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Unformatted text preview: 102 Chapter 2 Equations of value and yield rates examLE are Inveshnent period not equal to one year _' Problem: Mohammed had 520.000 io ltis investment account on August 15,
1999. On August 15, 2000 his balance was $11,200 and he deposited an
additional $5,000, giving him a new balance of $25,201 Clo August ‘.l5T “2.001l
Mohammed's account had a balance of $21,300. Assuming that there are no
other contributions to the accident, find the annual timeweightad'yield and
note that it is very close to the annual dollarweighted yield. ' Solution The balance grows as follows: $20,000 $21,200 $27,300 l / $20,200 Simona mom
{in the other hand. the dollarweighted yield i satisﬁes the equation of
value ' 1
Therefore a... = [(ﬂumxmmﬁi — 1 :1 .asaasstss. sedation + a}? + $5.000{1 + q = season. 'This equation may be solved using the quadratic equation, andi = .05 is the
only positive yield rate. I I EHMPLE 2.15 Timeweighted yield less than dollarweighted yield Problem: Astute Mr. Haywood notices that although the “Tomorrow F'und*I
has an excellent performance history, it performed less well when tlte price of
gasoline experienced a sharp rise lte decides to invest in the fund but, to the extent possible, move his money away from the fund during periods when he anticipates a sharp increase in gasoline prices. Go January 1, Mr. Haywood ' deposits $103,010 in the fund. Cln March 1 his balance is $102,111), and he withdraws $50,000. [in May 1 ltis balance is $52,501 and he deposits $50,000. At the end of the year Mr. HoytHood’s fund balance is $111,000. Find the
time—weighted yield for the “Tomorrow Fun d" for the year, and show that the
is lower than Mr. llaywood‘s dollarweighted yield. dewrev ;:_,. =,.a:'.'_I:i.sc.u—rs' '..:..vy:::..:.'.a.......1 “jinnmissu Sectian 23 Problems, Chapter 1 103 '. solution. .The balance grows as follows: ﬁrsat $100,000 ——r $102,000 I $52,500 $111,000 $51010  $102,500 I tii1_0.l_»_' 3 nightlife i... = rs. = {Mnﬂ‘ﬂnimmi —' 1 e 115205379. The 5 [03.110 £52,000 £102,503 " _ dollarweighted yield i satisﬁes ' _.‘.":I':.' ' It
itt11.Dﬂt}= Emmott + i] — $50,000Ij1 + ql—i _: sso,ona{1_+ rye. tifiqgoe $100,00tlﬂ + i}  ssanoou + £30 + 550,000{1 + i]%.Then
I? Inc ' I set $110,534.53 r'.' 111,000. Therefore, Mt. Haywood‘s dollarweighted yield
is higher than Em. In fact, using the “guess and check" method or Newton’s
method, one may determine that Mr. Haywood has a yield rate of approxi
matety .1103. More easily, the Elit ll Plus calculator Cash Flow worksheet mayr
he used to ﬁnd a monthly yield rate of 351201439 is' equivalent to an annual
yield rate of 1103092019110. The reason that Mr. Hayumd‘s dollar—weighted yield exceeds the
Tomorrow Fund’s Limeweighted yield is that he timed his deposits and
withdrawals well. Had they been poorly timed, his dollaraweighted yield
would have been worse than the Fund's timeweighted yield. I 2.3 PROBLEMS, CHAPTER 2
1191 Chapter 2 writing problems
: 11] [following Section (2.3)] Consider the equation Elem“ + o“ — smart + if + sauna + i} = asses. _ Describe a ﬁnancial situation for which this is the “associated time 4
Equation of value. Give detaiJs explaining the sources of any deposits
and the reasons for any withdrawals. [following Section [2.3)] Consider the equation .15 30 .
sauna“ + if“ = E $10,000u". it I m=l 104 Chapter 2 Equations of value and yield rates Describe a ﬁnancial situation for which this is tlte associated time 35
equation of value. (3) [following Section (23)] Write an adverdmm ent for an investment fund,
giving annual yields earned by the fund over each of ﬁve years and an
annual time—weighted yield for the period. {4) Learn about I.l'tree mutual funds and the portfolio focus of each fund.
