2.4 INVESTMENT RETURN


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1 2.4 INVESTMENT RETURN In Examples (2.2.5), (2.3.5), and (2.3.6), we determined an unknown interest rate. The rates we found are known as yield rates for the investments. More generally, consider the time τ equation of value (under compound interest at the rate i) (2.4.1) k C tk 1 C i τ t k D B 1 C i τ T. A rate of interest which satisfies (2.4.1) is called an (annual) yield rate or internal rate of return for the investment giving rise to (2.4.1). Yield rates are sometimes called dollarweighted yield rates to distinguish them from timeweighted yield rates that are introduced in Section (2.7). Think of a yield rate as an interest rate on savings and loans that would result in the contributions C tk accumulating to B at time T [as shown by setting τ D T in Equation (2.4.1)].
2 dvdriverdos /2/ : page # Chapter 2 Equations of value and yield rates Yield rates are a measure of how attractive a particular financial transaction may be. A lender wishes to have a high yield rate while a borrower searches for a low yield rate. However, as we will consider in Section (2.5) and have already noted in our discussion of Example (1.7.8), there are complications posed for the lender unless repayments may be invested at a rate equal to the original interest rate. Furthermore, if the signs of the contributions C tk fluctuate, there may not be a consistent borrower and lender. We shall see that Equation (2.4.1) does not always have a solution and when it does, that solution need not be unique. However, in (2.4.7) we discuss hypotheses under which uniqueness is guaranteed, and these hypotheses are frequently satisfied in real life applications. EXAMPLE A unique yield rate Problem: Gautam invested $1,000 on March 1, 1998 and $600 on March 1, In return he received $600 on March 1, 1999 and $1,265 on March 1, Show that i D 10% is the unique yield rate. Solution A March 1, 2001 equation of value describing Gautam s investment is $1,000 1 C i 3 $600 1 C i 2 C $600 1 C i $1,265 D $0. Let p x D 1,000x 3 600x 2 C 600x 1,265. Then i is a yield rate if and only if 1 C i is a real root of p x D 0. Note that p 1.1 D 0. Therefore, i D.1 D 10% is a yield rate, and x 1.1 divides the polynomial p x.infact, p x has the factorization p x D x 1.1 1,000x 2 C 500x C 1,150.The quadratic formula shows that 1,000x 2 C 500x C 1,150 D 0 has no real roots, and consequently 1.1 is the only real root of p x D 0. So,.1 = 10% is the only yield rate. We note that if a graphing calculator is available, you might use it to quickly locate this rate.. EXAMPLE No yield rate Problem: Ace Manufacturing agrees to pay $100,000 immediately and again in exactly two years in return for a loan of $180,000 one year from now (to be used to replace a piece of machinery). Their CEO is asked what yield rate is associated with this transaction, but he is unable to answer the question. Why must this be the case? Solution A time 2 equation of value describing this financial arrangement is $100,000 1 C i 2 $180,000 1 C i C $100,000 D $0. Equivalently, 10 1 C i C i C 10 D 0. Thus, by the quadratic equation, 1 C i D 18; Since < 0, this leaves no real 20 solutions to the yield equation..
