Question

Provide written responses to the following:

(a) Provide a clear explanation of what is meant by the phrase ‘the time value of money’. As part of your explanation supply examples to clarify your comments.

(b) The process of compounding and discounting in finance are related. Explain this relationship.

(c) ‘In order to compare two or more interest rates, they must be expressed on a suitable common scale.’ Explain the meaning of this quotation and give some numerical examples to illustrate your response.

(d) What is meant by the term ‘Redundant Constraints’? Provide an example in support of your discussion.

Answer 1

Provide a clear explanation of what is meant by the phrase ‘the time value of money’. As part of your explanation supply examples to clarify your comments.

The time value of money (TVM) is the idea that money available at the present time is worth more than the same amount 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.

Basic Time Value of Money Formula

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)}

Time Value of Money Example

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

The process of compounding and discounting in finance are related. Explain this relationship.

Discounting and compounding are two sides of the same coin. Both are used to adjust the value of money over time. They just work in different directions: You use discounting to express the value of a future sum of money in today's dollars, and you use compounding to find the value of a current sum of money in future dollars.

Compounding and discounting are integral to the economic concept of the "time value of money." This is the idea that a sum of money in the present time has more economic value than an equal sum of money at some point in the future. In simpler terms: A dollar today is worth more than a dollar tomorrow. Say you have a choice between receiving $100 now or $100 in one year. If you take the $100 now, you can invest it. Even if you put it in an account earning a scant 1 percent annual interest, you'd have $101 a year from now, compared with just $100 if you waited to receive the money. The $100 is worth more, therefore, if you take it today.

Compounding Into the Future

Compounding allows you to project what a given sum of money will be worth in the future. Say you have $100 and you want to know what it will be worth a year from now. Compounding requires you to make an assumption about the kind of return you can earn on your money if you invest it. Say you assume you can earn an average four percent annual return. In one year, therefore, you forecast that you will have $104, or $100 multiplied by 1.04. After another year, you'll have $108.16 — or $104 times 1.04. With compounding, each year's earnings become part of the next year's principal, which allows money to grow faster.

Discounting to Present Value

Discounting is the opposite of compounding. You're taking a sum of money from a point in the future and translating it to its value in today's dollars — which usually will be less. Continuing from the previous example, say you assume an annual return of four percent. If you were to invest $96.15 today at a four percent annual return, you would have exactly $100 a year from now. Therefore, $100 a year from now is really worth just $96.15 today. This is called discounting to present value.

Applications

Finance professionals use compounding and discounting all the time to evaluate investments. Since money changes in value over time, you must express all cash values in the "same" dollars to be able to compare them. Say you're considering a project that will require $100,000 in upfront costs now and deliver $25,000 a year in revenue for the next four years. When you discount that future revenue to present value, it will add up to less than $100,000, so the project is a money-loser. Similarly, a project that produces $100,000 in revenue now but will require a payment of $100,000 in five years is a money-maker, since the upfront payment will compound to well over $100,000 in the intervening years.

Formulas

The formulas for discounting and compounding are quite basic. In these formulas, "CF" is the cash flow, or the amount being converted; "n" is the number of years over which you're converting the amount; and "r" is the assumed average annual rate of return.

To discount a future cash flow to present value (PV): PV = CF / (1 + r)^n

To determine the future value (FV) of a cash flow after compounding: FV = CF * (1 + r)^n

The relationship between discounting and compounding is evident from the similarity between the formulas. When discounting, you divide the cash flow by the factor "(1 + r)^n," which reduces the present value of the cash flow. When compounding, you multiply the cash flow by the same factor, which increases the future value of the cash flow.