Present value
Present value is the concept that states an amount of money today is worth more than that same amount in the future. In other words, money received in the future is not worth as much as an equal amount received today.
Present value is important because it allows investors to compare values over time. PV can help investors assess future financial benefits of current assets or liabilities. Used in areas like financial modeling, stock valuation, and bond pricing, based on its future returns, investors can calculate present value.
Where,
PV = present value
FV = future value
r = rate of return
{n} = number of periods
Future value
Future value (FV) is the value of a current asset at a future date based on an assumed rate of growth. The future value (FV) is important to investors and financial planners as they use it to estimate how much an investment made today will be worth in the future.
Where,
PV = present value
FV = future value
r = rate of return
{n} = number of periods
Example 1
Suppose you are depositing an amount today in an account that earns 5% interest, compounded annually. If your goal is to have $5,000 in the account at the end of six years, how much must you deposit in the account today?
Solution
The following information is given:
Future value = $5,000
Interest rate = 5%
Number of periods = 6
We want to solve for the present value.
PV = FV/ (1 + r)t
Inserting the known information,
PV = 5,000 / (1 + 0.05) 6
PV = 5,000 / (1.3401)
PV = 3,731
Example 2
What is the present value of $1,000 received in two years if the interest rate is?
(a) 12% per year discounted annually
(b) 12% per year discounted semi-annually
Solution
- We want to solve for the present value.
PV = FV/ (1 + r)t
Inserting the known information,
1,000 / (1 + 0.12) 2
797.19
b. 1,000 / (1 + 0.12/2) 2*2
792.09
Example 3
Using present value formula
Rs7,000 for 10 years from now at 7% is worth how much today?
Solution
We want to solve for the present value.
PV = FV/ (1 + r)t
Inserting the known information,
7,000 / (1 + 0.07)10
Rs. 3558.45
Example 4
You are scheduled to receive Rs. 13,000 in two years. When you receive it, you will invest it for six more years at 8 percent per year. How much will you have in eight years?
Solution
We want to solve for the future value.
FV = PV/ (1 + r)t
Inserting the known information,
13,000 (1 + 0.08) 6
Rs. 20,629.37
Key takeaways –
- Present value states that an amount of money today is worth more than the same amount in the future.
- Future value (FV) is the value of a current asset at some point in the future based on an assumed growth rate.
Calculation of interest is one of the most basic uses of mathematics in finance.
We can define the simple interest as- “the price has to be paid for the use of a certain amount of money or principal for certain period, is called interest”
Amount- This is the sum of the principal and the interest at any time.
The rate of interest is denoted by ‘i’.
Let the principal is 100 rs. And the interest is 8 then we can say that the rate of interest is 8 percent per annum or we can write it as r = 8%
Suppose ‘P’ is the principal and ‘n’ is the time for which the principal is given and ‘r’ be the rate of interest per annum
‘I’ be the amount of interest
And ‘i’ be the rate of interest per unit
Then-
Here- i = r/100
We can calculate the amount as-
Example: Sushmita invested 5000rs. At the rate of 8 per cent per annum then what will be the value of the amount she invested in 5 years.
Solution
Here P = 5000, i = 8/100 = 0.08, n = 5, now
Hence the required amount is - 7000
Example: Find out the simple interest on 5600 rs. At 12 percent per annum from July 15 to September 26, 2020.
Solution
Here we calculate the time-
Total days from July 15 to sept 26 = 73 days or 73/365 = 1/5 years
And P = 5600, i = 12/100 = 0.12
Simple interest = P.i.n. = 5600
Hence the simple interest is – 134.40
Example: Harpreet invests 1200rs. At 10 percent per annum for some time and it becomes 1560 then find the principal when that will become 2232 at 8 percent p.a. in the same time.
Solution
Here in first situation- P = 1200, A = 1560 and i = 0.10
So that,
In second situation-
A = 2232, n = 3, i = 0.08
Compound interest-
In compound interest, the principal does not remain same but increases at the end of each interest period.
