## Frequency Asked Questions on Lump Sum

## Table of Contents

### What is a lump sum?

A lump sum refers to a single, complete payment of a sum of money, typically all at once rather than in instalments. It’s a one-time payment that covers the entire amount owed or agreed upon without the need for further payments. A lump sum is also defined as a fixed and complete amount of money paid at once, rather than being paid in instalments or regularly.

### What is the future value of a lump sum?

The future value of a lump sum refers to the estimated worth of a sum of money at a specific point in the future, considering potential interest. In other words, it is the value that a given amount of money will be at a certain point in the future if it earns a rate of interest. That is, The future value is the amount that the lump sum will grow to over time with interest, assuming it is invested or earns interest. A lump sum may earn a simple interest or compound interest.

### How is the Future Value of the Lump sum Calculated?

Future value is calculated using the formula:

\(FV = PV \times\left(1 + \frac{r}{m}\right)^{nm}\)

Where:**FV** is the future value of the lump sum**PV** is the present value or initial amount**r** is the interest rate (expressed as a decimal)**n** is the number of compounding periods**m** is the number of times the interest is compounded every year.

Let’s illustrate this with an example:

Suppose you have $5,000 that you want to invest in a savings account with an annual interest rate of 6%. You want to know the future value of this investment after 5 years if it is compounded annually, semi-annually, quarterly and monthly.

Then, you can solve it as follows:

Given PV = $5,000, r = 0.06 (6% as a decimal), n = 5 years,

1. **Annually (m = 1):**

\(FV = $5,000 \times \left(1 + \frac{0.06}{1}\right)^{1 \times 5} = $6,691.13\)

2. **Semi-annually (m = 2):**

\(FV = $5,000 \times \left(1 + \frac{0.06}{2}\right)^{2 \times 5} = $6,719.58\)

3. **Quarterly (m = 4):**

\(FV = $5,000 \times \left(1 + \frac{0.06}{4}\right)^{4 \times 5} = $6,734.28\)

4. **Monthly (m = 12):**

\(FV = $5,000 \times \left(1 + \frac{0.06}{12}\right)^{12 \times 5} = $6,744.25\)

As you can see, as the compounding frequency increases, the future value also increases due to the more frequent application of interest.

### What is Compound Interest?

Compound interest refers to the interest earned not only on the initial amount of money (the principal) but also on the accumulated interest from previous periods. In other words, it is an interest earned on the original amount (principal) plus accumulated interest. So, when a lump sum earns interest, it simply means that interest is paid on the principal plus previously earned interest.

### How does the Number of Years and Compounding Frequency affect Future Value?

Generally, there is a direct relationship between the number of years an investment is held and its future value. This means that the longer a lump sum is invested, the higher its future value will be. Similarly, there is also a direct relationship between the future value of a lump sum and the frequency with which it is compounded. For example, an amount invested for 5 years will have a higher future value if the interest is compounded quarterly rather than annually, just as we saw in the above example.

### What is Present Value of Lump Sum?

The present value of a lump sum refers to the current worth or value of a sum of money that is to be received or paid in the future, after accounting for the time value of money. It’s the amount that needs to be invested today to have a specific amount in the future, taking into consideration the potential earning capacity of that money over time. In essence, the present value is the amount that a lump sum is worth right now.

### How is the Present Value of a Lump sum Calculated?

The present value of a lump sum can be calculated using the formula:

\(PV=\frac{FV}{\left(1+\frac{r}{m}\right)^{n \times m}}\)

where r is the discount rate (interest rate used for discounting, expressed as a decimal).

Let’s illustrate this with an example:

Suppose you want to find the present value of an investment that will be worth $10,000 in 5 years, with an annual interest rate of 6%. You want to know the present value of this investment if it is compounded annually, semi-annually, quarterly, and monthly.

Then, you can solve it as follows: Given FV = $10,000, r = 0.06 (6% as a decimal), n = 5 years,

1. **Annually (m = 1)**:

\(PV = \frac{$10,000}{\left(1 + \frac{0.06}{1}\right)^{1 \times 5}} = $7,472.58\)

2. **Semi-annually (m = 2):**

\(PV = \frac{$10,000}{\left(1 + \frac{0.06}{2}\right)^{2 \times 5}} = $7,440.94\)

3. **Quarterly (m = 4):**

\(PV = \frac{$10,000}{\left(1 + \frac{0.06}{4}\right)^{4 \times 5}} = $7,424.70\)

4. **Monthly (m = 12)**:

\(PV = \frac{$10,000}{\left(1 + \frac{0.06}{12}\right)^{12 \times 5}} = $7,413.72\)

Higher compounding frequencies lead to lower present values because more frequent compounding requires a smaller present value to result in the same future value.

### What is Compounding?

Compounding is the process by which the value of an investment increases over time due to the interest earned on both the initial principal and the accumulated interest from previous periods. It is the process by which both the initial amount and the accumulated interest contribute to generating additional earnings in the future. For example, suppose you invest $100 in a savings account with an annual interest rate of 5%. After one year, your investment will have grown to $105 due to the interest earned. If you leave the money in the account for another year, you will earn interest not only on the initial $100 but also on the $5 of interest earned in the first year. This means that after two years, your investment will have grown to $110.25. As you can see, compounding increases the value of an investment over time.

### What is 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. It’s the opposite of compounding, where future values are reduced to their equivalent values in today’s terms. Discounting involves applying a discount rate to future cash flows or amounts to determine their current value.

### What is the difference between discounting and compounding?

Discounting and compounding are two sides of the same coin. While compounding is the process of calculating the future value of an investment based on its present value, discounting is the process of calculating the present value of an investment based on its future value. So, while compounding helps to determine how much an investment will grow over time, discounting helps to determine how much an investment is worth today.

### What is the difference between the discount rate and the interest rate in a lump sum?

When it comes to lump sum payments, the discount rate is typically used in situations where you want to determine how much a future lump sum is worth in today’s dollars whereas the interest rate is used when you want to calculate the future value of a lump sum. So, in essence, the discount rate is used to calculate the present value of a future amount while interest rate is used to calculate the future value of a present amount.

### What is Compounding Frequency

Compounding frequency refers to the number of times per year that interest is calculated and added to the principal of an investment. Common compounding frequencies include annually (once per year), semi-annually (twice per year), quarterly (four times per year), monthly (twelve times per year), and weekly (52 times per year). The more frequently interest is compounded, the faster the investment will grow due to the effects of compounding.