Think & Done

TVM formula explained with example calculation of present and future value of a lump-sum, ordinary annuity with end of period payments and annuity due with start of period payments. We discuss time value of money concepts such as finding present value of lump sum, ordinary annuity and annuity due with uniform series of cash flows. These topics are the cornerstone of study of advanced topics in financial management such as loan and mortgage payments, money earned by investing in savings account and pension funds, discounted cash flow analysis, and valuation of stocks and bonds. The time value of money formula is derived from TVM equation which can be written in 2 different ways yet the results derived from this equation are the same. The following paragraph lists and explains the two different TVM equations upon which I will build the rest of the discussion.

Time value of money Equation #1

TVM equation has 3 components: one of these is the compounded PV or present value, the second component is the compounded period payment (start of period payments or end of period payments), the third one is the future value or FV. To ensure that the TVM equation can be used for both start and end of period payments, we introduce a flag or indicatior variable called TYPE which is either set to 0 for end of period payments or set to 1 for start of period payments. Rate is represented as a variable of i, and number of periods is represented as n. Have a quick look at the following form of TVM equation:

PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV = 0

The sum of the three components of the TVM equation results in zero. To ensure the sum of these three parts is zero, one of the values of PV, FV or PMT must be negative and one of these values must be positive. From this TVM equation, we can derive the values or solve for any of the other variables such as finding the present value, future value, number of periods, and periodic payment. However the variable i for interest rate is not solvable from this equation in instance when there is a periodic payment; in such a case we will have to resort to making use of numerical methods to find an approximation of the interest rate.

Time value of money Equation #2

Let me now show you an alternative form of the same time value of money equation; this version of the TVM equation discounts the future value FV and the periodic payment. This equation produces identical results for present value, future value, periodic payment, number of periods and the interest rate. It is just a mutated form of the TVM equation I listed in the last paragraph. Here is a quick look at this TVM equation listed below

FV(1+i)^-n + PMT (1+i*TYPE)[1- {(1+i)^-n}]/i + PV = 0

This equation discounts the future value and the periodic payments. The sum of the three components of this TVM equation also results in zero when at least one of the three amounts of FV, PV and PMT is positive and at least one of these three amounts is negative. We can solve for any of the variables from this equation by algebraic manipulation. To solve for interest rate i when there is a periodic payment, we would have to make use of numerical methods that will find an approximation of the interest rate.

TVM Equation #3:

When the interest rate is zero, the two TVM equation will not hold true. As when the interest rate is 0%, the periodic payments discounted or compounded at 0% will yield the product of payment amount times the number of periods. And at 0%, the discounted FV or compounded PV will equal the original amount. Thus we will make use of a third TVM equaion listed below when the interest rate is 0%

PV + PMT * N + FV = 0

We will make use of the TVM equations listed earlier in this discussion to calculate present value and future value of a lump-sum, ordinary annuity with end of period payments, annuity due with start of period payments, number of periods n, periodic payments of an annuity, and the interest rate. To be able to compute these variables we will use rules of algebra to solve for any of the variables in the TVM equation. For occasions when it is not possible to solve for a variable with algebraic manipulation, we will resort to numerical methods to solve for such a variable.

Present Value of annuity

For a detailed discussion on the topic, view this tutorial that explains finding present value of annuity. The present value annuity formula is illustrated with example calculation. Using any one of the two TVM equations, we can calculate present value of an ordinary annuity PVA with end of period payments and present value of an annuity due with start of period payments. We will make an assumption that the annuity payment is a constant amount with uniform series of cash flows. To be able to find the present value of annuity, we will have to solve for the PV variable in the TVM equation. Follow the steps listed below in rearranging the TVM equations to calculate present value of ordinary annuity with start of period payments and oridnary annuity with end of period payments.

PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV = 0
PV(1+i)ⁿ - PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV = -PV(1+i)ⁿ
PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV = -PV(1+i)ⁿ
-PV(1+i)ⁿ = PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV
PV(1+i)ⁿ = -PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV
PV(1+i)ⁿ/(1+i)ⁿ = [-PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV] / (1+i)ⁿ
PV = [-PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV] / (1+i)ⁿ
PV = [-PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV] / (1+i)ⁿ

Now let us attempt to find present value of an ordinary annuity where deposits in amounts of $100 are made at the end of each month for the next 10 years. Assuming that interest rate is 10%, we are given the task to find present value of this ordinary annuity with end of period payments. We will use the following values to plug in to the PV formula we just derived out the TVM equation
PMT = -100
TYPE = 0 for end of period payments
FV = 0
N = 12x10 = 120
I = 10%/12 = 0.008333334
PV = ?
PV = [-PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV] / (1+i)ⁿ
PV = [100 (1+10%/12 * 0)[{(1+10%/12)¹²⁰} -1]/(10%/12) - 0] / (1+10%/12)¹²⁰
PV = [100 (1+0)[{(1+0.008333334)¹²⁰} -1]/0.008333334 - 0] / (1+0.008333334)¹²⁰
PV = [100 [{(1.008333334)¹²⁰} -1]/0.008333334] / (1.008333334)¹²⁰
PV = [100 [{2.707041491 -1}/0.008333334] / 2.707041491
PV = [100 [1.707041491/0.008333334] / 2.707041491
PV = [100 [204.84497890347 / 2.707041491]
PV = [100 [75.67116337]
PV = $7,567.12

