CDs Bought at a Bank verses CD’s Bought from a Brokerage ...

CDs Bought at a Bank verses CD's Bought from a Brokerage

Floyd Vest

CDs bought at a bank. CD stands for Certificate of Deposit with the CD originating in a FDIC insured bank so that the CD is insured by the United States government. Consider the following "extra simple" example where the CD buyer has opted to reinvest the interest payments back into the CD at the CD rate. We made this example very simple to demonstrate interest paid on interest.

Example 1 Assume a person can buy a two-year CD for $1000 that pays an annual nominal interest rate (APR, Rate) of 5%. Dividends are paid and compounded semiannually so a (simulated) semiannual rate is 5% = 2.5% = 0.025.

2 How much is the Final Balance of the CD and what is the Annual Percentage Yield (APY, Yield)?

Calculating the Final Balance. The semiannual dividends (interest payments) are 0.05

2 (1000) = $25.

The following is a drawn out derivation, using simulated bank postings, illustrating earning interest on interest: Posting 1: Dividend 1 = 1000(0.025) Postings 2, 3, and 4 include accumulated interest on interest and the current dividend. The factor of (1 + 0.025) results from compounding. (See the Compound Interest Formula in a Side Bar Note.) Posting 2: 1000(0.025)(1 + .025) + 1000(0.025) Posting 3: 1000(0.025)(1 + 0.025 )2 + 1000(0.025)(1 + 0.025) + 1000(0.025)

Posting 4: 1000(0.025)(1 + 0.025 )3 + 1000(0.025)(1 + 0.025 )2 + 1000(0.025)(1 + 0.025)

+ 1000(0.025) + 1000, where $1000 is the price of the CD. Posting 4 gives the Final Balance including principal plus interest.

We can calculate the Final Balance by factoring to get

(1) 1000(0.025)[(1 + 0.025 )3 + (1 + 0.025 )2 + (1 + 0.025) + 1] + 1000

We recognize the expression in brackets as the sum of an ordinary annuity. (See Sum of an Ordinary Annuity in the Side Bar Notes. For the basic Mathematics of Finance

formulas for Compound Interest and Annuities, see Luttman or Kasting in Unit 1 of this

course.) The sum is

(1+ 0.025)4 !1

(2)

= 4.1525 .

0.025

Then we calculate to get 1000(0.025)(4.152) + 1000 = $1103.81 as the Final Balance of

the CD. The bank will not quite do it this way as can be seen in an exercise. To

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summarize, the CD has earned interest of $103.81 and the principal was repaid. The interest on interest was $3.81. (See the exercises and Side Bar Notes for formulas.)

You Try It #1 (a) Do a derivation similar to the above Example 1 for a semiannual two-year bank CD of $10,000 that pays an annual 2% nominal rate. By this activity, you will be demonstrating earning interest on interest and deriving a formula for calculating the Final Balance. (b) Draw and label a time line. (c) Use the Formula for the Sum of an Ordinary Annuity given in a Side Bar Note to calculate the Final Balance. (d) How much interest was earned? How much interest on interest was earned? (e) Use a general Formula for the Sum of an Ordinary Annuity to give a general closed form formula for the Final Balance of a bank CD. Label all your variables.

Annual Percentage Yield (APY). What is the APY that the bank will report for the CD in Example 1? By definition the APY is annual rate that pays the same as the periodic rate compounded for a year. Thus APY = (1 + 0.025 )2 ? 1 = 0.0506 = 5.06%. (See an

explanation in the You Try Its.)

To summarize, the 5% nominal rate compounded semiannually yielded actual annual earnings of 5.06% compounded annually; i.e., 1000(1 + 0.056 )2 = $1103.81.

Also, we have earned interest on interest.

Typically, the bank will report the APY as a percent with two digits to the right of the decimal, and the annual nominal rate (APR, Rate) will be reported as a percent with three digits to the right of the decimal point.

You Try It #2 (a) For the CD you calculated in You Try It #1, calculate the APY. (b) Give a general meaning of APY. If i is the periodic rate compounded k times per year, give two basic formulas for APY.

The way banks accumulate interest in a CD. Consider a $150,000, nine-month CD paying interest monthly at the annual nominal rate of 2.440%. In the calculations, the bank carries the principal of $150,000 and accumulated interest forward each month to get a Balance brought Forward to the beginning of the month. Then, it calculates interest for the month on this balance and adds it to the Balance brought Forward to get the end of month Ending Balance.

For example, on 8/01 the Balance brought Forward is $150,576.26. Then on 8/31 the

Deposit

Dividend

=

0.02440 365

(150,576.26)

=

312.04

.

31

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The Deposit Dividend of 312.04 is added to the Balance brought Forward of 150,576.26 to get the Ending balance of $150,888.30 posted 8/31. Please note that the denominator is never equal to 12, the number of months in a year.

You Try It #3 (a) Do a derivation of a formula for Final Balance by following the above steps from the beginning of the investment of principal P, in a two year semiannual CD, with an annual nominal rate r, with the semiannual rate i = r . Start with the Beginning of the first six

2 months, then to the End of the first six months, etc. Factor and use exponents and label formulas at each step. You will end with a closed form formula for the Final Balance B. (b) Apply your formula to the CD in Formula 1 in the above discussion. You should get the same answer. Do you recognize the formula that you derived? You now have two formulas for calculating the Final Balance B of a bank CD.

