CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited States govermet. Cosider the followig extra simple example where the CD buyer has opted to reivest the iterest paymets back ito the CD at the CD rate. We made this example very simple to demostrate iterest paid o iterest. Example 1 Assume a perso ca buy a two-year CD for $1000 that pays a aual omial iterest rate (APR, Rate) of 5%. Divideds are paid ad compouded semiaually so a (simulated) semiaual rate is 5% =.5% = 0.05. How much is the Fial Balace of the CD ad what is the Aual Percetage Yield (APY, Yield)? Calculatig the Fial Balace. The semiaual divideds (iterest paymets) are 0.05 (1000) = $5. The followig is a draw out derivatio, usig simulated bak postigs, illustratig earig iterest o iterest: Postig 1: Divided 1 = 1000(0.05) Postigs, 3, ad 4 iclude accumulated iterest o iterest ad the curret divided. The factor of (1 + 0.05) results from compoudig. (See the Compoud Iterest Formula i a Side Bar Note.) Postig : 1000(0.05)(1 +.05) + 1000(0.05) Postig 3: 1000(0.05)(1 + 0.05 ) + 1000(0.05)(1 + 0.05) + 1000(0.05) Postig 4: 1000(0.05)(1 + 0.05 ) 3 + 1000(0.05)(1 + 0.05 ) + 1000(0.05)(1 + 0.05) + 1000(0.05) + 1000, where $1000 is the price of the CD. Postig 4 gives the Fial Balace icludig pricipal plus iterest. We ca calculate the Fial Balace by factorig to get (1) 1000(0.05)[(1 + 0.05 ) 3 + (1 + 0.05 ) + (1 + 0.05) + 1] + 1000 We recogize the expressio i brackets as the sum of a ordiary auity. (See Sum of a Ordiary Auity i the Side Bar Notes. For the basic Mathematics of Fiace formulas for Compoud Iterest ad Auities, see Luttma or Kastig i Uit 1 of this course.) The sum is () (1+ 0.05)4!1 = 4.155. 0.05 The we calculate to get 1000(0.05)(4.15) + 1000 = $1103.81 as the Fial Balace of the CD. The bak will ot quite do it this way as ca be see i a exercise. To Bak vs. Brokered CDs Sprig, 011 1
summarize, the CD has eared iterest of $103.81 ad the pricipal was repaid. The iterest o iterest was $3.81. (See the exercises ad Side Bar Notes for formulas.) You Try It #1 (a) Do a derivatio similar to the above Example 1 for a semiaual two-year bak CD of $10,000 that pays a aual % omial rate. By this activity, you will be demostratig earig iterest o iterest ad derivig a formula for calculatig the Fial Balace. (b) Draw ad label a time lie. (c) Use the Formula for the Sum of a Ordiary Auity give i a Side Bar Note to calculate the Fial Balace. (d) How much iterest was eared? How much iterest o iterest was eared? (e) Use a geeral Formula for the Sum of a Ordiary Auity to give a geeral closed form formula for the Fial Balace of a bak CD. Label all your variables. Aual Percetage Yield (APY). What is the APY that the bak will report for the CD i Example 1? By defiitio the APY is aual rate that pays the same as the periodic rate compouded for a year. Thus APY = (1 + 0.05 ) 1 = 0.0506 = 5.06%. (See a explaatio i the You Try Its.) To summarize, the 5% omial rate compouded semiaually yielded actual aual earigs of 5.06% compouded aually; i.e., 1000(1 + 0.056 ) = $1103.81. Also, we have eared iterest o iterest. Typically, the bak will report the APY as a percet with two digits to the right of the decimal, ad the aual omial rate (APR, Rate) will be reported as a percet with three digits to the right of the decimal poit. You Try It # (a) For the CD you calculated i You Try It #1, calculate the APY. (b) Give a geeral meaig of APY. If i is the periodic rate compouded k times per year, give two basic formulas for APY. The way baks accumulate iterest i a CD. Cosider a $150,000, ie-moth CD payig iterest mothly at the aual omial rate of.440%. I the calculatios, the bak carries the pricipal of $150,000 ad accumulated iterest forward each moth to get a Balace brought Forward to the begiig of the moth. The, it calculates iterest for the moth o this balace ad adds it to the Balace brought Forward to get the ed of moth Edig Balace. For example, o 8/01 the Balace brought Forward is $150,576.6. The o 8/31 the Deposit Divided = 0.0440 (150,576.6) = 31.04. 365 31 Bak vs. Brokered CDs Sprig, 011