Present Value of annuity due

As explained earlier, an annuity due has start of period payment and we denote this by using a value of 1 for TYPE in the TVM equation, thus if the payments in the previous example were made at the start of each month then finding present value of this annuity due will be nothing more than multiplying the result of our calculations with (1+i * TYPE) where i is the rate of interest per month and TYPE is 1. This will make the present value higher than what it is for ordinary annuity because the payment in the first month is not being discounted.
PV = [-PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV] / (1+i)ⁿ
PV = 100 (1+10%/12 * 1) [{(1+10%/12)¹²⁰} -1]/(10%/12) - 0] / (1+10%/12)¹²⁰
PV = 100 (1+0.008333334 * 1) [75.67116337]
PV = 100 (1.008333334) [75.67116337]
PV = 100 (76.3017564)
PV = $7,630.18

Present Value of a lump-sum

Present value of lump-sum is found using the same PV formula we derived earlier, the only aspect that will differ in finding a present value of a lump-sum has to do with PMT being zero. As there is no periodic payment thus we will only have to deal with two variables of PV and FV along with N and i. Let us take an example where we have to find the present value of $1000 at an interest rate of 5% for a period of 10 years. Follow the steps outlines in the following paragraph to see the use of TVM equation in action
FV = -1000
PMT = 0
I = 5% = 0.05
N = 10
PV = ?
PV = [-PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i - FV] / (1+i)ⁿ
PV = [-0 (1+5%*0)[{(1+5%)¹⁰} -1]/5% - (-1000)] / (1+5%)¹⁰
PV = [- (-1000)] / (1+0.05)¹⁰
PV = 1000 / (1.05)¹⁰
PV = 1000 / 1.62889462677744140625
PV = 613.91

Future Value of annuity

For a detailed discussion on the topic, view this tutorial that explains finding future value of annuity. The future value annuity formula is illustrated with example calculation. Now I will go ahead and use the first of the TVM equations to solve for future value of an ordinary annuity with end of period payments. The process starts with separating the FV variable to the left hand side of the equation thus making it possible for us to find FVA or future value of annuity. Let us take an example where you are depositing $100 at the end of each month for the next 10 months. Assuming the annuity pays an interest at the rate of 10% per year, our task is to find the future worth of this annuity with end of period payments. Follow the steps outlined in the following paragraph to see how to derive the FV formula from TVM equation and then to find the FVA.

PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV = 0
PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i + FV - FV = 0 - FV
PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i = - FV
- FV = PV(1+i)ⁿ + PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i
FV = -PV(1+i)ⁿ - PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i

Let us now plug in the following values in to the FV formula we just derived from the TVM equation:
PMT = -100
TYPE = 0 for end of period payments
PV = 0
N = 12x10 = 120
I = 10%/12 = 0.008333334
FV = ?
FV = -PV(1+i)ⁿ - PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i
FV = -0(1+10%/12)¹²⁰ - (-100) (1+10%/12 * 0)[{(1+10%/12)¹²⁰} -1]/(0.008333334)
FV = - (-100) (1+0.008333334 * 0)[{(1+0.008333334)¹²⁰} -1]/0.008333334
FV = 100 [{(1.008333334)¹²⁰} -1]/0.008333334
FV = 100 [2.707041491} -1]/0.008333334
FV = 100 [1.707041491]/0.008333334
FV = 100 [204.8449]
FV = $20,484.50

Future Value of annuity due

As explained earlier, an annuity due makes start of period payments and we denote this by using a value of 1 for TYPE in the TVM equation, thus if the payments in our last example were made at the start of each month then finding future value of this annuity due will be nothing more than product of result of our calculations and (1+i * TYPE) where i is the rate of interest per month and TYPE is 1. This will make the future value of annuity due higher than what it is for ordinary annuity because the payment in the last month is compounded for an extra period.
FV = -PV(1+i)ⁿ - PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i
FV = -0(1+10%/12)¹²⁰ - (-100) (1+10%/12 * 1)[{(1+10%/12)¹²⁰} -1]/(0.008333334)
FV = 100 (1+0.008333334 * 1)[{(1+0.008333334)¹²⁰} -1]/0.008333334
FV = 100 (1.008333334)[204.8449]
FV = 100 (206.55194084698966)
FV = $20,655.20

Future Value of a lump-sum

Future value of lump-sum is also found using the same FV formula we had derived a bit earlier, the only aspect that will differ in finding a future value of a lump-sum has to do with PMT being zero. As there is no periodic payment thus we will only have to deal with two variables of FV and PV along with N and i. Let me illustrate this with an example where we have to find the future value of $1000 at an interest rate of 10% for a period of 5 years. Follow the steps outlines in the following paragraph to see the use of TVM equation in action
PV = -1000
PMT = 0
I = 10% = 0.10
N = 5
FV = ?
FV = -PV(1+i)ⁿ - PMT (1+i*TYPE)[{(1+i)ⁿ} -1]/i
FV = -(-1000)(1+10%)⁵ - 0 (1+10%*0)[{(1+10%)⁵} -1]/10%
FV = 1000(1+0.10)⁵
FV = 1000(1.1)⁵
FV = 1000(1.61051)
FV = $1,610.51