For each fund, note the annual yield rates over various time periods.
include the period from the date of date of inception to a recent date and
a recent ﬁve—year period. Comment on similarities and differencrs in the
performance of U113 funds. Speculate on die census of any performance
discrepancies. {2.2} Equations of value for intrestments involving a single deposit made
under compound interest [1] Mr. Lopes. deposits SE in an accent paying 4% annual effective discount
The balance at the end of three years is $002. Find It". (2} Marianne deposiH $1,000 in a live—year certificate ofdeposit. At maturity
the balance is $2,530.64. Find the annual effective rate of interest
governing the account. {3} Susanne remembers that her only deposit into her savings account was a
$1,000 deposit She knows that the account has had a constant nominal
interest rate of 3.2 0's convertible monthly and that the balance is now
51,965.35 How long ago did Suzanne matte her deposit? [4) Use the rule of seventytwo to approximate the length of time it tallies
money to double at an annual effective interest rate of 5% and then at
an annual effective rate of 10%. Then find d'te exact time it takes for
ntoney to double at each of these interest rates. [5] Derive a "rule of rt" to approximate the length of tinte it takes for money
to triple. As in the derivation of the “rule of seventy—two," your rule
should be derived to give an especially good estimate wlten the annual
effective interest rate is 0%. After you have stated your role, compare
the approximations it gives for annual effective interest rates of 40.“: and
iii—ta with the true values at these rates. {2.3} Equations ofvalue for investments with multiple contributions fl} Sidney borrows $ 12000. The loan is governed by compound interest and
the annual effective rate of discount is 6%. Sidney repays $4,000 at thﬁ
end of one year, It at the end of two years, and $3,000 at the end of three
years in order to exactly pay oil' the loan. Find Jt'. Section 2.0 Problems, Cl'tapterl 105 _ (2} Rafael opens a savings account with a deposit of $1,501 He deposits
. 'r_ . . $500 one year later and $1,000 a year after that. Just after Rafael's
._ deposit of $1,000, the balance in his account is $3,0i‘0. Find the annual
effective interest rate governing the account " {a} Esteban borrows $10,000, and the loan is governed by compound interest
at an annual effective interest rate of 0%. Esteban agrees to repay d'tt:
loan by making a payment of $10,000 at the end of 1" years and a payment
of$12,000 at the end of 2'." years. Find '1". (4} Shaltm'i opens a savings account with a deposit of $3,500. She deposits
$500 six months later and $311} nine months after opening the aceounL
The balance in Shakari’s account one year after she opened it. is $5,012.
Assuming that the account grows by compound ittterest at a constant
annual effective interesr rate it, ﬁnd i. {5] A loan is negotiated with the lender agreeing to accept $0,000 after
__ one yEar, $0,0tlﬂ‘aftet two years. and $20,000 after four years lt1 full
repayment of the loan. The loan is renegotiated so that the borrower
'maltes a single payment of $37,111) at time T and this results ill the
same total present value of payments when calculated using an annual
effective rate of 5%. Estimate '1" using the method of equated time. Also
ﬁnd i" exactly. {t5} Anne and Frank Smith each borrow $12,000 from their father. Anne
and Mr. Smith have agreed that she will repay her loan in full by paying
$60111 in two years and $0000 in four years. Franlt prefers to make one
lump payment of $15000 to fully repay his loan. 1When should he make
that payment so that he and his sister will each have the same effective
interest rate?I {1"} Let 111,51. ...,lr,, be positive real numbers. Set a = (Eiﬂarlfﬂ. 013