3 dvdriverdos /2/ : page #104 Section 2.4 Investment return 89 EXAMPLE Undefined yield Problem: Banker Johnson is always on the lookout for opportunities to make money without risking any of his own funds. Johnson is able to borrow $10,000 for one year at an annual effective interest rate of 4%, then loan out the $10,000 for one year at an annual effective rate of 6%. What is Johnson s yield rate on this transaction? Solution Banker Johnson must pay.04 $10,000 D $400 of interest for the money he borrows. However, he receives interest of.06 $10,000 D $600 on the $10,000 he loans out. He thus makes $200 D $600 $400 on the transaction without tying up any of his own money. No finite yield rate describes this situation. It might be tempting to say that the yield rate is infinite. However, we refrain from that because this would not give us a way to distinguish Johnson s situation from the even more favorable situation in which he is able to loan out the $10,000 at a rate higher than 6%.. EXAMPLE Multiple yield rates Problem: Alice and Afshan are friends. Afshan agrees to give Alice $1,000 today and $1,550 two years from now if Alice will give her $2,500 in one year. What is the yield rate for this transaction? Solution A time two equation of value for the given situation is $1,000 1 C i 2 $2,500 1 C i C $1,550 D 0. This is quadratic in 1 C i and the quadratic formula tells us that 1 C i D 2,500 ; 6,250,000 6,200,000 2,000 D 2,500 ; 50,000 2,000 D 25 ; Therefore, i D 5C 5 L or i D 5 5 L We have two distinct positive interest rates.. EXAMPLE Three distinct yield rates Problem: Parties A and B agree that A will pay B $1,000,000 at t D 0and $3,471,437 at t D 2. In return, B will pay A $3,228,000 at t D 1 and $1,243,757 at t D 3. Find the possible yield rates, estimating each to at least the nearest 10,000th of a percent. Solution For those equipped with a BA II Plus calculator, the rate closest to zero may easily be found. (In general, if the cashflow worksheet gives you a yield rate, it is the one closest to zero.) The correct sequence of buttons to produce the approximate yield rate of % is
4 dvdriverdos /2/ : page # Chapter 2 Equations of value and yield rates CF 2ND CLR WORK ENTER C/ ENTER ENTER C/ ENTER IRR CPT. For those not using the BAII Plus, this first yield may be found using either Newton s method or guess and check, each of which might be facilitated by a good graph of the function f x D1,000,000 1 C x 3 3,228,000 1 C x 2 C 3,471,437 1 C x 1,243,757, perhaps obtained on a graphing calculator f x The graph is helpful because the yield rates are the roots of this polynomial, and for either Newton s method or the guess and check method, an initial approximation is needed. A graph shows that as well as being a root near 4%, there is a root close to 6% and another close to 12%. Here we use the guess and check method to find the root close to 6% and calculusbased Newton s method to find the root near 12%. The root near 6%: We successively calculate f.06 D 1.42, f.061 D.25, f.0611 D , f.0612 D , f D , f D , f D ,and f D So, the yield rate is between % and %. x
5 dvdriverdos /2/ : page #106 Section 2.4 Investment return 91 The root near 12%: Note that f x D 3,000,000 1 C x 2 6,456,000 1 C x C 3,471,437 and set x 0 D.12. Then x 1 D x 0 f x 0 f x 0 D L , x 2 D x 1 f x L L , x f 3 D x 2 f x 2 L x f x L , and x 4 D x 3 f x 3 L , f x , L So, the rate is approximately equal to %. Using the BA II Plus calculator for these methods: In the guess and check method and also in Newton s method, one is repeatedly evaluating the polynomial f. It is helpful to store the coefficients of f in successive memories and also to reserve a register for one plus the argument at which the function is evaluated. Likewise, in Newton s method, use successive memories for the coefficients of the derivative f. As one calculates f or f, store the partial results in an available register, remembering that the BA II Plus calculator allows you to add a displayed number to the value stored in register m by pushing STO C m.. EXAMPLE Three party example Problem: Brian, Filemon, and Harold are friends. Brian will pay Filemon $1,000 now. Filemon will pay Brian $300 and Harold $800 in exactly one year. Finally, Harold will pay Brian $900 two years from now. What is Brian s annual yield for this threeway transaction spanning two years? Solution Brian s time 2 equation of value is $1,000 1 C i 2 $300 1 C i $900 D $100[10 1 C i C i 9]. The quadratic equation then tells us that 1 C i D 3; 9C360. Since Brian invests $1,000 and receives $1,200 > $1,000, 20 he has a positive yield rate, namely i D 3C 9C360 1 L Note that the threeway financial transaction of Example (2.4.7) can be viewed in terms of twoparty loans. We could think of it as the following package of loans. (1) Brian loans Filemon $1,000 at t D 0 and receives $1,100 from Filemon at time t D 1 in full repayment of the loan. (2) Brian loans Harold $800 at t D 1 and receives $900 from Harold at time t D 2 in full repayment of the loan. Note that the interest rate paid by Filemon is $1,100 1 D.1 andthe $1,000
6 dvdriverdos /2/ : page # Chapter 2 Equations of value and yield rates interest rate paid by Harold is $900 1 D.125. Brian s yield rate is in between $800 these two rates. When computing Brian s yield rate in Example (2.4.7), we were not concerned with the source of the money coming in or the recipient of outgoing money. Rather, we took a bottom line approach. We were only concerned with the times and amounts of all contributions (positive or negative) by Brian. This approach is fundamental to the successful determination of yield rates. When computing the yield rate received by an investor, take a bottom line approach. Consider only the times and amounts of all contributions (positive or negative) by the investor.
