Time value of money
It is said that as dollar today is worth more than a dollar to be received tomorrow or at a later date. This is due to the uncertainty that it would actually be received tomorrow and some other variables such as inflation, interest rate and so on. It is only natural to prefer certainty over uncertainty that is why most people prefer to receive money now than to be promised the same at a later date in the future.
Financial decisions are affected by the timing of cash flows resulting from them that is why the concept of time value is very crucial in making financial decisions. Let us take an example from the above discussion. Remember we discussed some of the variables affecting the preference of a dollar today to a dollar In the future, and we said one of such variables is the interest rate. Assuming interest rates are positive and you receive one dollar today, you can take this dollar and invest it in an asset class such as bonds or money market instruments, and get back some returns on the initial dollar received.
From the concept of time value of money emanates future value and present value. Future value represents future value amounts resulting from current investments in an interest earning period. Present value on the other hand aims to assess the value of the investment and whichever future benefit that may arise as a result of current actions. These two concepts can be utilized when calculating loan amortization, yield to maturity of bonds, estimation of internal rates of return and so on.
Let us now take a look at the first of the two which is future value.
Future Value
As discussed earlier, it is the future amounts resulting from current investments in an interest earning period. Future value basically applies compounding to a present value now, that would materialize to some future amount. I am sure you may be wondering what compounding means. Compounding as it relates to future value is a situation where your interest receives interest. Confusing right?….Nah!… Simply put, it means that throughout the period, your interests are reinvested to generate their own earnings. Back to the discussion of the future value, we can say it is the sum to which a beginning amount of money would grow over n years at an interest rate of R% per annum.
From the above, we can deduce the formular for future value as:
FV = PV (1 + r)^{n}
Where,
FV= Future value at the end of the period
PV= Present Value or initial amount or principal
r= Interest rate
n= Number of years of period
Let’s take an example:
Example 1:
You arrange a fixed deposit contract with your bank on an initial amount of $5,000 that pays 10% interest compounded annually. How much will you receive at the end of 4 years.
Solution:
Recall that:
FV = PV (1 + r)^{n}
From the above question,
FV= ?
PV= $5,000
r = 10% or 0.1
n = 4 years
Our FV would then be:
FV= 5000 (1+ 0.10)^{4}
= 5000 (1.10)^{4}
= 5000 (1.4641)
= $7,320.5
At the end of four years, the amount you would receive would be $7,320.5
Another method that can be used to solve the above problem is the time line method
Amt @ the end of the period | |||||
5,000 (1.10) | 5,500 ( 1.10) | 6050 (1.10) | 6655 (1.10) | 7320.5 | |
end of yr 1 | end of yr 2 | end of yr 3 | end of yr 4 |
Explanation:
We start with the initial amount which is $5,000, then we multiply it by the interest for the year which is 10 %. For the first year we end up with $5,500 which is the initial amount of $5,000 + interest of 10%. We carry this over to the second year. In the second year we multiply our yearly interest to the 5,500 we carried over from the first year (this is the initial 5,000 + interest in the first year). This gives us $6,050 at the end of the second year. In the third year we repeat the same thing, we take the $6,050 of the previous year and multiply it by the same 10% interest and get $6655 at the end of the third year. In the fourth year, we also repeat the same process. We take the $6655 ending balance for the third year as our beginning balance In the fourth year, and then we multiply it by the interest rate of 10%, and end up with $7320.5 as our future value for the whole period.
Intra period compounding
There are sometimes when an amount is compounded more than once a year. We call it intra period compounding. Intra period compounding occurs when there is an increase in the number of times per year interest is compounded. The general formula for intra period compounding is:
FV = PV (1+ r/m) ^{m X n}
Semiannual compounding would then be:
FV = PV (1 + r/2)^{2 x n}
Quarterly compounding
FV = PV (1+ r/4) ^{4 x n}
Monthly compounding
FV = PV (1+ r/12) ^{12 x n}
Weekly compounding
FV = PV (1+ r/52) ^{52 x n}
^{ }
Continuous compounding:
FV = PVe^{rn}
Where e= 2.718
r= growth rate
n = number of period
Now lets take an example by using monthly compounding
Example: Suppose you deposit $1000 in a savings account that pays 10% interest compounded monthly, how much would you receive at the end of the 2 years?
Suggested Solution:
Recall that the formula for monthly compounding is
FV = PV (1+ r/12) ^{12 x n}
From the above question,
FV = ?