The Deposit Divided of 31.04 is added to the Balace brought Forward of 150,576.6 to get the Edig balace of $150,888.30 posted 8/31. Please ote that the deomiator is ever equal to 1, the umber of moths i a year. You Try It #3 (a) Do a derivatio of a formula for Fial Balace by followig the above steps from the begiig of the ivestmet of pricipal P, i a two year semiaual CD, with a aual omial rate r, with the semiaual rate i = r. Start with the Begiig of the first six moths, the to the Ed of the first six moths, etc. Factor ad use expoets ad label formulas at each step. You will ed with a closed form formula for the Fial Balace B. (b) Apply your formula to the CD i Formula 1 i the above discussio. You should get the same aswer. Do you recogize the formula that you derived? You ow have two formulas for calculatig the Fial Balace B of a bak CD. CDs bought from a broker. Brokered CDs origiate i a FDIC Isured bak but are sold by a broker. Typically, ew brokered CDs do ot offer the optio of reivestig the divideds (iterest) i the CD at the CD rate. Divideds are simply paid to the ower at the ed of each divided period. Brokered CD s typically pay divideds semiaually. Calculatig the Fial Balace of a Brokered CD. I our famous example we would buy a $1000 two-year brokered CD payig divideds (iterest) semiaually at the aual omial rate of 5%. Each divided would be 0.05 (1000) = $5. (Agai, we are doig a simulatio. See the You Try Its.) For Postigs 1,, 3, ad 4 for the CD, each would display a divided of $5. Postig 4 would also display the price of the CD of $1000 as see o the timelie i Figure 1. $5 $5 $5 $5 + $1000 0 1 P = $1000 Fig. 1: A cash flow timelie for a brokered CD. 1 1 1 years The Value at Maturity would be $1000. Total iterest eared is $100. The Fial Balace is $1100. See the exercises ad Side Bar Notes for other terms of a brokered CD. Bak vs. Brokered CDs Sprig, 011 3
Compariso of the two CD s. The two CDs were both $1000 two year CDs with a omial aual rate of 5%, ad with divideds paid semiaually. For the CD bought at the bak with the optio of reivestig divideds at the CD rate, Fial Balace = $1103.81. Total iterest = $103.81. Iterest o iterest is $3.81. If you are ot impressed with the $3.81, see the exercises. For the brokered CD, Value at Maturity = $1000. Total iterest = $100. For the brokered CD, there is o assumptio as to what happes i the future to the divideds. For this particular CD the Value at Maturity is the same as the price, but for CDs bought o the secodary market, this may ot be true. The divideds of a brokered CD are typically paid ito a sweep moey market fud i the ame of the CD ower. Of course, divideds might be reivested at a good rate. (See the Side Bar Notes.) However, for the small ivestor, this is ot easy to do ad ot coveiet. They could sped the moey, but the poit may be to accumulate savigs. The broker may charge a accout maiteace fee but ay commissio o the CD is et the omial iterest rate; i.e., it is ot subtracted from the aouced iterest paymets or the Value at Maturity. Termiology for a brokered CD. The terms used for CDs by a brokerage are aual omial rate (APR), Yield to Maturity (YTM), ad Aual Percetage Rate (APR). The term Aual Percetage Yield (APY) is ot used o this type of a brokered CD. You Try It # 4 Do the calculatios, describe the evets, draw a timelie, ad use brokered CD terms, for a brokered CD similar to the bak CD i You Try It #1. The meaig of Yield to Maturity (YTM). By defiitio, YTM is the aual omial rate that discouts the cash flows to the price of the CD. That is, i our case! (3) Price = 5 1+ YTM $ # % & '4 '1! + 5 1+ YTM $ # % & '! + 5 1+ YTM $ # % &! 