Some of my friends wanted to know more about the concepts and mechanics of time value, I have tried to put few examples along with it.
The process of discounting future cash flows converts them into cash flows in present value terms. Conversely, the process of compounding converts present cash flows into future cash flows.
Time Value Principle 1: Cash flows at different points in time cannot be compared and aggregated (i.e. added). All cash flows have to be brought to the same pointin time before comparisons and aggregations can be made.
We will consider working out PVs of five types of cash flows – simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities.
Simple Cash Flows
A simple cash flow is a single cash flow at a specified future point of time; it can be depicted on a time line:
Where, CF_{t} = the cash flow at time t. This cash flow can be discounted back to the present using a discount rate. Concurrently, cash flows in the present can be compounded to arrive at an expected future cash flow.
I. Discounting a Simple Cash Flow
Discounting a cash flow converts it into present value rupees and enables the user to add and compare the cash flows. Suppose some one offers you two choices, Choice A is Rs 2500 after one year and Rs 2500 after 4 years or alternatively choice B is Rs 2500 after two years and Rs 2500 after three years. Which one should you choose? You can only decide by determining the present values of the two choices and one having higher PV is clearly the desirable choice. Suppose, considering the inflation rate, your preference to have money today rather than later and the risk associated with the cash flow coming from the promisor is 11%. We can now compare the two as under:
t 
CF_{t} 
PV


1 year 
2500 
2500/1.11 
2252.25 
2 years 
2500 
2500/1.11^{2} 
2029.06 
3 years 
2500 
2500/1.11^{3} 
1827.98 
4 years 
2500 
2500/1.11^{4} 
1646.83 
Choice A = 2252.25 + 1646.83 = 3899.08
Choice B = 2029.06 + 1827.98 = 3857.03
This makes Choice A better and more desirable. Please note that for discounting (to get the PV) we have used the same formula used for compounding and getting the future value an amount lent today.
Present Value of Simple Cash Flow = CFt/(1+r)^t
Where CF_{t} = Cash Flow at the end of time period t and r = Discount Rate.The present value of a cash flow will decrease as the discount rate increases and continue to decrease the further into the future the cash flow occurs. We will illustrate this point with an example. Assume that you own Bandrasoft, a small software firm (located on BKC). You are currently leasing your office space from BMC, and expect to make a lump sum payment to them of Rs 2,00,00,000 ten years from now. Assume that an appropriate discount rate for this cash flow is 10%. The present value of this cash flow can then be estimated —
Present Value of Payment = 2,00,00,000/ 1.1^{10} = 77,10,866. We have worked out the PV with different discount rate :
II. Compounding a Cash Flow
Current cash flows can be moved to the future by compounding the cash flow at the appropriate interest rate. Future Value of Cash Flow = CF_{0 }(1+ r)^{t}
Where ; CF0 = Cash Flow now and r = Discount rate . Again, the compounding effect increases with both the discount rate and the compounding period.
The Rule of 72 : A Short Cut to estimating the Compounding Effect
In a pinch, the rule of 72 provides an approximate answer the question “How quickly will this amount double in value?” by dividing 72 by the interest rate; thus, a cash flow growing at 6% a year will double in value in approximately 12 years, while a cash flow growing at 9% will double in value in approximately 8 years.
III. The Frequency of Discounting and Compounding
The frequency of compounding affects both the future and present values of cash flows. Quite often cash flows are assumed to be discounted and compounded annually, i.e., interest payments and income are computed at the end of each year, based on the balance at the beginning of the year. In some cases, however, the interest may be computed more frequently, such as on a monthly or semiannual basis. In these cases, the present and future values may be very different from those computed on an annual basis; the stated interest rate, on an annual basis, can deviate significantly from the effective or true interest rate. The effective interest rate can be computed as follows
Effective Interest Rate = {1+ (Stated Annual Interest Rate/n)}^{n} – 1
Where, n = number of compounding periods during the year (2=semiannual; 12=monthly)
For instance, a 10% annual interest rate, with semiannual compounding, effective interest rate = (1+.10/2)^{2} 1 = 1.052 – 1 = .10125 or 10.25%
As compounding becomes more frequent, the EAR rate increases, and the PV of future cash flows decreases.
Annuities
An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. Defining A to be the annuity. An annuity can occur at the end of each period, as in the time line shown below or at the beginning of each period.
I. Present Value of an EndofthePeriod Annuity
The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present and then adding up the present values. Alternatively, a formula can be used in the calculation. In the case of annuities that occur at the end of each period, this formula can be written as
where
A = Annuity
r = Discount Rate
n = Number of years
Accordingly, the notation we will use in the rest of this book for the present value of an annuity will be PV(A,r,n).