PV= $1000
r = 10% or 0.10
m = 12 (compounding is monthly)
n = 2 years
So we say:
FV = 1000 (1 + 0.1/12) ^{12 X 2}
= 1000 (1.0083)^{24}
= 1000 (1.21942)
= $1219.42
Present value
We said earlier that a dollar today is worth more than a dollar to be received tomorrow or at a later date. This is the same as saying that the present value of one dollar to be received in the future is less than the value of one dollar with you today. What the present value of a dollar today would be is dependent on the earning opportunities of the receiver, as well as the point of time the money is to be received.
When determining present value, we ask ourselves the following question: “If I could earn r% on my money, what would I be willing to pay to be availed the opportunity of earning a future sum accruing in n periods from now?”
Unlike the calculation of future period where we compound, the calculation of present value requires discounting (which is the opposite of compounding). Discounting helps the decision maker determine the present value of a future amount, assuming he has an opportunity to earn a certain return, r on the money.
What then is the formula for deriving the present value? Simple! The formula is:
Where
Fvn = Future value for number of years
r= rate of interest
n = number of years
We can also write the formula as:
PV = FV (1+r) ^{–n}
Let us take an example:
Example:
Tolu wishes to find the present value of $5,000 that is due to him 10 years from now. His opportunity cost is 10%. What is the worth now.
Suggested Solution:
Recall:
PV = FV (1+r)^{-n}
From the question,
PV = ?
FV = 5,000
r= 10% or 0.10
n = no of years
Fixing in the figures into the formular, we have:
PV = 5000 (1 + 0.10)^{-10}
= 5000 (0.3855)
= 1,927.7
= $1,928 (approx.)
What if we have a situation of irregular streams of cash flows in various future years? Easy, we just need to determine the present value of each future amount and add all the individual present value.
Let’s take an example
Year | Cash flow | |
1 | 500 | |
2 | 700 | |
3 | 400 | |
4 | 300 | |
Assuming we have the above listed cash flows in their respective years earning a minimum of 8%. What is the most you should pay for the opportunity to receive the streams of cash flow?
Suggested Solution:
Year | Cashflow | PVIF _{k,n} | PV |
1 | 500 | 0.926 | 463 |
2 | 700 | 0.857 | 599.9 |
3 | 400 | 0.794 | 317.6 |
4 | 300 | 0.735 | 220.5 |
1601 |
This means that the maximum amount we should pay to receive a stream of cashflows of $500+$700+$400+$300 (total of $1900) in 4yrs time at the rate of 8% is $1601.
Note: PVIF_{k,n }is the present value of interest factor at a rate r for n number of years. It is derived with the formula PVIF = (1+r)^{-n}
Where
r = rate of interest
n= number of years
Determining the interest rate in a present value or future value formula.
From the previous discussions on present value and future value of a lump sum, the growth rate, r was always given in the question. What if we were told to find r if both present value and future value is given? I would take you through how to derive the interest rate through the table and mathematical formula.
Let’s assume that Franklin wishes to find the rate of interest or growth rate of the following cash flows occurring in these years.
Year | Cashflow |
2006 | 2674.11 |
2005 | 2546.78 |
2004 | 2425.5 |
2003 | 2310 |
2002 | 2200 |
Interest would have been earned for 4 years, and to find the rate at which this has occurred, we divide the amount received in the earliest year by that received in the latest year.
= 2200 / 2674.11 = 0.823
To determine the interest, we say: 1- (p)^{1/n}
Where
P = Figure derived by dividing the earliest year by the latest year.
n = Number of years
so we have
r = 1 – (0.823)^{1/4}
= 1 – (0.95)
= 0.05 or 5%
Using the mathematical formula of future value, we assume it is a lumpsum payment of $2200 in 2002 which earns interest for four years that results into $2674.11
With the formula =
FV = PV (1+r)^{n}
We can solve for r. Let’s assume from the table that:
Future Value (FV) = $2674.11
Present Value (PV) = $2200
N = 4 years
So we can say:
FV = PV (1+r)^{n}
^{ }
r = 1.05 – 1
r = 0.05 or 5%
I hope the above post was informative and helped you deal with problems in future value and present value calculations of a lumpsum money. For calculations on the future value and present value of an annuity, click here
Recommended Readings
Frank Fabozzi., Financial management and analysis.
Jae sim., Joel siegel., Schaum’s outline of financial management
Leave a Reply