5 1+ YTM $! # % & +1000 1+ YTM $ # % & We ca try a YTM of 5% (our aual omial rate) ad see what it does. (4) Price = 5(1.05) -1 + 5(1.05) - + 5(1.05) -3 + 5(1.05) -4 + 1000(1.05) -4. Let s multiply through by (1.05 ) 4 to get (Price)(1.05) 4 = 5[1.05 3 + 1.05 + 1.05 + 1] + 1000. '4. '3 + Bak vs. Brokered CDs Sprig, 011 4
Observig that the expressio i the brackets is the sum of a ordiary auity, we have (Price)(1.05) 4 = 5 (1+ 0.05)4!1% $ ' + 1000 = 5(4.15) + 1000 # 0.05 & 5(4.15) + 1000 Price = = $1000. 4 (1.05) So the broker is likely to report that YTM for the brokered CD is 5%. The problem. The problem is that the expressio i our YTM calculatio of 5[1.05 3 + 1.05 + 1.05 + 1] idicates that iterest is beig paid o iterest, compouded semiaually, at the omial aual rate of 5%. The brokered CD did t do this. I fiace books, calculator mauals, ad other publicatios, the precautio is usually idicated that YTM assumes divideds are reivested at the omial aual rate of the CD compouded per period. Actually, the brokered CD paid.5% per six-moth period with o idicatio of what happeed to the $5 divideds. It s just that people dealig with brokered CDs ad bods have to uderstad this issue. (See www.ivestopedia.com/terms/m/mirr.asp. Examiig the YTM (Yield to Maturity) cocept. To examie YTM, cosider our example of semiaual paymets, with YTM = i, where i is the iterest rate per sixmoth period. Let P = price of the CD (bod, stock, security, busiess ivestmet). Assume the followig timelie for cash flows cosistig of (divideds) iterest paymets I ad value at maturity M, for paymet periods. I 1 I I 3 I 4 I 1 + M... _ time P Fig. : Cash flow time lie for periods. We will divide price P ito parts so that P = P 1 + P + P 3 + + P. The ivest each P j at the rate i compouded per period to ear the cash flows. This gives: Multiplyig through by (1 + i) j gives: I 1 = P 1 (1 + i) P 1 = I 1 (1+ i )!1 I = P (1 + i) P (1 ) = I i + I 3 = P 3 (1 + i) 3 P 3 = I 3 (1+ i )!3...... M + I = P (1 + i) P = I (1 + i) + M (1 + i) I Bak vs. Brokered CDs Sprig, 011 5
Rememberig that P = P 1 + P + P 3 + + P, we get (4) P = I (1 + i) + I (1 + i) + I (1 + i) +... + ( I + M )(1 + i) 1 3 1 3 So YTM is the aual omial rate that discouts the cash flows to the price P. We have derived here the formula for YTM from basic priciples. If there are semiaual divideds, the YTM = i. Also remember that YTM assumes reivestmet of divideds at the aual omial YTM rate. Oe ca solve a YTM equatio for i by multiplyig through by (1 + i) to get a polyomial. If the Is are equal, you ca use the formula for the sum of a ordiary auity to assist. But you still will wat to use a computer program. If the Is are uequal, you simply use available calculators or computers to solve the th degree polyomial for a root (zero) betwee 1 ad, thus givig a i betwee 0 ad 1. Usig the Fiacial Fuctios i the TVM Solver o the TI83/84. We will cosider a two-year $10,000 bak CD above, with a aual omial rate of 5%, payig Divideds semiaually at the ed of each period, ad Divided of $50. We will solve for i = 0.05/. We will use the ames of variables ad the sig covetios of the TVM Formulas. I the TVM Solver we will eter 4 as N, (-)10000 as PV, 50 as PMT, ad 10000 as FV, ad calculate i. Code ad commets: d Fiace Eter 4 Eter to PV (-)10000 Eter 50 Eter 10000 Eter For P/Y write 1. Eter Select Pmt:Ed to highlight I% Alpha Solve You read.50 So i =.5% = 0.05 ad the aual omial rate = 5% = 0.05. Due to the curious ature of the defiitio of YTM for a brokered CD, this will also work for a brokered CD. See the Exercises. Bak vs. Brokered CDs Sprig, 011 6
