# Time Value of Money (TVM)

The two principals in wealth building that an investor must understand are the Time Value of Money and Leverage. This post describes TVM and a few things about financial calculators.

But first 5 definitions: Compounding(i) & Discounting(i), Present Value(PV), Future Value (FV), Payment (PMT), Holding Period (n).

(A)Let’s say you invest $1000 today (PV) at 10% annually for 5 years (n) without making any additional Payments. You let the interest re-invested in the account therefore Compound over the next 5 years.

How much the Future Value of your investment be after 5 years of Holding Period (n)?

n=0, PV=$1,000

n=1, FV=$1,100

n=2, FV=$1,210

n=3, FV=$1,331

n=4, FV=$1,464

n=5, FV=$1,610

Discounting is the reverse process. You have $1,000 today and would like to have $1,610 in 5 years. What would be the Discount Rate? In other words, at what rate the FV of $1,610 has to be Discounted to arrive at the PV of $1000. From the example above it is 10%

The discount rate computation is not as intuitive as the compounding calculation shown above.

(B)Now let’s say you would like to have $2,000 in 5 years. The discount rate then is 14.87%. I.e. the FV of $2,000 is discounted at 14.87% for 5 years arrives at $1,000 to be invested today.

(C)Another example, you know the discount rate for an investment vehicle is, say, 13%. Your would like to have $2,000 in 5 years. How much your initial investment (PV) should be? You must invest $1,085.52 today at 13% for 5 years to have $2,000 then.

You get the idea. The financial calculators make this computation simple. The financial calculators have 5 keys for the above definitions. They are FV, PMT, PV, n, and i.

You input 4 of the 5 variables and solve for the 5th one.

(D)Enter for FV=$2,000, n=5, i=13%, PMT=0. Solve for PV. You get -$1,085.52. It is a negative value to denote cash out of pocket invested today.

(E)Enter for FV=$2,000, PV=-$1,000 (a negative number since cash out of pocket), n=5, PMT=0. Solve for i. You get 14.87%.

(F)So far there has not been any payments throughout the 5 years. Say you add $600 a year to your account. What is the FV?

Enter for PV= -$1,000(cash out of pocket), PMT= -$600(cash out of pocket) , n=5, i=10%. Solve for FV. It is $5,640.

A word of caution: ensure your n and i are of the same unit, for example monthly or annually.

(G)Say, instead, you are adding $50 a month to your account. It is still $600 a year. How would that change the results? What do you punch in your calculator?

PV= -$1,000, PMT= -50, n=5×12, i=10%/12 (make i and n of the same unit, months, to match the monthly payments). Solve for FV.

FV would be $5,549.

Why F & G have different results? They both have added $600 every year! The assumption is that the payments are made at the beginning of each period. In F, $600 is added to the account at the beginning of the year. Where in G, $50 is added to the account at the beginning of each month.

The F & G examples illustrate the **Time Value of Money**.

Here is the theory:

- If you receive a dollar today it has a greater value than receiving it in the future.
- The dollar can be invested today. You lose the opportunity to invest if you receive the dollar in future (Opportunity Cost)
- you take no risk if you get your dollar today

The interest rate (or the discount rate) of an investment then be adjusted to take into account the factors above: time, risk, opportunity cost

(H) In conclusion, here is one to think about: You invest $65,000 today. It pays 2% a year, and $600 a month as dividend. The $600 is re-invested in another instrument paying 4% annually. What is your total rate of return on the investment after 10 years?

What is a series of payments to be received in future worth today? **Net Present Value (NPV)**. More on this topic in another post.