CFA Level 1 Time Value of Money LO d: TVM Problems for different Frequencies of Compounding

Hello and welcome back. 

In the previous blog, we talked about the impact of frequency of compounding on effective annual rate (EAR). If you haven’t read the previous blog, please see it first.

In this blog, we will talk about how we can solve TVM problems for different frequencies of compounding.

We talked about this in the previous blog when we discussed EAR and solved examples 3 and 4. We will briefly revisit the concepts here and then practice a few examples to ensure that you get comfortable with this.

I am sure you know this by now that:-

FV = PV(1 + I/Y)^N  where

• PV is used for Present value or Current Value or Value at time 0

• FV is used for Future value or Value at time t

• N signifies Number of compounding periods

• I/Y is used for Interest rate per compounding period

Also, we use PMT for Per period payments or Annuity payments, or constant periodic cash flow. 

Finally, you should be comfortable using the TVM keys on your calculator.

We also covered these important conclusions about TVM calculations:-

1. At the end of 1^st compounding period, Compound Interest value is same as the Simple Interest value. This is because you have not started earning interest on interest.

2. Higher the frequency of compounding, greater is the amount of interest earned.

3. Compound interest effects are pronounced when the interest rates are high and the duration of investment is long.

Let’s look at few examples:-

Example 1)

You are planning to invest $10,000 for 2 years and you want to compare the difference in received amounts for various compounding frequency. The rate of Interest is 20% per annum. Calculate the final amount for the following frequencies of compounding:-

A.)Annually

B.)Semi-Annually

C.)Quarterly

D.)Monthly

E.)Daily

F.)Continuous

We will solve this problem using two approaches: the TVM approach and the EAR approach.

A.) For Annual compounding

PV = -10,000  N=2  I/Y=20  PMT=0 CPTàFV  to get $14,400.00

B.) For Semi-annual compounding

PV = -10,000  N=4  I/Y=10  PMT=0 CPTàFV  to get $14,641.00

How do we solve this using EAR?

EAR _{Semi Annual Compounding} = (1 + 0.1) ² – 1 = 0.21 = 21%

So, after 2 years, FV = 10,000(1.21)² = 10,000*1.4641 = $14,641

C.) For Quarterly compounding

PV = -10,000  N=8  I/Y=5  PMT=0 CPTàFV  to get $14,774.55

How do we solve this using EAR?

EAR _{Quarterly Compounding} = (1 + 0.05) ⁴ – 1 = 0.21550625 = 21.550625%

So, after 2 years, FV = 10,000(1.21550625)² = 10,000*1.477455 = $14,774.55

D.) For Monthly compounding

PV = -10,000  N=24  I/Y=1.667  PMT=0 CPTàFV  to get $14,870.32

How do we solve this using EAR?

EAR _{Monthly Compounding} = (1 + 0.01667) ¹² – 1 = 0.21939 = 21.939%

So, after 2 years, FV = 10,000(1.21939)² = 10,000*1.4869119 = $14,869.12

The slight difference that you get in the answers from these two methods is because of rounding errors.

E.) For Daily compounding

PV = -10,000  N=730  I/Y=0.05479  PMT=0 CPTàFV  to get $14,916.12

How do we solve this using EAR?

EAR _{Daily Compounding} = (1 + 0.0005479) ³⁶⁵ – 1 = 0.221316 = 22.1316%

So, after 2 years, FV = 10,000(1.221316)² = 10,000*1.491612 = $14,916.12

F.) For Continuous compounding

EAR = e^r – 1 = e^0.2 -1 = 0.221403 = 22.1403%

So, after 2 years, FV = 10,000(1.221403)² = $14,918.25 

Example 2)  

You have a financial commitment that requires you to pay $10,000 2 years from now and you want to compare the difference in amounts that you will have to deposit today for various compounding frequency. The rate of Interest is 20% per annum. Calculate the initial amount (PV) for the following frequencies of compounding:-

A.)Annually

B.)Semi-Annually

C.)Quarterly

D.)Monthly

E.)Daily

F.)Continuous

We will solve this problem using two approaches: the TVM approach and the EAR approach.

A.) For Annual compounding

FV = 10,000  N=2  I/Y=20  PMT=0 CPTàPV  to get $6,944.45

B.) For Semi-annual compounding

FV = 10,000  N=4  I/Y=10  PMT=0 CPTàPV  to get $6,830.13

How do we solve this using EAR?

EAR _{Semi Annual Compounding} = (1 + 0.1) ² – 1 = 0.21 = 21%

So, after 2 years, 10,000 = PV(1.21)²

       So, PV = 10,000/1.4641 = $6,830.13

C.) For Quarterly compounding

FV = 10,000  N=8  I/Y=5  PMT=0 CPTàPV  to get $6,768.39

How do we solve this using EAR?

EAR _{Quarterly Compounding} = (1 + 0.05) ⁴ – 1 = 0.21550625 = 21.550625%

So, after 2 years, 10,000 = PV(1.21550625)²

                            So, PV = 10,000/1.477455 = $6,768.39

D.) For Monthly compounding

FV = 10,000  N=24  I/Y=1.667  PMT=0 CPTàPV  to get $6,724.80

How do we solve this using EAR?

EAR _{Monthly Compounding} = (1 + 0.01667) ¹² – 1 = 0.21939 = 21.939%

So, after 2 years, 10,000 = PV(1.21939)²

                            So, PV = 10,000/1.4869119 = $6,725.34

The slight difference that you get in the answers from these two methods is because of rounding errors.

E.) For Daily compounding

FV = 10,000  N=730  I/Y=0.05479  PMT=0 CPTàPV  to get $6,704.15

How do we solve this using EAR?

EAR _{Daily Compounding} = (1 + 0.0005479) ³⁶⁵ – 1 = 0.221316 = 22.1316%

So, after 2 years, 10,000 = PV(1.221316)²

    So, PV = 10,000/1.491612 = $6,704.15

F.) For Continuous compounding

EAR = e^r – 1 = e^0.2 -1 = 0.221403 = 22.1403%

So, after 2 years, 10,000 = PV(1.221403)²

                             So PV = 10,000/1.4918 = $6,703.31 

I hope now you are comfortable in solving TVM questions for any frequency of compounding.

You can watch a video explaining the points discussed in this blog here:-

This brings us to the end of learning outcome d for this reading. In the next blog, we will talk about learning outcome e: FV and PV of a single Sum, Ordinary annuity and Annuity due, perpetuity and unequal cash flows.

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