Effective Interest Rate - TVM

Hi Guys,

I am new to the CFA and just beginning my long journey to pass this sucker (and finance really), and I’ve come across more of an understanding problem rather than a how to solve problem.

My question is this: When solving for the Payment of a fixed term amortizing loan, such as a mortgage, I would only divide the annual interest rate (let’s say it is 10%) by 12, to get the monthly interest rate (.0083333). This rate is used to calculate the interest vs. principal payment, etc.

However when we are discounting cashflows using the Net Present Value, we cannot simply use the annual discount rate/12. For Example, let’s say we have an initial investment of $100,000. Then three monthly periods of cash flows of $25,000, $30,000, and $35,000. At the end of year 4, we sell the investment for $200,000.

To solve for NPV, the discount rate that we use (10% again) will not work if we do .10/12, as this does not include compounding. We need to use the Effective Interest Rate, which would be =1+.10)^(1/12)+1

So why do we use the effective interest rate when solving for NPV/Discounted Cash Flows, but not when we are using Rate to solve for a mortgage payment/interest expense, etc? I understand that the Effective Interest Rate over the 12 months will equal the Annual Rate, but I don’t know why it isn’t used in the mortgage example.

I’m just missing the bridge here so any help is appreciated. Thanks!

It’s simply a quoting convention.

You always discount using an effective interest rate (i.e., one that compounds). For NPV problems involving monthly cash flows, the annual discount rate is quoted as an effective rate, so you need to compound (i.e., (1 + r)^(1/12) − 1)) to get the monthly rate. For mortgages, the annual discount is quoted as a nominal rate, so you need to divide by 12 to get the monthly rate.

It’s the same way with LIBOR: it’s quoted as a nominal (annual) rate, not an effective (annual) rate.

Thank you for the reply. My follow up question is regarding the annual discount rate for discounting vs. loans/mortgages.

For example, lets use an example with an initial Investment of $8,000,000. Cash Flows are: Period 1 - $740,000, Period 2 - $745,000, Period 3 - $758,000, Period 4 - $752,000 Period 5 - $11,000,000.

The annual Disocount Rate is at 9%. From your answer above, if we wanted to view these as monthly cashflows, then I need to find the monthly effective interest rate, which would equal 9% if compounded monthly for 12 months. That answer would be 0.72% (rounding for simplicity below)

Check Answer: (1+.0072)^12 = 9.00%

However if we have a 9% annual interest rate that compounds monthly, then isn’t the effective interest rate that is being earned/paid (9%/12) = .0075 = (1+.0075)^12 - 1 = 9.38%

Or is what I am calculating dependent on the type of problem I am looking at? I don’t really understand why when discounting cash flows we take the 9% and find the monthly rate that will equal 9% when compounded for 12 months VS. finding the effective rate that we truly pay on a loan/mortgage with the same annual rate of 9%, with an answer that is slightly more than 9%.

Maybe using a real world example will help. What would you do for the rate calculation when trying to find the Present Value for a stream of values for a lease payment? There are 57 payments remaining at $1,511.81 per month. The discount rate is 5.50%. To find the effective interest rate to calculate the NPV, would we just divide 5.50%/12, or would we use the (1+5.50%)^(1/12)-1 to get the true monthly rate that would equal 5.5% over 12 months?

Are there key phrases/situations that tell us when to use either?

When you calculated the monthly effective rate as 0.72%, you were treating the 9% annual rate as an effective rate.

Here, you’re treating the 9% annual rate as a nominal rate.

You cannot have it both ways.

In the problem you pose, I would treat the 9% interest rate is a nominal rate.

Some interest rates are quoted as nominal rates. (Annual) mortgage rates (in the US) are quoted as nominal rates, as are annual rates for car loans and credit cards. I would expect that most (annual) rates quoted for monthly payments are nominal rates. LIBOR is quoted as a nominal rate.

There aren’t key phrases or situations that guarantee that the rate given is effective or nominal, unless they explicitly use one of those words. However, as a general rule, unless they describe an annual rate as an effective annual rate, I’d treat it as nominal.