Let-
P- Principal
A - Amount
i = interest on re. 1 for a year
n = interest period
Then the amount can be calculated as-
And
Note- By using algorithm the above formula can be written as-
Note- if the compound interest is paid half-yearly, quarterly, monthly instead of a year there will be different formulae as given in the table below-
Time | Amount |
Annually | |
Half-yearly | |
Quarterly |
Example: Aman invests 1000 rupees at 5 percent p.a for four years then find the compound interest on it.
Solution
Here P = 1000, i = 0.05 and n = 4
Then we know that-
On taking log, we get-
Compound interest will be-
Which is the required answer.
Example: A sum of money invested at C.I. payable yearly amounts to 10, 816 Rs. at the end of the second year and to 11,248.64 rs. at the end of the third year. Find the rate of interest and the sum.
Solution
Here A1 = 10,816, n = 2, and A2 = 11,248.64, n = 3
We know that
A = P (1 + i)n we get,
10,816 = … (i)
11,248.64 =… (ii)
Here on dividing equation (2) by (1)-
We get-
And
Hence the rate is 4 percent.
Now from first equation-
10,816 =
Or
Now-
P = antilog 4.000 = 10,000
Therefore the require answer- 10,000
Nominal rate of interest is the rate of interest per annum which is compounded yearly, half yearly, quarterly, monthly, n times in a year or continuously.
Effective rate of interest is the rate of interest per annum compounded only once in a year.
There is a relationships between nominal and effective rate of interest under two different conditions-
- If compounding is ‘n’ times in a year-
Where ‘r’ is nominal rate and ‘R’ is effective rate.
2. If compounding is continuous-
Or
Relationship between two nominal rates-
If interest is compounded quarterly at percent and the interest is compounded half yearly at percent, then the relationship between the two is-
Example: Ronak deposited Rs. 10,000 in a bank for 3 years. Bank gives two offers 10 percent compounded quarterly or 8% compounded continuously, then which offer is preferable for Ronak?
Solution
Balance after three years under first offer-
Balance after 3 years under second offer-
So that we can conclude that the first offer is preferable for Ronak.
When a sequence of payments of some fixed amount are made into or taken out of an account at equal intervals of time, we call this an annuity. When the payments are made at the end of each period rather than at the beginning, we call it an ordinary annuity.
We will assume all annuities are ordinary and, furthermore, are certain, meaning the payments are made over a set period of time. All annuities discussed will also have the number of payments equal the number of compounding periods and all deposits/withdrawals will be stated.
Where
A = Future value of accumulated amount
PMT = Periodic payment
r = Annual percentage rate (APR) changed to a decimal
t = Number of years
m = Number of payments made per year
Example
A payment of $50 is made at the end of each month into an account paying a 6% annual interest rate, compounded monthly. How much will be in that account after 3 years?
Solution: We see that 𝑅 = 50, 𝑟 = 0.06, 𝑚 = 12, and 𝑡 = 3
Using the formula for finding the future value of an ordinary annuity, we get
So there will be $1966.81 in the account after 3 years.
Example
Assume that you make monthly payments of $ 725 into an ordinary annuity paying 6% compounded monthly. How much will be in the account after 8 years?
Solution
There will be $89050.70 in the account after 8 years.
Key takeaways - An ordinary annuity is a series of equal payments made at the end of consecutive periods over a fixed length of time
You may want to save regularly to have a fixed amount available in the future. The account that you establish for your deposits is called a sinking fund. Because a sinking fund is a special type of annuity, it is not necessary to find a new formula. We can use the formula for calculating the future value of an ordinary annuity that we have stated earlier. In this case, we will know the value of A and we will want to find R
Example
Assume that you wish to save $1,800 in a sinking fund in 2 years. The account pays 6% compounded quarterly and you will also make payments quarterly. What should be your quarterly payment?
Solution
Recall the formula for finding the future value of an ordinary annuity:
We see that 𝐴 = 1800, 𝑟 = 0.06, 𝑚 = 4, = 2
The quarterly payment should be $213.45
Example
Suppose you have decided to retire as soon as you have saved $1,000,000. Your plan is to put $200 each month into an ordinary annuity that pays an annual interest rate of 8%. In how many years will you be able to retire?