CDs bought from a broker. Brokered CDs originate in an FDIC Insured bank but are sold by a broker. Typically, new brokered CDs do not offer the option of reinvesting the dividends (interest) in the CD at the CD rate. Dividends are simply paid to the owner at the end of each dividend period. Brokered CD's typically pay dividends semiannually.

Calculating the Final Balance of a Brokered CD. In our famous example we would buy a $1000 two-year brokered CD paying dividends (interest) semiannually at the annual

0.05 nominal rate of 5%. Each dividend would be 2 (1000) = $25.

(Again, we are doing a simulation. See the You Try Its.) For Postings 1, 2, 3, and 4 for the CD, each would display a dividend of $25. Posting 4 would also display the price of the CD of $1000 as seen on the timeline in Figure 1.

$25

$25

$25

$25 + $1000

__|__________|__________|___________|___________|_______________

0

1

1

11

2

years

2

2

P = $1000

Fig. 1: A cash flow timeline for a brokered CD.

The Value at Maturity would be $1000. Total interest earned is $100. The Final Balance is $1100. See the exercises and Side Bar Notes for other terms of a brokered CD.

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Comparison of the two CD's. The two CDs were both $1000 two year CDs with a nominal annual rate of 5%, and with dividends paid semiannually.

For the CD bought at the bank with the option of reinvesting dividends at the CD rate, Final Balance = $1103.81. Total interest = $103.81. Interest on interest is $3.81. If you are not impressed with the $3.81, see the exercises.

For the brokered CD, Value at Maturity = $1000. Total interest = $100. For the brokered CD, there is no assumption as to what happens in the future to the dividends. For this particular CD the Value at Maturity is the same as the price, but for CDs bought on the secondary market, this may not be true.

The dividends of a brokered CD are typically paid into a "sweep" money market fund in the name of the CD owner. Of course, dividends might be reinvested at a "good" rate. (See the Side Bar Notes.) However, for the small investor, this is not easy to do and not convenient. They could spend the money, but the point may be to accumulate savings.

The broker may charge an account maintenance fee but any commission on the CD is net the nominal interest rate; i.e., it is not subtracted from the announced interest payments or the Value at Maturity.

Terminology for a brokered CD. The terms used for CDs by a brokerage are annual nominal rate (APR), Yield to Maturity (YTM), and Annual Percentage Rate (APR). The term Annual Percentage Yield (APY) is not used on this type of a brokered CD.

You Try It # 4 Do the calculations, describe the events, draw a timeline, and use brokered CD terms, for a brokered CD similar to the bank CD in You Try It #1.

The meaning of Yield to Maturity (YTM). By definition, YTM is the annual nominal

rate that discounts the cash flows to the price of the CD.

That is, in our case

(3)

Price

=

25!"# 1 +

YTM 2

$ %&

'1

+

25!"#

1

+

YTM $ '2 2 %&

+ 25!"#1+

YTM $ '3 2 %&

+

25!"# 1 +

YTM 2

$ %&

'4

+ 1000!"#1+

YTM 2

$ %&

'4

.

We can try a YTM of 5% (our annual nominal rate) and see what it does. (4) Price = 25(1.025)-1 + 25(1.025)-2 + 25(1.025)-3 + 25(1.025)-4 + 1000(1.025)-4.

Let's multiply through by (1.025 )4 to get

(Price)(1.025)4 = 25[1.0253 + 1.0252 + 1.025 + 1] + 1000.

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Observing that the expression in the brackets is the sum of an ordinary annuity, we have

(Price)(1.025)4

=

" (1+ 0.025)4

25 $ #

0.025

! 1%' &

+

1000

=

25(4.152)

+

1000

Price = 25(4.152) +1000 = $1000. (1.025)4

So the broker is likely to report that YTM for the brokered CD is 5%.

The problem. The problem is that the expression in our YTM calculation of 25[1.0253 + 1.0252 + 1.025 + 1] indicates that interest is being paid on interest, compounded semiannually, at the nominal annual rate of 5%. The brokered CD didn't do this. In finance books, calculator manuals, and other publications, the precaution is usually indicated that YTM assumes dividends are reinvested at the nominal annual rate of the CD compounded per period. Actually, the brokered CD paid 2.5% per six-month period with no indication of what happened to the $25 dividends. It's just that people dealing with brokered CDs and bonds have to understand this issue. (See terms/m/mirr.asp.

Examining the YTM (Yield to Maturity) concept. To examine YTM, consider our example of semiannual payments, with YTM = 2i, where i is the interest rate per sixmonth period. Let P = price of the CD (bond, stock, security, business investment). Assume the following timeline for cash flows consisting of (dividends) interest payments I and value at maturity M, for n payment periods.

I 1

I 2

I 3

I 4

...

In-1

In +M

_| ______|______|______|______|____________|_______|________

time

P

Fig. 2: Cash flow time line for n periods.

We will divide price P into n parts so that P = P1 + P2 + P3 + ... + Pn. Then invest each Pj at the rate i compounded per period to earn the cash flows.

This gives:

I1 = P1(1 + i) I2 = P2(1 + i)2

I3 = P3(1 + i)3

. . . M + In = Pn (1+ i)n

Multiplying through by (1 + i)j gives: P1 = I 1(1+ i )!1 P2 = I2 (1+ i)-2 P3 = I 3(1+ i )!3

. . . Pn = In (1+ i)-n + M (1+ i)-n

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