7 dvdriverdos /2/ : page # Chapter 3 Annuities (annuities certain) 3.10 YIELD RATE EXAMPLES INVOLVING ANNUITIES In Chapter 2 we introduced the yield rate. We now look at several yield rate problems that involve annuities. EXAMPLE Problem: Melanie pays $24,000 and receives a fifteenyear annuity with endofyear payments of $2,100. What is Melanie s annual yield on this investment? Solution The time 0 equation of value describing Melanie s investment is $24,000 D $2,100a 15 i. Recalling (3.2.4), we see that this equation is equivalent to the equation 24,000i 2, C i 15 D 0. So, the unknown yield rate is the root of the function f x D 240x C x 15. This root may be approximated by the guess and check method or by Newton s method. More easily, it may quickly be found in two different ways using financial buttons on the BA II Plus calculator. All four solutions are now presented. Solution by the guess and check method Melanie receives a single payment of $24,000 and her repayments total 15 * $2,100 D $31,500. Were she to have made one lump repayment of $31,500 at the end of eight years (the average of her repayment times), the yield rate would have been (31,500/24,000) 1 8 L 3.5% so it is reasonable to start the guess and check method with an initial guess of 3.5%. We calculate that f.035 L.065. The guess is therefore close, but since f evaluated at this initial guess is negative, looking at the definition of f indicates we should raise our guess for i a little. Check that f.036 L.0056, f.0361 L.00056, and f L So, the rate is between and.0361, probably closer to If need be, we can continue refining our guesses to get any needed degree of accuracy. Solution by Newton s method Once again, we begin with an initial guess x 1 D.035. Note that f x D C x 16, x 2 D x 1 f x 1 L L , and x 3 D x 2 f x f x 2 f x 1 L L One may continue to find x 4 D x 3 f x 3 L , so f x is a very good estimate. In fact f L
8 dvdriverdos /2/ : page #172 Section 3.10 Yield rate examples involving annuities 157 Solution using the TVM InEndModewithC/Y=P/Y=1,key C/ PV 1 5 N PMT 0 FV CPT I/Y to obtain I/Y = Solution using the Cash Flow worksheet Push CF 2ND CLR WORK to open and clear the Cash Flow worksheet. Thenpress C/ ENTER ENTER 1 5 ENTER IRR CPT to obtain IRR = The next two examples involve reinvestment (partial or complete) of annuity payments received. EXAMPLE Problem: Gian Carlo invests $58,000 and receives an annuity of $7,000 at the end of each year for thirteen years. Each time Gian Carlo gets a $7,000 payment, he immediately deposits $4,000 in a savings account that earns 9%. Find the annual yield received by Gian Carlo. Solution Recall that one takes a bottom line approach when computing a yield rate. So, the first task is to determine the amounts and times of all expenditures and receipts by the investor. The only expenditure is the initial investment of $58,000. We denote the time of this payment time 0. As for receipts, he receives $3,000 D $7,000 $4,000 at times 1, 2, He also receives the balance of the 9% savings account at time 13. This balance is $4,000s 13.09, or to the nearest penny $91, We therefore have the time 0 equation of value $58,000 D $3,000a 13 i C $91, v 13. As in the previous example, the yield rate may be found using the guess and check method, Newton s method, or more easily, using the annuity buttons on a business calculator. Solution by the guess and check method By (3.4.2), the fact that i satisfies the above equation of value is equivalent with i being a root of the function f x D 3,000 58,000x C 1 C x 13 91,813.54x 3,000. The investor contributes $58,000 and has repayments totaling 13 * $7,000 D $91,000. The average time of repayment is seven years so looking for an initial guess of the yield rate, we might calculate (91,000/58,000) L 6.6%.