l 1.. arithmetic mean of the numbers, and G = (Hg; 1 ed}, the geometric
mean of the numbers. The objective of this problem is to establish that
d .3 G and that A 3: 6 whenever i). , fag, , is, are not all equal. [a] Write the poiat—slope equation for the tangent line to y = tax at
(A, In A ]. (in) Use {a}. and concavity to show that. 1111' =5 A—1{.r — A] + Inst
for all positive 1. Moreover, shoalr that equality holds if and only. if
x = A. in] Use (to to provetltatlnG 2: igy=1rrsﬂ — a} + tan and that
this is a strict inequality unless all the his are equal. (d) straw that 23:, alra — A} + In n =1“. {e} Concludc that G c A and G if. A unless all the his are equal. 106 Chapter 2 Equations of value and yield rates [0) Let Cg denote the contribution in cents made at distinct times rt,
k = 1.2,... ,n. Suppose that these are all positive so that we have
deposits, but no withdrawals. Then the (3,, ’s are positive integers. As in {2.3.9},1et r = lntjiﬂf—‘lﬂlﬂn u. As in {2.3.10}, lat? = Eggtgtja. The objective of dais problem is to use the result of Problem (2.3.?) to
salablish tltat T 3 T and that this is a strict inequality unless a = . (a) Consider Cg quantities each equal to a“, = 1, 2. ...n. Then in all
we are considering C = C}, + Cu + "If," quantities. Use the
result of Problem [2.3.'F(e]] to show that Cris” + can“! + + Cynu‘l" 31;].—
C‘ and that this is a strict inequality unless I: = l. [in] Use {a} to show that the present value of the deposits is at least as
large as the present value given by the method of equated time and
that it is strictly larger Unless n = l . {1:} Show thatT .3: T with strict inequality unless u = 1. [9} Suppose that you pay $1,000 at time 0, get $4,000 at time t, and pay
$2.000 at time 1. Let Co = $l.000, C1 = —$4,000, and C1 = $2,000.
Set C = C3 + Cl +171 = $1,001 — $4,030 + $2,000 = —$l,000.
Find T such that getting an inﬂow of —C at time T has the same present
value as the above sequence of ﬁnancial transactions, assuming that the growth of money is governed by compound interest ati = 1%. Show that T is greater than the weighted average T = L—g'ﬂ + Eel1 + 5% ['Ihis shows that Inequality {2.3.1 1} need not hold if you have a negative
contn' bntiou.] {10) [recommended For those with a BA Il Plus calculator] Dax borrows $300,030 and the loan is governed by contpound interest at
an annual effective interest rate of 4.15%. Dax agrees to repay the loan
by ten equally spaced paymenLa, the ﬁrst 1' our of which are for $23,000
and the next six of which are for $40,000. When should he mate the ﬁrst
payment"? {11] Find the amount to be paid at the end of eight years that is equivalent to
a paytnent of $400 now and a payment of $300 at the end of [our years {a} if 6% simple interest is earned from the date each payment is made
and use a comparison date of right now. (b) ifﬁ‘l’u simple interest. is earned from the date each payment is made
and use :1 comparison date of eigltt years from now. Section 2.0 Problems, Chapter 2 10? ' [a] EIpiﬂlt’l why the fact you get different answers in parts (a) and
[b] does not contradict the fact that equations of value at dit‘l'ernnt
times are equivalent equations. {:1} Repeat parts {a} and (b) except replace “simple interest” with
f‘cornpound interest." '[calculus needed] Lise Newto n‘s method to solve the problem of Example (233}, More speciﬁcally, set ﬂ!) = 525(1.1}—2' + 525(1.1}“’ —
_:::'_":1,000, matte an initial guess T1 for a root T, and ﬁnd a sequence of
_1'_' __ approximations {T1} to T that allow you to obtain T to the nearest ' hundredth of apercent. Why might T. = .3 has reasonable initial guess":I {ﬁgs} investment return '(1}'=Payrnents of $3,000 now and $3,000 eight years from now are equivalent
' 's to a payment of $10,000 four years from now at either rate r' or rate j.