Exercises For all exercises, show all your work, label iputs, give formulas, label outputs, ad summarize. 1. For the $1000 Brokered CD discussed above, use the TVM Solver o the TI 83/84 to calculate i (the semiaual rate). What is YTM? Substitute the umbers ito a appropriate formula. (See Exercise 6.) There is o closed form formula for solvig for i i a auity.. Calculate the APY to compare the followig five-year bak CDs. CD No.1 pays.3% omial, compouded mothly. CD No. pays.4% omial compouded semiaually. 3. Use two formulas, oe i You Try It #1, ad the other i You Try It #3, to calculate the Fial Balace for a five-year $4000 bak CD compoudig mothly at a omial rate of 6%. 4. Use the formulas for Fial Balace B i You Try It #1 ad i You Try It #3 to derive the formula 0 = P + R (1+ i )!1% $ '! P (1+ i ). This is close to oe of the TVM # i & formulas i the Appedix for the TI 83/84 that is used by the TVM Solver. Look it up ad make the compariso i variables, G i, ad sig covetios. 5. Formula 4 above suggests the eed for a closed form formula for the Sum (Preset 1! (1+ i )! % Value) of these discouted values, R. Derive the formula A = R $ ', which # i & does this. 6. Show from Formula 4 that for a brokered CD, a geeral formula for Price is 1! ( 1+ i )! % Price = R $ ' $ i ' + M ( 1+ i )!, where R is the semiaual paymet, # & i = YTM/. = (the umber of years). M is the Value at Maturity, which ca be the same as Price. 7. Explai why the term APY is ot appropriate for brokered CDs. 8. Cosider a $00,000, five-year bak CD, at 10% omial rate compouded mothly. (a) Calculate the mothly iterest divideds. (b) Calculate the future value of the iterest paymets with iterest reivested at the CD rate. (c) Calculate the Fial Balace of Pricipal plus Iterest. (d) Calculate the total Divideds. (e) Calculate the iterest o iterest. Bak vs. Brokered CDs Sprig, 011 7
9. To summarize, give two formulas for the Fial Balace B for a Bak CD where P is the iitial deposit, r is the omial aual rate (yield), is the umber of years, k is the umber of compoudig periods per year, ad R is the periodic divided. 10. To summarize, give a formula for the Fial Balace B for a ewly issued Brokered CD where Price P = Value at Maturity M, with semiaual divideds R, for semiaual YTM periods, years, a APR = YTM = omial yearly rate, ad R = M. 11. To summarize, give the formula for the Price P of a ewly issued Brokered CD with semiaual divideds R, semiaual periods, years, Value at Maturity M, ad i = YTM as the semiaual rate where the aual omial rate = YTM = APR Bak vs. Brokered CDs Sprig, 011 8
Side Bar Notes Geeral Formula for the Sum of a Ordiary Auity with paymets R at the ed of each (1 + i) 1 period, iterest rate i per period, for periods: S = R. i Compoud Iterest Formula for the sum S, from ivestmet of Pricipal P, ivested for periods, with i the iterest rate per period: S = P(1 + i). Shoppig for CD rates. The followig are istructios for usig the Iteret to shop for ad compare CD rates from baks ad brokerages. For CDs bought from a bak: go to www.bakrate.com. Click o the Fid a CD or MM rate box. Select the terms you wat. Click Natioal. You ca do busiess atioally just as easily as locally. Click o Sort, ad APY to sort by APY. Write dow the 1-, -, ad 5-year rates. Write dow