The Advanced Arithmetic and Theorems of Mutual Fund Statements. Floyd Vest

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1 The Advanced Arithmetic and Theorems of Mutual Fund Statements Floyd Vest Millions and millions of Americans have investments in mutual funds This gives them professional management of their money invested in stocks, bonds, and other investments Stock mutual funds usually pay dividends approximately every six months An annual mutual fund statement Table is a simulation and compression, with explanation, of an annual stock mutual fund statement Trade date Dividend per share Dollar amount of this transaction Share Price Shares Transacted Total shares owned Account Value 2/3/previous year ,305 6/ /25 $020 $ $25, / /5 $30 $ $25, /3/year end $25,88578 Table Simulated annual stock mutual fund statement The following explanation of the treatment of dividends by stock mutual funds is adapted from the source cited below: When a dividend is paid out to share holders, its NAV, or share price, will be reduced by the amount of the dividend per share For example if you have 00 shares in your account when the NAV is $0 a share, your account is $000 If the fund pays a dividend of $2 a share (or $200 since you have 00 shares) the NAV drops to $8 a share and your original 00 shares is now worth $800 However, if you automatically reinvest the dividend, the $200 distribution buys $200 worth of shares at $8 per share The distribution therefore adds 25 shares ($200 divided by $8) to your account, so you own 25 shares worth $8 each, for a total of $000 which was your original account value before the dividend was paid (americanfundscom/funds/effectcapital-gainhtm) You Try It # Calculate and substitute the numbers from Table into the previous explanation Start on the date 6/24 You Try It #2 Fill in the Account Value in two places in the above table Fill in number of Shares transacted in one place in the table Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20

2 The algebra of the stock fund statement Table 2 displays the beginnings of the algebra of the above stock fund statement We will use intermediate variables P, P, P 2, and P 3 for Share prices The meanings of the variables and expressions are given by their position in Table 2 Most of the additional algebraic expressions for entries in the table are too long to include in the table They are presented in the following discussion The symbol P stands for Share price at the beginning of the year T represents Total shares owned at the beginning There are three appreciation rates of Share prices: a, a 2, a 3 P is the Share price on the Day before dividend D and P = P ( + a ) P 2 is the Share price for the Day before dividend D 2 and P 2 = (P D )( + a 2 ) P 3 is the Share price at the end of the year and P 3 = (P 2 D 2 )( + a 3 ) See Table 2 Trade date Dividend per share Dollar amount of transaction Share Price Shares Transacted Total shares owned Account Value 2/3/previous year P (T)(P ) Day before D P = P ( + a ) Dividend day ofd, Day D T(D ) P D Day before D 2 P 2 = (P D )( + a 2 ) Dividend day of D 2, Day 2 D 2 P 2 D 2 2/3/year end P 3 = (P 2 D 2 )( + a 3 ) Table 2 The algebra of the stock fund statement For example, from Table the appreciation rates are a = = a 2 = = a 3 = = The Dividend D = $020 per share The Dividend D 2 = $30 per share The key Share prices are P = $286, P = $3266 = P ( + a ), P 2 = $3246 = (P D )( + a 2 ), P 3 = $2984 = (P 2 D 2 )( + a 3 ) For Total share owned at the beginning (2/3), T = shares ( T) D Shares transacted on Day (day of dividend D ) = ( ) P + a D ( T) D Total shares owned on Day = T + ( ) P + a D Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 2

3 ( T) D Account Value on Day = T + P ( ) [P ( + a ) D ] + a D For Day 2 (day of dividend D 2 ), the Dollar amount of the transaction ( TD ) = T + ( D2 ) P D For Day 2, the number of shares transacted = T( D ) T + ( D2 ) P D P D 2 2 T( D ) T( D ) T + ( D2 ) P D Formula : The Total shares owned on Day 2 = T + + P D P2 D2 D + D2 D P D = (T) + + P D P2 D 2 At this stage of the development, we will make some observations and simplifications One simplification is Theorem Theorem : On Day, the Total Account Value is (T)(P )( + a ) Proof: From the above, Total Account Value after appreciation a and the dividend D = ( TD ) T + [ P ( + a) D] P ( + a) D T [ P ( + a ) D ] + ( T) D = [ P ( + a ) D ] P ( + a) D = T[ P ( + a ) D + D ] = T(P )( + a ) Theorem 2: Total Account Value on the day before Day equals Total Account Value on Day The dividend subtracts out Proof: The Total Account Value after appreciation and before Dividend day of D is (T)(P )( + a ) From Theorem, the Total Account Value on Day is also (T)(P )( + a ) Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 3