silt:1 Find .i — jl. Explain why the yield rate is not unique in this case. [2] Sumess, Inc enters into a. ﬁnancial arrangement in wltich it agrees to  pay $100,000 today and $101,000 two years from new in exchange for
' ' $200,000 one year from now. Show that there is on yield rate that can be assigned to this two—year transaction. {3} Sigmund, Inc. agrees to pay $150,000 today and $40,000 [our years from
today in remrn for $110,000 two years from today. What is the yield rate
for this fonriyear ﬁnancial arrangement"? [4] Ftrrns A, B, C, and D enter into a ﬁnancial arrangement. Money ﬂush
ﬁrm A will pay expanding ﬁrms B and C each $1,000,030 today. B will
pay D $2,200,000 three years from today. C will pay 13 SSW,“ two years
from today and D $350,000 two years from today. Finally, D will pay A
$3,200,001 six years from today. Calculate the yield rate or interest rate,
to ﬂ'te nearest hundredth of a percenL. that each ﬁrm experiences over
llle period of their involvement {6 years [or A, 3 years for B, 2 years for
C. and 4 years [or D]. [5) 3015a instcsts $0,572.00 at r = n and season at r = 1. in return, she
receives $21,004 at r = 1 and $10,000 at r = 3. Write down a time 0
equation of value and verify that it is satisﬁed for u = .94, n = .95, and . " U = .05. Find the corresponding threc yield rates. {5] Pedro inves1s $100,000 at t' = 0 and $60,000 at t = 2. In return he gets
$60,000att = 1 and $116,500 at! = 3. Write down a time 3 equation of
value describing Pedro‘s investment. Explain why there is a unique yield
rate and find it. 1uﬂ Chapter: Equations ot value and yield rates in Show thatil' —1 a y a egg: Cult + rors = Egg Catt + pie
and Bull]. Bali} , .,.. B,n_][r'] are all positive [with Huit'} as given in
[24.8)], then j = r'. (0] [remmmended For those with a BA ll Plus calculator]
On January I, Ezequiel opens an account at Friendly Bank. His opening
deposit is for $50 and he makes deposits at the end of each quarter for
[our years, then maltes no tnore deposits. He closes the account exactly
seven years after he opens it and receives $3423.23. Find his annual
yield rate for this seven—year period if his quarterly deposits were $00
in the ﬁrst year. $25 in the second year. $50 in the third year, and were
successively $30!], $450. $000. and $240 in the fourth year. {‘3} [recommended for those with a BA ll Plus calculator]
A loan of 520.0001: to be repaid by thirtythree ehdof—rnonth payntents.
The ﬁrst payment is $400 and then eaclt payment is $25 more than the
previous paymenL Find the annual yield rate correct to the nearest
hundredth of a percent. (HINT: The Cash Flow worksheet only accepts
twentyfour payments. or thirty—two if you have a BA l Plus Professional
calculator. If you are nothing with the EA ll Plus calculator. suppose that
the payments beyond the twenty—fourth+ which you do not have registers to
accommodate, are made along with the twenty—fourth. Now use the "gums and check" method, obtaining your first estimate by using  This is a challenging problem. especially for those of you with only 24
registers. Hoerevermhen performing the Euccive calculations required by
the "guess and check" method. you may make judicious u5e of the “PU
subworItsheet to decrease your warlL] [2.5) Reinvestment considerations {1) Angela loans Kathy $0.000. Kathy repays the loan by paying $0,000 at
the end of one and a half years and $4,000 at the end of three years. The money received at r = 1% is immediately reinvested at an annual eﬂective interest rate of “in. Find Kathyis annual clieclive rate of
interest and Angela’s annual yield. [2] Kurt loans Randy 514.000.]1andy repays the loan by paying Kurt $4.005
at the end of one year and 56.000 at the end of two years and as well
as at the end of three years. The money received at r = 1 and at t = 1
is immediately reinvested at an annual effective interest rate of 3%.
Correct to the nearest tenth of a percent. tind Randy’s annual reflective
inlerest rate and Kurt‘s annual yield (3} On January IS, 2000. Enterprise A loans $15,000 to Enterprise l3 and
511000 to Enterprise (.7. Enterprise 13 repays Enterprise A $2,000 on
January 15. 2002 trnd this money is reinvested at a 5‘55. annual et'feclllt‘i Section 2.0 Problemsr Chapter 2 109 rate. Enterprise [3 repays Enterprise A $22,500 on January 15, 2W4.
