- Introduction - Time Value of Money
- Interest Rates
- Interest Rate Equation
- Nominal Interest Rate and Effective Yield
- Time Value of Money for Different Compounding Frequencies
- Future Value of a Single Cash Flow
- Present Value of a Single Cash Flow
- Future Value and Present Value of Ordinary Annuity
- Present Value and Future Value of Annuity Due
- Present Value of a Perpetuity
- Present Value and Future Value of Uneven Cash Flows
- Annuities with Different Compounding Frequencies
- Using a Timeline to Solve Time Value of Money Problems

# Time Value of Money for Different Compounding Frequencies

Let’s first review the time value money concept using a very simple example.

### Example 1

Let’s say you have $2000 to invest. You decide to invest it for 3 years in an account that pays you an interest of 6% per annum. How much will your investment grow to in 3 years?

We are calculating the future value of an investment after 3 years. This will be calculated as follows:

FV = $2000*(1.06)3 = $2382.03

### Example 2

Your target is to have $10,000 saved in your account in 5 years. How much money should you invest now to reach your target in 5 years when your investment account earns you 8% per annum?

We are calculating the present value of a future cash flow. This will be calculated as follows:

PV = $10,000/(1.08)^5 = $6805.83

This means that if you invest 46805.83 now for 5 years at 8% interest rate per annum, you will receive $10,000 at the end of 5 years.

A common assumption in both the above problems was that the frequency of compounding was annual. That is, the interest is compounded only annually. However, this is not always the case. The frequency of compounding could be anything, most commonly being, monthly, quarterly, semi-annually, or annually. Let’s look at how our future value and present value will change if we use a different frequency of compounding.

### Example 1 (With Quarterly Compounding)

In our first example, if the compounding frequency was quarterly, then how much will our investment grow to?

**Step 1: Calculate the quarterly rate**

Quarterly rate = 6%/4 = 1.5%

**Step 2: Calculate number of compounding periods**

Compounding periods = 3 years * 4 = 12 periods

**Step 3: Calculate Future Value**

FV = $2,000*(1.015)^12 = $2391.24

As you can see, the future value based quarterly compounding is more than future value based on annual compounding.

Note that we could also calculate the effective annual yield and then calculate future value as shown below:

EAY = (1.015)^4 – 1 = 6.13635%

FV = $2,000(1.0613635)^3 -1 = $2,391.24

Note that both the methods produce same results.

### Example 2 (With Monthly Compounding)

In our second example, if the compounding frequency was monthly, how much should we invest now to reach our target of $10,000 in 5 years with an annual interest rate of 8%?

The monthly rate is 8%/12 = 0.667% and the number of compounding periods is 5*12 = 60.

PV = $10,000/(1.00667)^60 = $6710.77

As you can see, with monthly compounding we need to invest less to reach our target.

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