Assignment 8513 (Q NO.3)

Q.3

Discuss the following:                                                         (20)

1.       Time Value of Money

2.       Compounding

3.       Discounting

4.       Annuities

5.       Perpetuities

Ans part 1                       

Time Value of Money

The time value of money (TVM) is the concept that money available at the present time is worth more than the identical sum in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. TVM is also sometimes referred to as present discounted value.

The time value of money draws from the idea that rational investors prefer to receive money today rather than the same amount of money in the future because of money's potential to grow in value over a given period of time. For example, money deposited into a savings account earns a certain interest rate, and is therefore said to be compounding in value. 

Further illustrating the rational investor's preference, assume you have the option to choose between receiving $10,000 now versus $10,000 in two years. It's reasonable to assume most people would choose the first option. Despite the equal value at time of disbursement, receiving the $10,000 today has more value and utility to the beneficiary than receiving it in the future due to the opportunity costs associated with the wait. Such opportunity costs could include the potential gain on interest were that money received today and held in a savings account for two years.

Depending on the exact situation in question, the TVM formula may change slightly. For example, in the case of annuity or perpetuity payments, the generalized formula has additional or less factors. But in general, the most fundamental TVM formula takes into account the following variables:

• FV = Future value of money
• PV = Present value of money
• i = interest rate
• n = number of compounding periods per year
• t = number of years

Based on these variables, the formula for TVM is:

FV = PV x [ 1 + (i / n) ] ^{(n x t)}

Assume a sum of $10,000 is invested for one year at 10% interest. The future value of that money is:

FV = $10,000 x (1 + (10% / 1) ^ (1 x 1) = $11,000

The formula can also be rearranged to find the value of the future sum in present day dollars. For example, the value of $5,000 one year from today, compounded at 7% interest, is:

PV = $5,000 / (1 + (7% / 1) ^ (1 x 1) = $4,673

Ans part 2

Compounding

Compounding is the process in which an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time. This growth, calculated using exponential functions, occurs because the investment will generate earnings from both its initial principal and the accumulated earnings from preceding periods. Compounding, therefore, differs from linear growth, where only the principal earns interest each period.

Compounding typically refers to the increasing value of an asset due to the interest earned on both a principal and accumulated interest. This phenomenon, which is a direct realization of the time value of money (TMV) concept, is also known as compound interest. Compound interest works on both assets and liabilities. While compounding boosts the value of an asset more rapidly, it can also increase the amount of money owed on a loan, as interest accumulates on the unpaid principal and previous interest charges.

To illustrate how compounding works, suppose $10,000 is held in an account that pays 5% interest annually. After the first year, or compounding period, the total in the account has risen to $10,500, a simple reflection of $500 in interest being added to the $10,000 principal. In year two, the account realizes 5% growth on both the original principal and the $500 of first-year interest, resulting in a second-year gain of $525 and a balance of $11,025. After 10 years, assuming no withdrawals and a steady 5% interest rate, the account would grow to $16,288.95.

The formula for the future value (FV) of a current asset relies on the concept of compound interest. It takes into account the present value of an asset, the annual interest rate, and the frequency of compounding (or number of compounding periods) per year and the total number of years. The generalized formula for compound interest is:

FV = PV x [1 + (i / n)] ^{(n x t)}, where:

• FV = future value
• PV = present value
• i = the annual interest rate
• n = the number of compounding periods per year
• t = the number of years

Example

The effects of compounding strengthen as the frequency of compounding increases. Assume a one-year time period. The more compounding periods throughout this one year, the higher the future value of the investment, so naturally, two compounding periods per year are better than one, and four compounding periods per year are better than two.

To illustrate this effect, consider the following example given the above formula. Assume that an investment of $1 million earns 20% per year. The resulting future value, based on a varying number of compounding periods, is:

• Annual compounding (n = 1): FV = $1,000,000 x [1 + (20%/1)] ^{(1 x 1)} = $1,200,000
• Semi-annual compounding (n = 2): FV = $1,000,000 x [1 + (20%/2)] ^{(2 x 1)} = $1,210,000
• Quarterly compounding (n = 4): FV = $1,000,000 x [1 + (20%/4)] ^{(4 x 1)} = $1,215,506
• Monthly compounding (n = 12): FV = $1,000,000 x [1 + (20%/12)] ^{(12 x 1)} = $1,219,391
• Weekly compounding (n = 52): FV = $1,000,000 x [1 + (20%/52)] ^{(52 x 1)} = $1,220,934
• Daily compounding (n = 365): FV = $1,000,000 x [1 + (20%/365)] ^{(365 x 1)} = $1,221,336

As evident, the future value increases by a smaller margin even as the number of compounding periods per year increases significantly. The frequency of compounding over a set length of time has a limited effect on an investment's growth. This limit, based on calculus, is known as continuous compounding and can be calculated using the formula:

FV = PV x e ^{(i x t)}, where e = the irrational number 2.7183.