Solution
We see that 𝐴 = 1000000, 𝑅 = 200, 𝑟 = 0.08, 𝑚 = 12
So you will be able to retire in 44.3486 years
Key takeaways - A sinking fund is a fund containing money set aside or saved to pay off a debt or bond.
If the payments are made at the beginning of each period rather than at the end, we call it an annuity due
Present Value of an Annuity Due
The present value of an annuity due uses the basic present value concept for annuities, except we should discount cash flow to time zero.
The formula for the present value of an annuity due is as follows:
Or
Where:
- PMT – Periodic cash flows
- r – Periodic interest rate, which is equal to the annual rate divided by the total number of payments per year
- n – The total number of payments for the annuity due
Future Value of an Annuity Due
The future value of an annuity due uses the same basic future value concept for annuities with a slight tweak, as in the present value formula above.
To calculate the future value of an ordinary annuity:
Where:
- PMT – Periodic cash flows
- r – Periodic interest rate, which is equal to the annual rate divided by the total number of payments per year
- n – The total number of payments for the annuity due
Example
An individual makes rental payments of $1,200 per month and wants to know the present value of their annual rentals over a 12-month period. The payments are made at the start of each month. The current interest rate is 8% per annum.
Solution
FV of the Investment = $1,200 x 11.57
FV of the Investment = $13,886.90
Example
A company wants to invest $3,500 every six months for four years to purchase a delivery truck. The investment will be compounded at an annual interest rate of 12% per annum. The initial investment will be made now, and thereafter, at the beginning of every six months. What is the future value of the cash flow payments?
Solution
FV of the Investment = $3,500 x 10.49
FV of the Investment = $36,719.61
Key takeaways - If the payments are made at the beginning of each period rather than at the end known as annuity due
Annuity-
An annuity is a sequence of equal payment or a sequence of regular payment at regular intervals or in other words, An annuity is a fixed sum paid at regular intervals under certain conditions.
The length of time during which the annuity is paid can either be until the death of the recipient or for a guaranteed minimum term of years, irrespective of whether the annuitant is alive or not.
The time between payments is called the payment interval, and the time which the money is to be paid is called the term of the annuity.
Amount of an annuity:
Amount of an annuity is the total of all the installments left unpaid together with the compound interest of each payment for the period it remains unpaid.
The formulas to find the amount of annuity are given below-
When annuity is payable annually and interest is also compounded annually | |
annuity is payable half-yearly and interest is also compounded half-yearly | |
annuity is payable quarterly and interest is also compounded quarterly |
Present value of an annuity-
Present value of an annuity is the sum of the present values of all payments (or installments) made at successive annuity periods
The formulas to calculate present value ‘V’ of an annuity P are given below-
When V of an annuity P payable annually | |
When V of an annuity P payable half-yearly | |
When V of an annuity P payable quarterly |
The future value is calculated as-
Example: Sundar decides to deposit 20,000rs. at the end of each year in a bank which pays 10% p.a. compound interest.
If the installments are allowed to accumulate, what will be the total accumulation at the end of 9 years?
Solution
Suppose A be the total accumulation at the end of 9 years. Then we get-
Here P = 20,000 rs.,i = 10/100 = 0.1 and n = 9
Then
Hence the total required accumulation is 2,71,590rs.
Example: Rajeev purchased a flat valued at 3,00,000rs. He paid 2,00,000rs. at the time of purchase and agreed to pay the balance with interest of 12% per annum compounded half yearly in 20 equal half yearly installments.
If the first installment is paid after six months from the date of purchase, find the amount of each installment.
[Given log 10.6 = 1.0253 and log 31.19 = 1.494]
Solution
Here2,00,000 has been paid at the time of purchase when cost of the flat was 3,00,000, we have to
consider 20 equated half yearly annuity payment P when 12% is rate of annual interest compounded half
yearly for present value of 1,00,000rs.
So that-
Or
Then-
Hence the amount of each installment = 8,718.40
Suppose,
Taking log-
Hence
X = 0.3119
Key takeaways-
- Annuity-
An annuity is a fixed sum paid at regular intervals under certain conditions.
The length of time during which the annuity is paid can either be until the death of the recipient or for a guaranteed minimum term of years, irrespective of whether the annuitant is alive or not.