9 dvdriverdos /2/ : page # Chapter 3 Annuities (annuities certain) But the investor s reinvestment rate is 9%, so a reasonable initial rate is somewhere inbetween these rates, say their average, 7.8%. Now calculate that f.078 L Due to the equation of value, the fact that the evaluated value is positive leads us to consider a higher guess. Calculate that f.08 L So, the yield rate is inbetween.078 and.08, and.079 is apt to be close. Check that f.079 L.866 and f.0791 L 3.4 so the yield rate is between 7.9% and 7.91%, probably closer to 7.9%. Continue to the needed degree of accuracy. Solution by Newton s method As in the previous solution, we begin with an initial guess x 1 D.078. Note that f x D 58, C x 14 91,813.54x 3,000 C 91, C x 13, and x 2 D x 1 f x D.078 L This is already f x 1 42, an excellent estimate, but we could repeat to find the yield rate with greater accuracy. Solution using the TVM In END mode with C/Y = P/Y = 1, push C/ PV 1 3 N PMT FV CPT I/Y to obtain I/Y = We note that for this last method to work, it was essential that Gian Carlo closed the savings account at time 13, the time of his last annuity payment. Solution using the Cash Flow worksheet Push CF 2ND CLR WORK to open and clear the Cash Flow worksheet. Then press C/ ENTER ENTER 1 2 ENTER ENTER IRR CPT to obtain IRR = The justification for entering 12 for C01 is that the total inflows to the investor at times 1, 2,..., 12 are all $3,000, but at time 13 the inflows total $3,000 C $91, D $94, Thus the proper entry for C02 is 94, EXAMPLE Problem: Serena invests a total of $10,000. She uses part of the $10,000 to purchase an annuity with payments of $1,000 at the beginning of each year for ten years. The purchase price of the annuity is based on an effective interest rate of 8%. As annuity payments are received, they are reinvested at an annual effective interest rate of 7%. The balance of Serena s $10,000 is
10 dvdriverdos /2/ : page #174 Section 3.10 Yield rate examples involving annuities 159 invested in a tenyear certificate of deposit with a nominal annual interest rate of 9% convertible quarterly. Calculate the annual effective yield on the entire $10,000 investment over the tenyear period. Solution The value of the annuity is $1,000$a L $7, Since the purchase price must be an integral number of cents, it is $7, The amount invested in the certificate of deposit is therefore $10,000 $7, D $2, At time 10, since Serena invests her annuity payments at 7%, she has accumulated $1,000$s L $14, plus the value of the certificate of deposit which is $2, C L $6, The total accumulation is therefore $14, C $6, D $21,487.94, and her yield is i where $10,000 1 C i 10 D $21, So, i D (21, /10, 000) L %.. In the next example, it is the expenditures that form an annuity. EXAMPLE Problem: Hideo deposits $1,500 at the beginning of each year for sixteen years in a fund earning an annual effective rate of interest of 6%. The interest from this fund is paid out annually and can only be reinvested at an effective annual rate of 5.2%. At the end of twenty years, Hideo liquidates his assets. What is Hideo s yield rate for the twenty yearperiod? Solution The balance in the 6% account at time 20 is 16 * $1,500 D $24,000 since interest is paid out from this account. The balance in the 5.2% account is more interesting. At time k, k {1, 2, 3,..., 20}, there is an interest payment from the 6% account deposited in the 5.2% account. If k {1, 2, 3,..., 16}, the amount of this deposit is $1,500k *.06 D $90k. Thisisbecausefor k {1, 2, 3,..., 16}, the amount on deposit in the 6% account in the kth year is $1, 500k. On the other hand, if k {17, 18, 19, 20}, the amount on deposit in the 6% account in the kth year stays constant at 16 * $1,500 D $24,000 and the amount deposited to the 5.2% account is.06 * $24,000 D $1,440. PAYMENT: 1 * $90 2 * $90 16 * $90 $1,440 $1,440 TIME: Payments to the 5.2% account
