Introduction to Microeconomics 12 Week

August 22, 2017

Time Value of Money

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.

Simple Interest

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:
\[ FV=P+I=PV(1+Ni) \]

Example

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.

Compound Interest

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 \[ FV=PVe^{iN} \]

Debt Estimation

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

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.

Example

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. Draw a cash flow diagram for the five-year life of the project. Use the corporation's viewpoint. Should proposal be implemented.

Finding Loan Parameters Given FV

To find loan parameters, the solution can be derived from the $FV$ formula
For interest rates, $i=\sqrt[N]\frac{FV}{PV}-1$.
For number of periods, $N=\frac{\ln\frac{FV}{PV}}{\ln(1+i)}$.
For present value, $PV=\frac{FV}{(1+i)^N}$

Examples

What us the present equivalent of $18,000 to be received in 15 years when the interest rate is 7% per year?
How long does it take (to the nearest whole year) for $1,000 to quadruple in value when the interest rate is 15% per year?
The first U.S. congress in 1789 set the president's salary at $25,000 per year. In 2014, the president's salary is $400,000 each year. What us the compounded average annual increase in the president's salary for the past 225 years?

Annuity and its Present and Future Values

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

Finding FV given Annuity value

To find $FV$, we need to bring each annuity payment to the common period base.
In case of $FV$, the common base is period $N$, thus each annuity payment is accrued to this period.
\[ FV=A+A(1+i)+A(1+i)^2+\dots+A(1+i)^N=\\A\sum\limits_{j=0}^N (1+i)^j \]
Simplifying terms, we can get simpler formula to calculate $FV$.
\[ FV=A\frac{(1+i)^N-1}{i} \]

Finding PV given Annuity value

To find $PV$, we need to bring each annuity payment to the common period base.
In case of $PV$, the common base is period $N$, thus each annuity payment is discounted to this period.
\[ PV=A+\frac{A}{(1+i)}+\frac{A}{(1+i)^2}+\dots+\frac{A}{(1+i)^N}=\\A\sum\limits_{j=0}^N (1+i)^{-j} \]

Finding PV given Annuity value cont.

Simplifying terms, we can get simpler formula to calculate $PV$.
\[ PV=A\frac{(1+i)^N-1}{i(1+i)^N} \]
Using formulas for $PV$ and $FV$ we can find $FV=PV(1+i)^N$.

Example

If a certain machine undergoes a major overhaul now, its output can be increased by 20%, which translates into additional cash flow of $20,000 at the end of each year for 5 years. If $i=15\%$ per year, how much can we afford to invest to overhaul this machine?
You can buy a machine for $100,000 that will produce a net income, after operating expenses, of $10,000 per year. If you plan to keep the machine for 4 years, what must the market (resale) value be at the end of 4 years to justify the investment? You must make a 15% annual return on your investment.

Finding Loan Parameters Given PV or FV

We can find the value of annuity when we are given $PV$ or $FV$, given respective simplified formulas.
\[ A=FV\frac{i}{(1+i)^N-1}=PV\frac{i(1+i)^N}{(1+i)^N-1} \]
Similarly, we can find value for number of periods
\[ N=\frac{\ln(\frac{FVi}{A}+1)}{\ln (1+i)}=\frac{\ln{1-\frac{PVi}{A}}}{\ln{(1+i)}} \]
The value for interest rate however cannot be explicitly found from the formula. Interpolation techniques, such as bisection method can be used to obtain $i$.

Example

An expenditure of $20,000 is made to modify a material-handling system in a small job shop. This modification will result in first-year savings of $2,000, a second-year savings of $4,000, and a savings of $5,000 per year thereafter. How many years must the system last if an 18% return on investment is required? The system is tailor made for this job shop and has no market (salvage) value at any time.

Deferred Annuities

If the cash flow does not begin until some later date, the annuity is known as a deferred annuity.
To find the values of deferred annuity, simply use stardard formula and discount the value by number of periods it is deferred.

Example

Jason makes six EOY deposits of $2,000 each in a savings account paying 5% compounded annually. If the accumulated account balance is withdrawn 4 years after the last deposit, how much money will be withdrawn?

Nominal and Effective Interest Rates

If interest period (i.e. time between successive compounding) is less than one year, often the interest rate is still quoted on annual basis (i.e. year).
Thus if we say "12% compounded semiannually", it means 12% yearly rate is spread out over 2 semi-annual periods. This rate is known as nominal interest rate.
If we calculate the insterest rate at which the accrural happened, we get the effective interest rate. Effective interest rate is the exact rate of interest earned on the principal during one year.

Example

Compute the effective annual interest rate in each of these situations:
10% nominal interest, compounded semiannually.
10% nominal interest, compounded quarterly.
10% nominal interest, compounded weekly.

Wrap up

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.
We analysed annuities and derived formulas for its present and future values.
We introduced nominal and effective interest rates.