4 Theorem 3: On Day, the Total profit is (T)(P )(a ) The increase in Account Values comes from appreciation of the Share prices Proof: Profit on Day = (Total Account Value on Day ) (Total Account Value at the beginning) = T(P )( + a ) T(P ) = T(P )[ + a ] = T(P )( a ) Final Share price Theorem 3: The final Share price on 2/3/end of year is P 3 = {[P ( + a ) D ]( + a 2 ) D 2 }( + a 3 ) = P ( + a )( + a 2 )( + a 3 ) D ( + a 2 )( + a 3 ) D 2 ( + a 3 ) Proof: From Table 2, the 2/3/end Share price = P 3 = (P 2 D 2 )( + a 3 ) = {!" P D # $ ( + a 2 ) D 2 }( + a 3 ) =! " P + a { ( ) D # $ ( + a 2 ) D 2 }( + a 3 ) ( )( + a 2 ) D ( + a 2 ) D 2 ( ) ( )( + a 2 )( + a 3 ) D ( + a 2 )( + a 3 ) D 2 ( + a 3 ) { } + a 3 = P + a = P + a Theorem 32: On the day before Day 2, Total Account Value = ( T )( P )( + a )( + a 2 ) Proof: Total Account Value on the day before Day 2 =[ ( ) ( T) D P + a D] ( + a 2 ) T + P ( ) + a D D = (T)[ P ( + a ) D ]( + a2) + P ( + a) D T a [ P ( + a ) D + D ] = T = ( )( ) + 2 Theorem 4: On Day 2, the Total Account Value is T ( )( P )( + a )( + a 2 ) ( )( P )( + a ) + a 2 Proof: From Theorem 32 and Theorem 2, Total account value on Day 2 = (T)(P )( + a )( + a 2 ) Another proof of Theorem 4: Total account value on Day 2 ( " D + % " D = ( P 2 D 2 )( T ) + % $ # P! D ' & + # $ P! D & ' (D ) + * 2 - * - * P 2! D - * 2 - )*,- ( ) Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 4

5 ( " D = ( T ) (P 2! D 2 ) + % $ # P! D ' & + " + D % $ # P! D ' & (D ) + * 2 - )*,-, & D = (T)( T ) "# (P! D )(+ a 2 )! D 2 $ % + ) ( ' P! D + * + & + D ) - ( ' P! D + * (D ) 0 2 / 2 ( D2 )( D ) ( D2 )( D ) = (T) ( P D )( + a2) + ( + a2) D D2 + D2 + P D P D + a ) P ( + a ) = (T)( + a 2 )( P D + D ) = (T)( ( ) 2 = (T)(P )( + a )( + a 2 ) Corollary: Total account value on the day before Day 2 is equal to the Total account value on Day 2 Theorem 5: The profit on Day 2 is (T)(P )[( + a )( + a 2 ) ] Proof: The profit on Day 2 =(Total Account Value on Day 2) (Total Account Value at the beginning) = (T)(P )( + a )( + a 2 ) (T)(P ) = (T)(P )[( + a )( + a 2 ) ] D Theorem 6: If a > then the Share price on Day is greater than the beginning Share P Price P D Proof: a > P P a ( ) > D since P > 0 ( ) > D + P P + P a Share price on Day = P! D = P + a ( ) D > P Theorem 7: If the Share price on Day is greater than P, then a > Proof: Share price on Day = P! D = P + a P + P ( a ) > P + D P a End of year theorems D P ( ) D > P P ( + a ) > P + D D ( ) > D a > P since P > 0 Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 5