the Natioal Average rates ad compare with the top rates. Do some math to compare the differeces o a $100,000 CD from age 30 to 70. For CDs bought from a brokerage, go to www.vaguard.com. Click o Go to Persoal Ivestors Site. Uder Research Fuds ad Stocks, click Stocks, Bods, ad CDs. Chose your terms ad jot dow the rates. Do some more math ad comparisos. Summarize your observatios ad discuss with your teacher ad your class. At oe time, CD rates were as high as 14%. What is differece i earigs betwee 14% ad 1% CDs over time? Why so much differece? Do t get trapped i the rate race. This was advice published i large prit o the cover of a brochure by a local Credit Uio. Do you thik it is good advice? FDIC isured bak ad brokered CDs. See http://www.fdic.gov/deposit/deposits/isured/. Iterest paid by brokered CDs. Brokered CDs typically pay iterest o the Pricipal P at! $ # YTM & the YTM rate by the formula Divideds = # 365 & P # & x % the six moth period. You otice that the deomiator is seldom, except possibly durig leap year. ( ), where x is the umber of days i Bak vs. Brokered CDs Sprig, 011 9
Modified Iteral Rate of Retur (MIRR). The basic defiitio of YTM is the same as that of Iteral Rate of Retur (IRR). IRR has the same problems as YTM. For brokered CDs oe problem is the assumptio that divideds are reivested at the CD rate. To compesate, modificatios of IRR (YTM) such as Modified Iteral Rate of Retur (MIRR) are used. Spreadsheets have built i fuctios to calculate MIRR. If you use MIRR, you will be asked for a reivestmet rate for positive cash flows ad a fiace rate for egative cash flows. Cosider the followig $1000, two-year brokered CD with YTM = 5% ad payig divided of $5 at the ed of every six moths. Cosider reivestmet of divideds at 4% YTM semiaually up to the maturity date of the CD. What is the MIRR? See http://e.wikipedia.org/modified_iteral_rate_of_retur. APR ad APY: Why Your Bak Hopes You Ca t Tell The Differece. See http://ivestopedia.com/articles/basics/04/10904.asp. Bak vs. Brokered CDs Sprig, 011 10
Aswers to You Try Its. You Try It #1 (a) (b) (c) Fial Balace = 100 (1+ 0.01)4!1% $ ' + 10,000 = $10,406.04. # 0.01 & (d) Total iterest = 10,406.04 10,000 = $406.04 Iterest o iterest = $6.04 (e) Oe versio of a geeral formula for the Fial Balace B for a bak CD is the followig: B = R (1 ) + i 1 + P where R is the Divided paymet, i is the iterest rate i per Divided period, is the umber of compoudig periods (Divided periods), ad P is the Price of the CD. You Try It # (a) 1 + APY = (1 + 0.01) 4, APY = (1 + 0.01) 4 1 = 0.001 =.01% (b) The APY is the aual iterest rate that yields the same as the periodic rate compouded for a year. If i is the periodic rate compouded k times per year, the APY is such that (1 + APY) = (1 + i) k ad APY = (1 + i) k 1. The APY (sometimes called Effective rate) is used to compare the rates of differet ivestmets icludig differet bak CDs. You Try It #3 (a) You should get the formula for the Fial Balace B = P(1 + i) 4, where i = r/ ad r is the aual omial rate, ad P is the pricipal ivested i the CD. I geeral, if i is compouded k times per year, the i = r/k. If is the umber of years, the B = P(1 + r k )k. Exercise: From this formula, give a formula for P the price of a CD. Solve for r. Solve for k. k! (b) B = 1000 1+ 0.05 $ # % & Iterest Formula. 4 = $1103.81. You have derived a versio of the Compoud You Try It #4 For this brokered CD, i = r/ = YTM = 0.0/ = 0.01 is the six moth rate. M = $10,000 = Price = Value at Maturity. Divided = R = 0.01(10,000) = $100 per six moths. Fial Balace = 4R + M = 4(100) + 10,000 = $10,400. YTM = r = %. Bak vs. Brokered CDs Sprig, 011 11
