Present value and future value of cash flows

CFA level I / Quantitative Methods: Basic Concepts / The Time Value of Money / Present value and future value of cash flows

The present value (PV) is the discounted value of all the future cash flows today. The future value (FV) is the compounded value of all the cash flows at a future date.

Example 1: Present value of a single sum of money

Matt wants to invest in a security that will pay $1,000 after a period of five years. The required return on the security is 12 percent per annum. How much amount should Matt pay for the security today?

Solution:

To solve such problems we need to identify what is given to us in the problem statement and what is asked of us to calculate. Every time value of money problem will have cash flows (PV, FV, and PMT), the number of periods (N), and periodic interest rate (I/Y).

In this problem, the $1,000 will be received after five years. So, it is a future value (FV). The period frequency is annual and a total number of periods is five (N). The periodic interest rate (I/Y) is 12 percent.

Using Financial Calculator:

FV = 1,000
N = 5
I/Y = 12
CPT-> PV = $567.43.

So, Matt should pay $567.43 for the security to earn 12 percent of interest rate compounded annually for five years.

Example 2: Future value of a single sum of money

Todd has just deposited $20,000 in a bank that has promised him to pay an interest rate of 6 percent per annum compounded annually for three years. What will be the value of his money after three years?

Solution:

In this problem, we have been given present value (PV) which equals $20,000. The periodic interest rate (I/Y) is 6 percent. The number of periods equals three, and we need to calculate the future value.

Using Financial Calculator:

PV = 20,000
I/Y = 6
N = 3
CPT->FV = 23,820.32

Annuity: Annuities are the equal payments at regular interval of time. The interval length should be same for all the payments. For example - EMIs (equally monthly installments).

Annuities can be of two types: Ordinary Annuity and Annuity due. When the equal payment occurs at the end of each period, then the annuity is called as an ordinary annuity. When the equal payment occurs at the beginning of each period, then the annuity is called as an annuity due.

The future values of ordinary annuity and annuity due are given below:

FV_OA = (A/r)[(1+r)^N - 1]
FV_AD = (A/r)[(1+r)^N - 1](1+r)

where: A= Annuity amount

If we know the future value of an annuity, then it is extremely easy to get the present value. All we need to is to discount the future value to the present date. Dividing the FV by (1+r)^N, we get

PV_OA = (A/r)[1-(1+r)^-N]
PV_AD = (A/r)[1-(1+r)^-N](1+r)

For the exam, you need not remember these formulas because the financial calculator makes it very easy to solve annuity problems.

Solving annuity problem using the financial calculator: Solving ordinary annuity problem is very simple. All you need to do is to put the value of the annuity and press the button PMT. But you should be careful about the sign of annuity. Depending on your signs of cash inflows and cash outflows, you should keep the sign of annuity as annuity can be both cash outflow as well as a cash inflow.

If it is annuity due, then you need to change the setting of the financial calculator and change it to BGN mode. To change it to BGN mode, you need to press [2nd][BGN][2nd][SET]. Please remember to bring the calculator again to the END (default) mode by pressing those four buttons again after solving the problem. Otherwise, you can get the wrong answers for the other questions. Once you have set the BGN mode, then the problem will be solved just like the ordinary annuity problem.

The periodic interest rate in the annuity problems will be the interest rate for the period whose length is equal to the length between two consecutive annuity payments.

If you don't want to change the settings of the calculator, then there is another way to solve annuity due problem. You just need to solve the problem as if it were an ordinary annuity problem and then apply the following formulas to get the result for the annuity due.

FV_AD = FV_OA(1+r)
PV_AD = PV_OA(1+r)
PMT_AD = PMT_OA/(1+r)

The above formulas are very intuitive. We know that we make all the payment one period earlier for the annuity due. Therefore, its present value and future value will increase by that one period i.e. multiplied by (1+r). When the PV or FV are the same, then also, we can easily intuitively deduce that we need to make less annuity payment for the annuity due as compared to an ordinary annuity.

Example 3: Present value of an ordinary annuity

Paul wants to support salary one factory worker in an NGO for five years. The monthly salary of the worker is expected to remain constant at $3,500 per month for next five years. The salary will be paid at the end each month. How much money should Paul deposit in a bank today that can pay the worker for five years? Assume that the bank pays an interest rate of 3 percent per annum compounded monthly?