11 6.8 YIELD RATE EXAMPLES Bonds provide a series of cashflows. While we have concentrated on bonds with level coupons, a portfolio of bonds may give a nonlevel cashflow pattern due to staggered purchase or redemption times [see Examples (6.8.2) and (6.8.3)]. Bonds may be resold prior to maturity one or more times [See Example (6.8.4)], and coupons may be partially or fully reinvested [see Example (6.8.5)]. An investor may purchase a bond at a premium and desire to recoup the amount of the premium in a sinkingfund account, as in Example (6.8.1). These practical situations provide a wealth of new problems involving yield rates. EXAMPLE Recouping a premium in a sinking fund account Problem: Alicia bought a newly issued $1,000 20% tenyear bond, redeemable at $1,100 and having yearly coupons. It was bought at a premium at a price of $1, Alicia immediately took a constant amount D from each coupon and deposited it in a savings account earning 8% effective annual interest, so as to accumulate the full amount of the premium a moment after the final deposit. At the end of the ten years, Alicia closed out her savings account. Find the yearly effective yield rate earned by Alicia for her combined tenyear investment.
12 dvdriverdos /2/ : page #308 Section 6.8 Yield rate examples 293 Solution In order to accumulate the premium $1, $1,100 D $ in her savings account, each year Alicia needs to deposit $ L$ s 10 8% So, if we let D D $20.71, at the end of the ten years Alicia will receive $ when she closes out her savings account. Since the annual coupon amount is.2 * $1,000 D $200, she keeps $200 $20.71 D $ at the ends of years 1, 2,..., 10. But at time 10 she also gets the redemption $1,100 and the savings account liquidation $300.02, and these total $1, Thus she earns $ each year while keeping her principal of $1, intact and her yield is $ L %. Of course, since a 1, i D 1 1Ci 10, this i yield could also be obtained by solving for i in the time 0 equation of value $1, D $179.29a 10 i C $1, C i 10.. EXAMPLE Finding the yield of a laddered portfolio of bonds Problem: On January 1, 1978, Lenny Ladderman purchased a portfolio of twelve $1,000 parvalue bonds for $10, The portfolio consisted of 6% bonds, each with annual coupons, The twelve maturity dates for the bonds were January 1 of the years from 1983 through Find Lenny s yield rate. Solution The annual coupon payment resulting from each bond not yet redeemed is.06 $1,000 D $60. Through January 1, 1983, when the first redemption takes place, Lenny had twelve $60 coupons for a total of $720, and after that the contribution from coupons decreases by $60 annually. Each year from 1983 through 1994 there is a $1,000 redemption in addition to the coupon contribution. So, we have cashflows as indicated in the following figure. PAYMENT: $720 $720 $1,720 $1,660 $1,600 $1,060 TIME: (Jan. 1) The yield rate j satisfies the time 0 equation of value $10, D$720a 4 j C I 1,720, 60 a 12 j 1 C j 4 D$720a 4 j C($1,720a 12 j C 60 j (a 12 j 12 1Cj 12 )) 1Cj 4. You may now use the guess and check method to find the yield rate j, beginning with a rate somewhat greater than the coupon rate, say 7%, since the package of bonds was bought at a discount (due to the fact that the purchase price was less than the sum of the redemption values). In fact, j L