2. Amount of an annuity:
Amount of an annuity is the total of all the installments left unpaid together with the compound interest of each payment for the period it remains unpaid.
3. Present value of an annuity-
Present value of an annuity is the sum of the present values of all payments (or installments) made at successive annuity periods
Depreciation-
Depreciation is defined as the expensing of the cost of an asset involved in producing revenues throughout its useful life.
Depreciation expense affects the values of businesses and entities because the accumulated depreciation disclosed for each asset will reduce its book value on the balance sheet. Depreciation expense also affects net income. Generally the cost is allocated as depreciation expense among the periods in which the asset is expected to be used. Such expense is recognized by businesses for financial reporting and tax purposes.
Depreciation does not actually represent any kind of cash transaction. Instead, it simply represents how much of an asset's value has been used up over time and can be deducted as an expense.
The total depreciation over a period of time is known as "accumulated depreciation". The "book value" of an asset is calculated by deducting the accumulated depreciation from the original purchase price. The book value is what is reflected as the asset's value on the balance sheet.
There are four criteria used to calculate depreciation-
1- The initial cost of the asset.
2- The expected residual value (also known as salvage value) - this is the value of asset at the end of its useful life, which may be zero.
3- The estimated useful life of the asset.
4- An appropriate method of apportioning the cost of the useful life of the asset.
Equated monthly installments-
An equated monthly installment(EMI) is a fixed payment amount made by a borrower to a lender at a specified date e ach calendar month. Equated monthly installments are used to pay off both interest and principal each month so that over a specified number of years, the loan is paid off in full.
Note-
- An equated monthly installment (EMI) is a fixed payment made by a borrower to a lender on a specified date of each month.
- EMIs allow borrowers the peace of mind of knowing exactly how much money they will need to pay each month toward their loan.
- EMIs can be calculated in two ways: the flat-rate method or the reducing-balance method.
The EMI flat-rate formula is calculated by adding together the principal loan amount and the interest on the principal and dividing the result by the number of periods multiplied by the number of months.
The EMI reducing-balance method is calculated using the formula shown below, in which P is the principal amount borrowed, I is the annual interest rate, r is the periodic monthly interest rate, n is the total number of monthly payments, and t is the number of months in a year.
Example: A man buys a machine for Rs.20,00. What its value be after 6 years, if it is assumed to depreciate at a fixed rate of 12% per annum.
Solution
Here we have P = 20,000, n = 6 and r = -0.12 (rate of interest is negative in depreciation)
Then,
So that the value of the machine in 6 years will be = 9288.08
Example: A machine valued at 500,000 depreciates at 6% per annum then in how many year its value will reduce to 100,000?
Solution Here P = 500,000, r = 0.06, n = ? and the final value is = 100,000
We know that-
Taking log on both sides-
So that-
Therefore approximately it will take 26 years for the value decline to 100,000
Example: Ayesha invests Rs. 3000 initially and then Rs. 1800 at the end of the first, second and third years and finally Rs. 600 at the end of 4th year.
If the interest is paid annually at the rate of 6.5% then find the value of the investment at the end of 5th year.
Solution
Rs. 3000 is invested for 5 years and grows to-
The three sums of Rs. 1800 are invested for 4, 3 and 2 years and grow in total-
And Rs. 600 is invested for 1 year and grows to-
Then the total value at the end of 5 years will be 11,280.81
Key takeaways-
- Equated monthly instalments- An equated monthly instalment (EMI) is a fixed payment amount made by a borrower to a lender at a specified date e ach calendar month. Equated monthly installments are used to pay off both interest and principal each month so that over a specified number of years, the loan is paid off in fullfill
- An equated monthly installment (EMI) is a fixed payment made by a borrower to a lender on a specified date of each month.
- EMIs can be calculated in two ways: the flat-rate method or the reducing-balance method.
- EMIs allow borrowers the peace of mind of knowing exactly how much money they will need to pay each month toward their loan.
References
- Practical business mathematics by S.A Bari
- Mathematics of commerce by K. Selvakumar
- Business mathematics with application by Dinesh Khattar and S.R Arora
- Statistical methods by Gupta SP