Today, we will understand that why it's crucial to include the time reference in your investments. How money changes its value at different timestamps and a bit about interest rates.
So what's the time value of money?
The money you have now is worth more than the identical sum in the future because of its potential earning capacity.
If you are given an opportunity to take 100 dollars now vs 100 dollars after 1 year. You should choose the first option. That is because the future value of money will definitely be more than its present value. For example -
This link is established through interest rates.
Interest rates
1. Its interpretations
We will understand it by taking an example. Say you lend 100 today and receive 110 after a year.
The required rate of return – You require a 10% interest rate or an increase in value to engage in this transaction.
Discount rate – The amount 110 is being discounted by 10% to get the present value of 100
Opportunity cost – The cost you forego. Say instead of investing, you spent those 100 dollars to eat at your favorite restaurant. Here, you have foregone the opportunity to earn 10% on those 100 dollars.
2. Investor's perspective
Investors can see interest rates as a sum of multiple components
Real risk-free interest rate – the rate you get on a security that has no risk, extremely liquid and we assume there’s no inflation.
Inflation premium – expected annual inflation in the upcoming period.
Default risk premium – the amount investor requires because of the risk of default. Say, two of your friends Akash and Ashish are asking for 100 dollars each. You know that Akash has previously taken money from you and hasn’t returned yet. He might not pay this time as well. So, you charge a higher interest rate to him than Ashish although the amount they are asking is the same.
Liquidity premium – premium an investor demands because of the lack of liquidity of the investment.
Maturity premium – Say you have an option to choose to invest among 2 bonds. Bond A has a maturity of 3 years, and Bond B has a maturity of 30 years. You would obviously require high-interest rates from Bond B because you will have to wait for 30 more years to get repaid. A lot can happen in that timeline like financial troubles, risk of not fully paid, increase in interest rates, and bond values might fall.
Future Value of Money
Future value is the value of cash or an asset at a particular date in the future. It shows you the amount to which a current asset would grow over some time.
Future value of a single cash flow
Now, assume if it were simple interest, with 100 as initial investment, the value after 2 years would have been 120. So why is there a difference?
The answer lies in compounding. In the above example, the 10 dollars which is the interest after 1 year is also receiving an interest of 1 dollar. So, the total amount after two years is 121 dollars.
Continuous Compounding
Continuously compounded interest assumes interest is compounded and added back into the balance an infinite number of times, or in other words, infinite compounding periods per year
The concept of continuously compounded interest is important in finance even though it’s not possible in practice.
Stated and effective rates
Effective rates can vary from the stated rates. Effective rates depend on the number of compounding periods.
a. with a discrete number of compounding periods:
b. with continuous compounding:
Future value of a series of cash flows
Few terms that we need to know before moving further.
Annuity: a finite set of level (equal) sequential (at an equally spaced time) cash flows.
Ordinary annuity: An annuity where the first cash flow occurs one period from today.
Annuity due: here the first cash flow occurs immediately.
Perpetuity: a set of level never-ending sequential cash flows with the first cash flow occurring one period from today.
Future value of the ordinary annuity
We will take an example here.
Say Annuity payments = 1000, rate = 5% and N = 5, what is the FV5 ?
Brute force technique: for each cash flow, we can calculate its value at T=5 using FV of a single cash flow technique, and then sum all those five up.
However, there is a formula that reduces this redundant process.
Unequal cash flows
Say we have a scenario where we have to find FV3,
Here, we can simply compound individual cash flows using the brute force method and add all three up.
Present Value of Money
Present value (PV) is the current value of a future sum of money or stream of cash flows given a specified rate of return. Present value takes the future value and applies a discount rate or the interest rate that could be earned if invested.
Present value of a single cash flow
Note: For a given discount rate, the farther in the future the amount to be received, the small the amount’s present value. You can clearly see it in the formula of PV. As the N increases, the denominator increases, hence the PV decreases.
Taking holding time constant, the larger the discount rate, the smaller the present value of a future amount. You can also comprehend that by looking at the PV formula. As r increases, the denominator increases, hence PV decreases.
Present value of a series of cash flows
Present value of a series of equal cash flows
Say we have an ordinary annuity with A = 10, r = 5% and N = 5
One technique could be brute force, where we can compute PV for each cash flows and add them up.
However, for simplification, we have a formula as well -
Present value of a series of cash flows (annuity due)
Taking the same previous example here -
The first technique would be brute force. After calculating, you would observe that the Present value in annuity due is greater than the Present value in the ordinary annuity. This is because in annuity due, we are receiving money faster than an ordinary annuity. You can also see it in the image.
The second way would be to find PV using a formula.
Present value of a perpetuity
PV = A/r
Where A is the annuity amount and r is the rate.
Example – Say A = 10 and r = 0.05
Present value indexed at times other than t= 0
An annuity or perpetuity beginning sometime in the future can be expressed in present value terms one period prior to the first payment.
Example = say A = 10 and r = 10%
Here, the PV of this perpetuity at year 4 = 10/0.1 = 100
Present value of a series of unequal cash flows
This is rather basic. Here you just have to find the present value of each individual cash flows and sum them up.
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