Aswers to Exercises 1. i =.5%. YTM = 5%.!. APY for No. 1 = 1+ 0.03 $ # 1 % &! APY for No. = 1+ 0.04 $ # 1 % & 1 '1 = 0.0344. '1 = 0.04144. No. pays a better rate. 3. CD for 5 years, 6% compouded mothly, P = $4000, i = 0.06/1 = 0.005. R = 0.005(4000) = $0. B is the Fial Balace. Substitutig i oe formula gives B = 0 (1+ 0.005)60!1% $ ' + 4000 = $5395.40. # 0.005 & Substitutig ito aother formula gives B = 4000(1 + 0.005) 60 = $5395.40. 6. From the cash flow timelie i Figure 1, ad from Formula 4 ad Exercise 5, the 1 (1 + i) preset value of the Rs is R, ad the preset value of M is M(1 + i) -. i 1 (1 + i) So the price of the brokered CD is P = R + M(1 + i) -. i 8. (a) Mothly divideds = 0.10 (00,000) = $1666.67. With a bak CD, the customer 1 has the optio of takig the divideds or reivestig them i the CD at the CD rate. (b) Future Value of iterest paymets = $19,06.04. (c) Fial Balace = 19,06.04 + 00,000 = $39,06.04 (d) Total of the divideds = 60 1666.67 = 100,000. (e) Total iterest o iterest = $9,06.04. 9. For a Bak CD, for the variables defied i the problem, where P is the iitial deposit, r is the omial aual rate (yield), is the umber of years, k is the umber of compoudig periods per year, ad R is the periodic divided, k 1+ r k (! $ + * r Fial Balace B = P 1+ k. Also B = R # k % & '1- * - * r - + P * - k ) *, - Bak vs. Brokered CDs Sprig, 011 1
10. For a ewly issued Brokered CD, for the variables defied i the problem, where Price P = Value at Maturity M, with semiaual divideds R, for semiaual periods, YTM years, ad APR = YTM = omial yearly rate, ad R = M, YTM Fial Balace B = R + M = M + M. 11. For a ewly issued Brokered CD, for the variables defied i the problem, where i is YTM the semiaual rate =, where the aual omial rate = YTM = APR for semiaual periods for years, with semiaual divideds R, ad Value at Maturity M, Price P = 1 (1 + i) R M (1 i) i + +. Bak vs. Brokered CDs Sprig, 011 13
Teachers Notes Shoppig for CD rates. Oe of the Side Bar Notes gives istructios for shoppig ad comparig the rates paid by brokered CDs ad CDs bought at a bak, ad shoppig for top rates paid atioally. You might have some studets report their research i this area, ad pass out copies of their reports. Retiremet savigs are ofte doe i IRAs, 401ks, ad 403bs with a isurace compay or a mutual fud family. Some will have a brokerage widow where you ca ivest i CDs. At times CDs pay better rates tha fixed accouts ad moey market fuds. At a brokerage widow, you ca also ivest i idividual bods ad stocks. Buyig ad holdig a collectio of bods to maturity ca at times pay better tha a fixed accout or a bod fud. The advatage of owig your ow bods over a bod fud is you kow their worth at maturity while the NAVs of a bod fud vary uexpectedly with market iterest rates. See articles i this course o bods icludig iterest rate risks ad bod duratio. A lifetime file o persoal fiace. Have your studets set up ad maitai a lifetime file that icludes articles from this course. Hopefully, some day, i the ear future, they will use them to maage successfully their ow fiaces. For example, this article gives some good web sites. There are hudreds of advatages to kowig fiacial mathematics ad persoal fiace. Have your studets keep a accumulatig list of the useful thigs they leared i this course. As you teach this course, you ca accumulate items for pre- ad post-tests ad questioaires o fiacial kowledge, attitudes, ad persoal values, ad mathematics kowledge ad skills. Have studets use this list to evaluate the course at the ed. The basic formulas for Bak ad Brokered CD s are summarized i the aswers to Exercises 9, 10, ad 11. Bak vs. Brokered CDs Sprig, 011 14