13 dvdriverdos /2/ : page # Chapter 6 Bonds The Cash Flow worksheet of the BA II Plus calculator may be used to obtain this yield rate relatively quickly. The cashflows shown in the above timeline may be entered by keying CF 2ND CLR WORK C/ ENTER ENTER 4 ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER IRR CPT. This sequence of keystrokes sets CF0= 10,568.61,C01=720, F01= 4,C02=1,720, F02=1, C03=1,660, F03=1, C04=1,600, F04=1, C05=1,540, F05=1, C06=1,480, F06=1, C07=1,420, F07=1, C08=1,360, F08=1, C09=1,300, F09=1, C10=1,240, F10=1, C11=1,180, F11=1, C12=1,120, F12=1, C13=1,060, and F13=1, and then the calculator computes the yield rate to be %.. EXAMPLE Pricing a laddered portfolio of bonds Problem: Nicolae Miloslav prefers a laddered portfolio of bonds. He purchases eight $20,000 7% parvalue bonds, each having semiannual coupons. The terms of these bonds are 3, 4, 5, 6, 7, 8, 9, and 10 years. Calculate the total price of these bonds to yield Mr. Miloslav a nominal rate of 8% convertible semiannually. Solution Each coupon of a $20,000 7%, bond with semiannual coupons is for $20,000 (.07 ) L $700. Therefore, for the first three years, at the end of each 2 halfyear Mr. Miloslav receives coupons totaling 8 * $700 D $5,600. During the subsequent seven years, each year he holds one fewer bond and hence the total of the semiannual coupons he receives goes down by $700. In addition to coupons, Mr. Miloslav receives redemption amounts of $20,000 at the end of years 3 through 10. So Mr. Miloslav s incoming cashflows consist of (1) semiannual coupons totaling $5,600 at each of the times 1, 1, 1 1, and (Calculated using a nominal interest rate of 8% convertible semiannually, these have a time 0 value of $5, 600a 4 4%.) (2) Midyear coupon payments totaling $5,600,$4,900, $4,200, $3,500, $2,800, $2,100, $1,400, and $700 at times t D 2 1, 3 1, 4 1, 5 1, 6 1, 7 1, 8 1, and 9 1, respectively. (Calculated using a nominal interest rate of 8% convertible
14 dvdriverdos /2/ : page #310 Section 6.8 Yield rate examples 295 semiannually, these have a time 0 value given by the expression of I $5,600, $700 a ) (3) Endofyear coupon payments totaling $5,600, $4,900, $4,200, $3,500, $2,800, $2,100, $1,400, and $700 at times 3, 4, 5, 6, 7, 8, 9, 10 respectively. (Calculated using a nominal interest rate of 8% convertible semiannually, these have a time 0 value of I $5,600, $700 a ) (4) Redemption payments of $20,000 at times 3, 4, 5, 6, 7, 8, 9, 10. (Calculated using a nominal interest rate of 8% convertible semiannually, these have a time 0 value of $20,000a ) Therefore, the total price of Mr. Miloslav s portfolio of bonds to provide him with a nominal yield of 8% convertible semiannually is $5,600a 4 4% C 1.04 C I $5,600, $700 a % C $20,000a % L $20, C $19, C $97, L $152, EXAMPLE Multiple resellings of a bond Problem: Audrie purchases a $10,000 fifteenyear 6% bond with semiannual coupons and a redemption value of $9,800. He pays a price P A. After sixandahalf years, just after the thirteenth coupon, Audrie sells the bond to Barry at a price P B. Barry holds the bond for five years (ten coupon payments), then sells it to Carl for a price P C, and Carl holds the bond through its redemption. Denote the semiannual yield rates earned by Audrie, Barry, and Carl by j A, j B, and j C respectively. It is known that j A D 2.9%, j B D 4.2%, and j C D 3.5%. Find the discount or premium on the bond purchased by Audrie. Solution The bonds purchased by all three investors have semiannual coupons of $10, D $300. Since Audrie gets thirteen coupons 2 and Barry receives ten, Carl s bond has D 7 coupons. So, according to the basic price formula, Carl s price P C satisfies P C D $300a 7 3.5% C $9, Calculating and rounding, one finds P C D $9, Since Carl pays this amount to Barry at the time Barry receives his tenth coupon payment, Barry s time 0 equation is P B D $300a % C $9, and thus P B D $8, Finally, Audrie receives $8, at the time of his thirteenth coupon, and P A D $300a % C $8, So, P A D $9, and the discount is $9,800 $9, D $
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