Time Value of Money

August 22, 2017

One of the factors of production is capital.
The price entrepreneur has to pay to use the capital is known as interest.
Assuming that the capital does not depreciate, interest has to be paid because the capital has a time value.
In general, time value of money is a concept stating that the value generated today (such as profit) worth more than the value generated in the future.
The reason why we have time value of money is in the uncertainty over future payments that a project can generate.

Interest

Interest is an opportunity cost of using capital (or money) in the proposed undertaking.
Interest rate is interest expressed as a portion of underlying capital. We call value of the capital as principal. Usually quoted as per defined time period (days, months, years).
Time Horizon defines the amount of time periods at which principal and interest is paid in full.
A payment made at any given time is referred as a cash flow.

When the total interest earned or charged is linearly proportional to the initial amount of the principal, the interest rate, and the time horizon for which the principal is committed, the interest and interest rate are said to be simple.
Total payment (principal and interest) is calculated as follows:
\[ FV=P+I=PV(1+Ni) \]

Suppose the loan valued at $15,000 is taken at 10% simple interest. Suppose, the loan has to be paid in 10 years, what would be the final payment. Assume no intermediate payments are made.
Total debt of a firm is $15,000. The interest on a corporate debt is calculated as annual rate 10% with payments made semi-annually for 10 years. Find total amount of interest paid on this debt.

Whenever the interest charge for any interest period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period, the interest is said to be compound.
Total payment (principal and interest) is calculated as follows:
\[ FV=P+I=PV(1+i)^N \]
A special case of compound interest is continuous compounding. It assumes that interest is continuously compounded over time horizon.
The formula for continuous compounding is \[ FV=PVe^{iN} \]

Sometimes, we consider a single project that requires financing under different alternatives (different time horizons, interest, accrual periods). The only requirement is to fully repay any outstanding amount at the end of the time horizon. Which scheme would be better?
If principal is given, we can find total payment at the end of the period. We refer to it as present value ($PV$) of a loan.
If final payment is given, that is, a principal plus interest, we can calculate the present value of the loan. We refer to final payment as future value ($FV$) of a loan.

Cash flow diagram is a one-dimensional representation of all payments made on a loan.
The horizontal line is a time scale, with progression of time moving from left to right.
The arrows signify cash flows and are placed at the end of the period. If a distinction needs to be made, downward arrows represent expenses and upward arrows represent receipts.
The cash flow diagram is depended on the point of view. For example, it can be seen by perspective of a lender or a borrower. In this case, the payment direction would be reversed.

Solve previous examples using compounded interest.
Before evaluating the economic merits of a proposed investment, the XYZ Corporation insists that its engineers develop a cash flow diagram of the proposal. An investment of $10,000 can be made that will produce uniform annual revenue of $5,310 for five years and then have a market value of $2,000 at the end of year five. Annual expenses will be $3,000 at the end of each year for operating and maintaining the project. Interest rate is compunded annually at 10%. Draw a cash flow diagram for the five-year life of the project. Use the corporation's viewpoint. Should proposal be implemented?

A series of uniform receipts, each amount $A$, occuring at the end of each period for $N$ periods with interest $i$ per period is called annuity.
$PV$ is a present equivalent value that occurs one interest period before the first $A$.
$FV$ is a future equivalent value that occurs at the same time as the last $A$, and $N$ periods after $PV$.

To find $PV$ and $FV$, we need to convert all payments to the same time basis.
For $FV$, all future payments must be accrued to the period N.
\[ FV=A(1+i)+A(1+i)^2+\dots+A(1+i)^N=A\sum\limits_{j=1}^N (1+i)^{j} \]
For $PV$, all future payments must be discounted to the period 1.
\[ PV=\frac{A}{(1+i)}+\frac{A}{(1+i)^{2}}+\dots+\frac{A}{(1+i)^N}=A\sum\limits_{j=1}^N (1+i)^{-j} \]

Simplifications for the formulas exist. For instance, $FV$ can be simplified into
\[ FV=A\frac{(1+i)^N-1}{i} \]
For PV: \[ PV=A\frac{(1+i)^N-1}{i(1+i)^N} \]
Using two formulas above we can get a corresponding relation between $PV$ and $FV$.
\[ FV=\frac{PV}{(1+i)^N} \]

Finding $A$ given $PV$ and $FV$ is straightforward using the formulas for $PV$ or $FV$.
\[ A=FV\frac{i}{(1+i)^N-1}=PV\frac{i(1+i)^N}{(1+i)^N-1} \]
Finding $N$ and $i$ is not trivial however. It requires usage of numerical methods such as various interpolation techniques.

If payments starts sometime at future date, the annuity is known as a deferred annuity.

We considered two ways of calculated the interest on capital: simple and compound.
We introduced cash flow diagram as a visual representation of cash flows made in a project.