Illustration : Estimating the Present Value of Annuities
Assume again that you are the owner of Vashisoft, and that you have a choice of buying a copier for 150,000 cash down or paying 50,000 a year for 5 years for the same copier. If the opportunity cost is 12%, which would you rather do?
PV of annuity = 50000 (11/1.12^{5})/0.12 = 180239
The present value of the installment payments exceeds the cashdown price; therefore, you would want to pay Rs 1,50,000 in cash now. Alternatively, the present value could have been estimated by discounting each of the cash flows back to the present and aggregating the present values.
Illustration : Present Value of Multiple Annuities
Suppose you are the pension fund consultant to Tata AIG, and that you are trying to estimate the present value of its expected pension obligations, which amount in nominal terms to the following:
Years Annual Cash Flow
1 – 5 Rs 200 crores
6 – 10 Rs 300 crores
11 – 20 Rs 400 crores
If the discount rate is 10%, the present value of these three annuities can be estimated as follows:
Present Value of first annuity = Rs 200 crores * PV (A, 10%, 5) = Rs 758 crores
Present Value of second annuity = Rs 300 million * PV (A,10%,5) / 1.10^{5} = Rs 706 crores
Present Value of third annuity = Rs 400 crores * PV (A,10%,10) / 1.10^{10} = Rs 948 crores
The present values of the second and third annuities can be estimated in two steps. First, the standard present value of the annuity is computed over the period that the annuity is received. Second, that present value is brought back to the present. Thus, for the second annuity, the present value of Rs 300 crores each year for 5 years is computed to be Rs1,137 crores; this present value is really as at the end of the fifth year. It is discounted back 5 more years to arrive at today’s present value which is Rs 706 crores.
Cumulated Present Value = Rs 758 crores + Rs 706 crores + Rs 948 crores = Rs 2,412 crores
II. Amortization Factors – Annuities Given Present Values
In some cases, the present value of the cash flows is known and the annuity needs to be estimated. This is often the case with home and automobile loans, for example, where the borrower receives the loan today and pays it back in equal monthly installments over an extended period of time. This process of finding an annuity when the present value is known is examined below —
Illustration : Calculating The Monthly Payment On A House Loan
Suppose you are trying to borrow Rs 20,00,000 to buy a house on a 20year mortgage with monthly payments. The annual percentage rate on the loan is 9.75%. The monthly payments on this loan can be estimated using the annuity due formula:
Monthly interest rate on loan = 0.0975/12 = 0.008125
Monthly mortgage payment = A(PV,r,n) = 20,00,000 x 0.008125 / (11/1.008125^{300}) =17,822.75
This monthly payment is an increasing function of interest rates. When interest rates drop, homeowners usually have a choice of refinancing, though there is an upfront cost to doing so. We examine the question of whether or not to refinance later in this lesson.
III. Future Value of EndOfThePeriod Annuities
In some cases, an individual may plan to set aside a fixed annuity each period for a number of periods and will want to know how much he or she will have at the end of the period. The future value of an endoftheperiod annuity can be calculated as follows:
FV of an annuity = FV(A,r,n) = A{(1+r)^{n} – 1)}/r
Suppose you are saving every month Rs 17,000 in your provident fund account which pays 8.5% interest every half year. How much would your fund make after 30 years when you plan to retire?
Interest rate = 8.5%/2 = 0.085/2 = 0.0425
Your savings every half year = 7,000 x 6 = 42,000
FV of your fund = 42,000{(1.0425)^{60} 1}/0.0425 = 1,10,18,479
IV. Annuity Given Future Value
Individuals or businesses who have a fixed obligation to meet or a target to meet (in terms of savings) some time in the future need to know how much they should set aside each period to reach this target. If you are given the future value and are looking for an annuity – A(FV,r,n) in terms of notation:
Illustration : Sinking Fund Provision on a Bond
In any balloon payment loan, only interest payments are made during the life of the loan, while the principal is paid at the end of the period. Companies that borrow money using balloon payment loans or conventional bonds (which share the same features) often set aside money in sinking funds during the life of the loan to ensure that they have enough at maturity to pay the principal on the loan or the face value of the bonds. Thus, a company with bonds with a face value of Rs 100 crores coming due in 10 years would need to set aside the following amount each year (assuming an interest rate of 8%):
Annually setting aside= Rs 100,00,00,000 x 0.08/{(1.08)^{10} 1} = 6,90,29,489
The company would need to set aside Rs 6.9 crores at the end of each year to ensure that there are enough funds (Rs 100 crores) to retire the bonds at maturity.