6 Theorem 8: The Total account value at the end of the year = (T)(P )( + a )( + a 2 )( + a 3 ) Proof: y Theorem 4: Total account value on Day 2 = (T)(P )( + a )( + a 2 ) = (P 2 D 2 )(Total number of shares on Day 2) ( T )( P )( + a)( + a2) = (Total number of shares on Day 2) P2 D2 Total Account Value at end of year ( T)( P )( + a)( + a2) = (P 3 ) P2 D2 ( T)( P )( + a)( + a2) = ( P 2 D 2 )( + a 3 ) P2 D2 = ( T )( P ) + a ( )( + a 2 )( + a 3 ) Theorem 9: Profit at the end of the year = ( T )( P )!( + a )( + a 2 )( + a 3 ) Proof: Profit for the year = T ( )( P )( + a )( + a 2 )( + a 3 ) ( T )( P ) = ( T )( P )!( + a )( + a 2 )( + a 3 ) # " Theorem 0: The rate of return for the year (Total return) = ( + a )( + a 2 )( + a 3 ) Proof: Total return = = ( + a )( + a 2 )( + a 3 ) $ [ ] ( T)( P ) ( a )( a )( a ) 2 3 ( T)( P ) You Try It #3 Apply the formula for Shares transacted on Day to get the value in Table You Try It #4 Apply the formula for Total shares owned on Day to get the value in Table You Try It #5 Apply Theorem to Table, to calculate the Total Account Value on Day Check it against Table You Try It #6 Apply Theorem 3 to Table to calculate the profit on Day Check it against Table " # $ Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 6

7 What is the effect of the increased number of shares? In explaining the return on a mutual fund investment, Gitman (p 590) discuses what happens if dividends and capital gain distributions are reinvested in the fund The result is that the investor receives additional shares (See Table ) Then he calculates the holding period (rate of) return = ( OriginalNumber of shares + additional shares)( P3 ) ( OriginalNumber of Shares)( P ) ( OriginalNumber of Shares)( P ) This formula suggests that the return comes partly from the additional shares This is a common explanation given in finance Our discussion calculates the holding period return in terms of appreciation of Share prices You Try It #7 Apply the above formula for the holding period Rate of return (Total return), for the year, using the entries in Table Compare your results with that of Theorem 0 Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 7

8 Exercises: Show all your work Put on units Label numbers Summarize Do proofs in general terms Label answers Due to rounding to the nearest cent in Table, numbers may differ by as much as 0002 Give a general formula, for the year, that gives the total dollars paid in dividends 2 Given a Share price at the beginning P = $0, and two dividend periods with $6 dividends each, and each with an appreciation rate of a, what must be true of the rate a for the Share price on Day 2, to be greater than the beginning Share price of $0? Discuss the graph 3 Given two dividend periods each with a the same dividend D and appreciation rate a, what must be true of the Share price on Day 2 for it to be greater than the beginning Share price P? Do the derivation in general terms 4 Use Formula to prove Theorem 8 D2 5 Prove that if a2 > then the Share price on Day 2 is greater than the Share price P D on Day D2 D 6 Prove that if a2 > and a > then Share price on Day 2 is greater than P P D P D ( + a2)( + a3) + D2 ( + a3) 7 Prove that if P3 > P then rate or return for the year > P 8 A commonly used formula for Total rate of return, for the year, for the investor is P3 P + D + D2 TR = P (a) Apply this formula to calculations in Table Compare the results for TR calculated from Account Values and using Theorem 0 Report your results (b) Prove a general version of this formula for the first dividend period from P to the end of Dividend Day : TR for the first dividend period = Share Price on Day2! P + D P (c) Do the calculations to check this formula with Account Values in Table Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 8

9 Side ar Notes Advanced Arithmetic The world runs on advanced arithmetic Our traditional attitude as ordinary mathematics teachers and students is to disparage advanced arithmetic as intellectually inferior to more advanced high school and college mathematics ut, again advanced arithmetic runs the world, especially with the general access to computers and calculators including such packages as spreadsheets You can name a dozen more such packages With spreadsheets and arithmetic, you can do much of high school and college mathematics Why do mutual funds distribute dividends? These distributions are made primarily for tax reasons Mutual funds that operate exclusively in a tax-sheltered environment such as that of a 40K or 403, do not report on annual statements the kind of distribution of dividends displayed in Table Investing in Stocks The S&P 500 is a major stock market index holding a basket of 500 large cap stocks From 988 to 200, the annual rate of return averaged 95% ut since 999 the index has suffered losses From 999 to 2002 is suffered a drop of approximately 38%, averaging a loss of about 46% a year From 2007 to 2008, it dropped 39% in about one year As of May 7, 20, the S&P was at and still below values of 530 on May, 2007 and 58 on August, 2000 Question: Calculate the average rate of return from August, 2000 to May 7, 20 What percentage of P dollars has it lost in these eleven years? Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 9

