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We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year.PowerPoint Presentation

We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year.

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We will deal with 3 different rates:

iNom = nominal, or stated, or

quoted, rate per year.

iPer = periodic rate.

EAR = EFF% = .

effective annual

rate

- iNom is stated in contracts. Periods per year (m) must also be given. This is also frequently referred to as an Annual Percentage Rate (APR).
- Examples:
- 8%; Quarterly
- 8%, Daily interest (365 days)

- Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.
- Examples:
8% quarterly: iPer = 8%/4 = 2%.

8% daily (365): iPer = 8%/365 = 0.021918%.

- Effective Annual Rate (EAR = EFF%):
Our bank offers a Certificate of Deposit (CD) with a 10% nominal rate, compounded semi-annually. What is the true rate of interest that they pay annually (the effective annual rate)?

0

6mo

12mo

5%

5%

$1

(1 + )

iNom

m

m

EFF% = - 1

How do we find EFF% for a nominal rate of 10%, compounded semiannually?

(1 + )

2

0.10

2

= - 1.0

= (1.05)2 - 1.0

= 0.1025 = 10.25%.

Any PV would grow to the same FV at 10.25% annually or 10% semiannually. In both instances, our money earns 10.25 percent each year.

For a given rate, more frequent compounding increases returns

12 percent compounded annually

EARA = 12%.

EARs = (1 + 0.12/2)2 - 1 = 12.36%.

EARM = (1 + 0.12/12)12 - 1 = 12.68%.

EARD(365) = (1 + 0.12/365)365 - 1 = 12.75%.

12 percent compounded semi-annually

12 percent compounded monthly

12 percent compounded daily

What is the FV of $100, if you earn 10 percent compounded semi-annually?

When using semi-annual periods we use the periodic rate:

6

5

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

Cpt = 134.01

When using annual periods we use the EAR:

10.25

100

0

INPUTS

3

N

I/YR

PV

PMT

FV

OUTPUT

Cpt = 134.01

The nominal rate is never used in calculations!

Can the effective rate ever be equal to the nominal rate? semi-annually?

- Yes, but only if annual compounding is used, i.e., if m = 1.
- If m > 1, EFF% will always be greater than the nominal rate.

When is each rate used? semi-annually?

The use of the periodic rate and the effective annual rate is determined by the choice of N. If N is the number of years, the EAR is used. If N is the number of periods (shorterthan 1 year), the periodic rate is used. The nominal rate is not used in calculations. As shown later, the choice of N will be determined by the number of payments (PMT).

Written into contracts, quoted by banks and brokers. Not used in calculations or shown

on time lines.

iNom:

Nominal, Periodic, or Effective Annual Rate? semi-annually?

- We never use the nominal rate calculating future or present values. We always use either the effective annual rate or the periodic rate of return.
- If PMT is 0, either the periodic rate or the effective rate may be used.
- Ex: Find the present value of $100 to be received in 5 years. The nominal rate is 6 percent, compounded monthly.Periodic: FV = 100, N = 5*12, I = _______; compute PV= $74.14E.A.R.: FV = 100, N = 5, I = __________; compute PV= $74.14

6/12

6.1678

- If PMT has a value, N must equal the number of payments. Since N is determined by the number of payments, we must make I (the interest rate) consistent with how we defined N.
- Ex: Find the present value of $100 per year for 5 years. The nominal rate is 6 percent, compounded monthly.PMT = 100, N = 5, I = _________; compute PV = $419.32
- Ex: Find the present value of $100 per month for 5 years. The nominal rate is 6 percent, compounded monthly.PMT = 100, N = 5*12, I = ________; compute PV = $5,172.56

6.1678

6/12

A few other points to note. Since N is determined by the number of payments, we must make I (the interest rate) consistent with how we defined N.

- If the rate is 12% compounded monthly, your EAR is 12.6825%, this is used if the N is in number of years.
- If the rate is quoted at 12% compounded monthly, the periodic rate would be 1% (12%/12). The 1% is used if the N is in number of months.
- If the rate is 12% compounded monthly, you cannot divide the 12% by anything besides its number of compounding periods (in this case 12 monthly periods), or you change the interest rate.

Example Since N is determined by the number of payments, we must make I (the interest rate) consistent with how we defined N.

- What is the future value of 4 quarterly payments of $100 each, given a nominal rate of 12 percent, compounded monthly.
- Note that the quarterly payments and monthly compounding are different, so we have to transform the monthly compounded rate to a quarterly compounded rate. This takes some work.First, find the EAR of the monthly rate:EARM = (1 + 0.12/12)12 - 1 = 12.68%.Second, find the nominal rate with quarterly compounding that has an effective annual rate of 12.68%(1 + Inom/4)4 - 1 = 12.68%.Inom = 12.12%Third, compute the future value of the paymentsN = 4, I = 12.12/4, PV = 0, PMT = 100, cpt FV = $418.55
- See the next slide for the Excel Example

Excel Calculation Example Since N is determined by the number of payments, we must make I (the interest rate) consistent with how we defined N.

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