in California for $500,000. But since this amount is too big, John decides to go to the bank for a mortgage loan and requests for a loan. John proposes that the bank give him $500,000 and he’ll return the loan amount over next 20 years by paying monthly instalments. The bank screens John’s application and after performing all the due diligence, agrees to give the amount to John at an annual interest rate of 4.8%.
Now, both bank and John (and of course us), are excited to know the monthly payment amount which John will pay to the bank. To calculate this payment, say PMT, we can first notice that John will pay a total of 20*12 = 240 payments. These payments will be discounted by a monthly interest rate of 4.8/12 = 0.4% = 0.004. Also, the sum of PV of all these payments must be equal to the original principal loan amount. Therefore, the equations looks as below:
John realises that this payment structure is the same which his dad used to buy an annuity contract for him. Since he helped his dad in investing his retirement fund in an annuity scheme, he understands how to calculate the PV of an annuity. Therefore, he uses the same approach and calculates PMT by making it the subject of the annuity formula. Let’s understand how he did it.
Let’s call our original loan amount as Principal, number of months as n, monthly interest rate as r and monthly payment as PMT. Then using the annuity formula, we can write below equation
Next, we make PMT as subject of the above equation and calculate it as follows:
By using the values of Principal loan amount, monthly interest rate and number of months, we can calculate the value of monthly payment as $3,245. This is the amount which John needs to pay per month for next 20 years to clear his mortgage loan of $500,000. Note that mortgage loans are backed by residential property, in this case John’s home, as collateral. So, if John defaults on his payments, the banks can take possession of his home.