10 References Gitman, Lawrence J, and Michael D Joehnk, Fundamentals of Investing, Harper and Row, 984, p 590 For an examination of this same Mutual Fund Statement from the point of view of profit from shares awarded, see the following article in this course: Vest, Floyd, The Profit from Additional Shares Awarded from Dividends in a Stock Mutual Fund, June 20 For the basics of mathematics of finance, see in this course: Kasting, Martha, Concepts of Math for usiness: The Mathematics of Finance (UMAP) , COMAPcom, 34 pages Provides the basics of mathematics of finance Starts with a review of algebra Does mathematics of finance on pages Has Compound Interest, Annuities, and Amortization with derivations, examples, exercises, sample test, and answers Luttman, Frederick W, Selected Applications of Mathematics to Finance and Investment (UMAP) 983, COMAPcom (5 pages of the basics of mathematics of finance, 24 formulas, derivations, examples, exercises, model test, and answers) Has some financial mathematics of real estate investments, and some calculus Does Compound Interest, Continuous Compound Interest, Annuities, Mortgages, Present Value of an Annuity, and APR The calculus can be left out Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 0

11 Answers to You Try Its You Try It # For example if you have shares in your account when the NAV is $3266 a share, your account is $25,46938 If the fund pays a dividend of $020 a share (or $5597 since you have shares) the NAV drops to $3246 a share and your original shares is now worth $25,334 However, if you automatically reinvest the dividend, the $5597 distribution buys $5597 worth of shares at $3246 per share The distribution therefore adds 4805 shares ($5597 divided by $3246) to your account, so you own shares worth $3246 each, for a total of $ , which was your original account value before the dividend was paid You Try It #2 6/24 account value: $ = $25, /4 account value: $ = $24, /5 number of shares transacted: You Try It #3 ( T) D Shares transacted on Day = P ( ) + a D = (020) 286( ) 020 = 4805 You Try It #4 Total shares owned on Day = = using numbers from You Try It #3 You Try it #5 y Theorem, Total Account Value on Day = (T)(P )( + a ) = (286)( ) = $25,46938 You Try It #6 y Theorem 3, Profit on Day = (T)(P )(a ) = (779834)(286)(045589) = $35833 Checking against Table, Account value on 6/25 = = 25,46935, 25, ,305 = You Try It #7 Holding period rate of return (Total return) for the year = [(Number of shares at the end)(final Share price) (Number of shares at the beginning)(share price at the beginning)]/[(number of shares at the beginning)(share price at the beginning)] = [867486(2984) (2862)] / [779834(286)] = y Theorem 0, Total return = ( + a )( + a 2 )( + a 3 ) = ( ( + 0)( ) = Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20

12 Answers to Exercises 2 For Share price on Day 2 to be greater than P, a > = 545% The graph is concave upward ( ) 3 For Share price on Day 2 to be greater than P, a > D + D2 + 4P D + P! 2P 8 (b) Theorem: For the first dividend period, TR = Share Price on Day! P + D P ( P D ) P + D P P P ( + a ) P Proof: y the formula, TR = = = = a y P P P Theorem 3, TR for period one = a So the formula gets the right answer Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 2

13 Teachers Notes Students discuss a mutual fund statement efore this lesson, divide the class into small groups Have each group appoint a secretary Give them thirty minutes to an hour to examine the following simulated mutual fund statement and answer the following questions: What would the average adult think about the form? What would they calculate, question, not understand? Have the students label their observations to specific dates, headings, columns, rows, and terms They could even express attitudes, values, and life goals Then, have them combine their results for a report to the class They could include observations from family and friends Trade date 6/26 2/6 Transaction Annual Stock Mutual Fund Statement Account value Dollar Amount On 2/3 previous On 2/3 year end of year $22,3885 $27,3445 Share price Shares transacted Total shares owned alance 2/3 previous year $ Income dividend Income dividend Short term 2/6 capital gain /6 Long term alance 2/3 end of year $ After this lesson, divide again with the same table and ask them to record and demonstrate al least fifty items of knowledge, skill, attitudes, values which they have learned and compare their results with the first set of observations Observations, no matter how detailed or seemingly redundant, should be accepted Of course, this activity could be done with other lessons For a free download course in financial mathematics see Just click on an article in the annotated bibliography, download, and teach it Advanced Arithmetic and Theorems of Mutual Fund Statements Spring, 20 3