In the above example, the future value with continuous compounding equals: FV = $1,000,000 x 2.7183 ^{(0.2 x 1)} = $1,221,403.

Ans part 3

Discounting

Discounting is the process of determining the present value of a payment or a stream of payments that is to be received in the future. Given the time value of money, a dollar is worth more today than it would be worth tomorrow. Discounting is the primary factor used in pricing a stream of tomorrow's cash flows.

For example, the coupon payments found in a regular bond are discounted by a certain interest rate and added together with the discounted par value to determine the bond's current value.

From a business perspective, an asset has no value unless it can produce cash flows in the future. Stocks pay dividends. Bonds pay interest, and projects provide investors with incremental future cash flows. The value of those future cash flows in today's terms is calculated by applying a discount factor to future cash flows.

In general, a higher the discount means that there is a greater the level of risk associated with an investment and its future cash flows. For example, the cash flows of company earnings are discounted back at the cost of capital in the discounted cash flows model. In other words, future cash flows are discounted back at a rate equal to the cost of obtaining the funds required to finance the cash flows. A higher interest rate paid on debt also equates with a higher level of risk, which generates a higher discount and lowers the present value of the bond. Indeed, junk bonds are sold at a deep discount. Likewise, a higher the level of risk associated with a particular stock, represented as beta in the capital asset pricing model, means a higher discount, which lowers the present value of the stock.

Ans part 4

Annuities

An annuity is a series of equal payments made at equal intervals during a period of time. In other words, it’s a system of making or receiving payments where the payment amount and time period between payments is equal.

Example

Many people play the lottery in hopes to cash in on the big jackpot. Unfortunately, most people don’t win it big, but an extremely small percentage of people do. After they win, they often have to make the choice whether to be paid in a lump sum or in an annuity. For example, a million dollar jackpot could be paid out immediately in one lump sum of $600,000 or in $5,000 monthly installments for 15 years.

This option takes the time value of money into consideration. Notice that neither option actually pays out a full $1,000,000. This is because over time money should earn interest. Thus, $600,000 today will equal $1,000,000 in the future after interest is added up over the years. The same is true for the annuity payments.

Loans are also set up as annuities. Sometimes people don’t think of them as annuities because they are not receiving the payments. Remember annuities are just agreements with equal payments and time intervals. When a business signs a loan with a bank, it agrees to make a payment each month for specific amount. The payments are due each month until the loan principle is paid off.

In present value calculations, an annuity is a series of equal cash amounts occurring at equal time intervals. The identical cash amounts are sometimes referred to as payments, receipts, or rent.

Some examples of business transactions that form an annuity include:

1.   The equal amounts of interest paid every six months by the issuer of debt securities known as bonds.

2.   The monthly payments required by a lease agreement for equipment or a vehicle.

3.   The annual payments required by a purchase agreement.

The annuity payments are often discounted to arrive at their present value. The annuity payments can also be used to determine the effective interest rate that is embedded in an agreement.

Depending on the starting point of the first payment, an annuity will be further identified as an ordinary annuity, an annuity in advance, a deferred annuity, etc.

Ans part 5

Perpetuities

Perpetuity in the financial system is a situation where a stream of cash flow payments continues indefinitely or is an annuity which has no end. In valuation analysis, perpetuities are used to find the present value of a company’s future projected cash flow stream and company’s terminal value. Essentially, perpetuity is a series of cash flows that keep paying out forever.

Perpetuity refers to an infinite amount of time. In finance, perpetuity is a constant stream of identical cash flows with no end. The present value of a security with perpetual cash flows can be determined as:

The concept of a perpetuity is also used in a number of financial theories, such as in the dividend discount model (DDM).

An annuity is a stream of cash flows. Perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate. An example of a financial instrument with perpetual cash flows is the the British-issued bonds known as consols. By purchasing a consol from the British government, the bondholder is entitled to receive annual interest payments forever. Although it may seem a bit illogical, an infinite series of cash flows can have a finite present value. Because of the time value of money, each payment is only a fraction of the last.

Specifically, the perpetuity formula determines the amount of cash flows in the terminal year of operation. In valuation, a company is said to be a going concern, meaning that it goes on forever. For this reason, the terminal year is perpetuity, and analysts use the perpetuity formula to find its value.

Example

For example, if a company is projected to make $100,000 in year 10, and the company’s cost of capital is 8%, with a long-term growth rate of 3%, the value of the perpetuity is:

= [Cash Flow_{Year 10} x (1 + g)] / (r - g)

= ($100,000 x 1.03) / (0.08 - 0.03)

= $103,000 / 0.05

= $2.06 million

This means that $100,000 paid into a perpetuity, assuming a 3% rate of growth with an 8% cost of capital, is worth $2.06 million in 10 years. Now, a person must find the value of that $2.06 million today. To do this, analysts use another formula referred to as the present value of a perpetuity.