Solution:

The equal monthly payment of $3,500 will be made at the end of each month. So, it is a case of an ordinary annuity.

Using financial calculator:

PMT = 3,500
I/Y = 3/12 = 0.25 (periodic interest rate)
N = 5*12 = 60 (60 monthly periods in five years)
CPT->PV = $194,783.25

Paul needs to deposit $194,783.25 in his account for supporting the worker.

Example 4: Future value of an ordinary annuity

Kim plans to deposit $1,500 for three years in a bank at the end of every month. The bank is providing a return of 8 percent per annum compounded monthly on the deposits. What will be the worth of Kim's account in the bank at the end of three years?

Solution:

We need to identify the inputs for the financial calculator. Since $1,500 will be deposited every month, it is an annuity, and an ordinary one as the money will be deposited at the end of every month.

PMT = 1,500
I/Y = 8/12 = 0.666 (We need to make it periodic rate and period is monthly)
N = 3*12 = 36 (We have 36 monthly periods in 3 years)
CPT-> FV = 60,803.34

Her account will be worth $60,803.34 after three years.

Example 5: Present value of an annuity due

Roger's son has just joined a college for under-graduation. He will require $3,000 at the beginning of each month for his college expenses for four years. How much money should Roger deposit in a bank account today to take care of his son's college expenses assuming that the bank offers an annual return of 6 percent compounded monthly?

Solution:

His son requires the money at the beginning of each month. So, it is an example of an annuity due. We can solve it using two ways. One is to solve the problem using ordinary annuity and then multiply the present value by (1+r) to get the equivalent value of an annuity due. The other way is to change the setting of the calculator to the beginning (BGN) mode and then simply solve the problem.

Changing the settings to BGN mode: [2nd][BGN][2nd][SET]

PMT = 3,000
I/Y = 6/12 = 0.5
N = 4*12 = 48
CPT->PV = 128,379.67

Roger needs to deposit $128,279.67 in his bank account to take care of his son's college expenses.

Note: Please do not forget to change the settings of calculator to the default mode again after solving the problem by pressing the same buttons again - [2nd][BGN][2nd][SET]

Example 6: Future value of an annuity due

Justin wants to save up for his retirement. He plans to save $20,000 each year starting from the beginning of the current year. He will save the money for 15 years till his retirement. What will be the total amount in his retirement account assuming that the retirement account earns 8 percent per annum?

Solution:

This problem is an annuity due problem as the money is saved at the beginning of each year.

Using the financial calculator:

First, change the setting to beginning mode by pressing these buttons: [2nd][BGN][2nd][SET]

PMT = 20,000
I/Y =8
N =15
CPT->FV = 586,485.66

Justin will have $586,485.66 in his account at retirement.

A perpetuity is an ordinary annuity for perpetual time i.e. a set of never ending equal cash flows (A) that will start from one period from now.

The present value of a perpetuity is calculated by summing the present value of all future cash flows.

PV = A/(1+r) + A/(1+r)² + A/(1+r)³ + ...... ∞
PV = A/(1+r)[1 + 1/(1+r)+ 1/(1+r)² + ...... ∞]

This is an infinite geometric progression with a common ratio less than one. The sum of such progression (1+k+k² +....∞) is 1/(1-k)

Using that we will get, PV = A/r

Example 7: Present value of perpetuity

The annual membership charges for a club is $5,000 to be paid at the end of each year. What will be the charges of the lifetime membership assuming a discount rate of 10 percent?

Solution:

Lifetime membership is an example of perpetuity. Though it is not exactly a perpetuity as human life does not last forever. But assuming it to be a perpetuity, the present value will be equal to A/r = 5,000/0.10 = $50,000.

The answer is intuitive. You should pay that much amount so that the club can earn interest on that amount that is equal to the annual membership. On $50,000, the club will get an interest of $5,000 per annum at a rate of 10 percent per annum.

Series of unequal cash flows: For a series of unequal cash flows, we need to take each cash flow separately and then discount/compound that to the current/future date to get the present/future value and then add all of such values together to get the answer.

It is a long process and will take time on calculator doing that. However, there is a function named NPV that can do it quickly. We will be dealing with that function in the next reading. You are less likely to get a question from this.

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