What is the annual yield received by Enterprise A over the fouryear
interval. Compare it to the annual eﬂective interest rates paid by
Enterprises H and C. {2,6} Approximate dollarweighted yield rates {1} Sandra invests 310.332 in the Wise Investment Fund. Three moutlts
later her balance has green to $11,902 and she deposits $2,000. Two
months later her fund holdings are 514.300 and she wideraws $2,000.  Two years after her initial investment she learns that her holdings are
worth $12,500. {a} Write an equation of value involving die exact dollar—weighted
annual yield r' over the twosyeat period. [b] Compute the.appro:ltimate dollar—weighted annual yield over the
investment period using {2.6.5} and again using {2.6.3}. {2) On February 1, Arsltalc’s investment account has a balance of $19,300.
He deposited $1,200 on April 1 and $2,000 on May 1. He withdrew
$0,400 on J uly I. On November 1, Arshalt's balance was $14,320. Find
Anhalt’s approximate dollar—Weighted annual yield for this nine—month
period using [2.0.5]. ' [3) Franklin's investment fund had a balance of $290000 on January 1. 1995
and a balance of $440,000 two years later. The amount of interest earned
during the two years was $34.00}. and the annual yield rate on the fund
was 5.4%. Estimate the [dollarweighted} average date of contributions
to die account. [4] The investment balance of a litnt is 3155,00101100 at the beginning of a
two—year period and $2,000,000 at the end. The ﬁrm makes a. single
nontribution during the twoyear interval of $200,000. What is the
difference between the approximate annual dollar—weighted yield earned
by the ﬁrm if the contribution is made after t‘i months as opposed to it
being made after one year? {2.2} Food perfonnante ill CHI January ' :1930, Antonio invests 59.400 in an investment land. [in
January 1, ltlﬂEl his balance is $0,600 and he deposit: $2,400. On July 1.
1930 his balance is M49100 and he withdraws $1,000. [in January I, 1092 llis balance is 5P. Express: his annual timeweighted yield as a function
Of P. {2} Arthur buys Slllllllwortlr ol' stoolt. Six months later, the value of the
stock has risen to $2,2tltl and Arthur buys another 3  .000 worth of stock. 11u ChaptErZ Equations civalue and yield rates After another eight months, Arthur’s holdings are worth $2.900 and lie _
sells off $000 of them. Ten months later, Arthur ﬁnds that his stock has ' a value of $2,100.  [a] Colupule the annual timeweighted yield rate of the stock over the twoyear period. [it] Compute the annual dollarweighted yield for Arthur over the _' 2 twoyear period.
{c} Should the answer in part {a} or part {b} be larger? IWhy? {3} Bright Future [nvesnnent Fund has a balance of “305,000 on January I l. {in May 1, the balance is $1,230,001 Immediately alter this balance is
noted. $300,000 is added to the fund. If there are no further contribuLlons to the fund for the year and the limeweighted annual yield for the fund
is 16%, what is the fund balance at the end of die year? Chapter 2 review problems [I] Sohail makes an initial investment of $20,013. In return, he receives
$4,003 at the end of one year and another $10,003 at the end of three
years. [a] ﬁssilming that the investment is made at simple inlerest at rate 1",
write down an equation of value for the investment and ﬁnd r. {b} Assuming that the investment is made at compound interest at
effective interest rate t', Inilrile down an equation of value for the
invesnnent and justify the smlernent that there is a unique yield rate. Use the “guess and checlt method" to estimate f to the nearest
hundredth of a percent. [c] Starting with Ilie same initial guess I'ori that you Md in {b}, check
you answer using Newlon‘s method. {d} [recommended for those with a 0A ll Plus caICulator]
Use the Cash Flow mrltsheet In ﬁnd i to the nearest millionth. [1) {:1} Suppose now that invesunents are governed by compound interest
at an effective inlerest rate i 3 0. 0y how much does Lhe sum of the
time a value oil 55K paid at Lime 0 and the time a value of“: paid all
tlute 2n exceed EEK? Express your answer as a function...
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