CFA Level 1 (2009) - 1

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

PV3 becomestheFV that you need three years from today from your three equal end-of-year deposits. To determine the amount of the three payments necessary to meet this funding requirement, be sure that your calculator is in the END mode,

· input the relevant data, and compute PMT..

N =3; I1Y =10; FV =-4,169.87; CPT -+ PMT =$1,259.78

· The second part of this problem is an ordinary annuity. If you changed your calculator to BGN mode and failed to put it back in the END mode, you will get a PMT of

· $1,145, which is incorrect.

Example: Funding a retirement plan

Assume a 35-year-old investor wants to retire in 25 years at the age of 60. She expects (0 earn 12.5°!tl on her investments prior to her retirement and 10% thereafter. How much must she deposit ar the end of each year for the next 25 years in order to be able to withdraw $25,000 per year at the beginning of each year for the 30 years from age 60 to 90?

Answer:

This is a two-step problem. First determine the amount that must be on deposit in the retirement account at the end of year 25 in order to fund the 30-year, $25,000 annuity due. Second, compute the annuity payments that must be made to achieve the required amount.

Step 1: Compute the amount required to meet the desired withdrawaLs.

The required amount is the present value of the $25,000, 3D-year annuity due at the beginning of year 26 (end of year 25). This can be determined by entering the relevant data, with the calculator in the END mode, and computing PY.

N = 29; I1Y = 10; PMT =-$25,000: CPT --4 PV = $234,240 (for 29 years)

Now add the first annuity payment to get $234,240 + $25,000 = $259,240. The investor will need $259,240 at the end of year 25.

Please note that we could have also performed this computation with our calculator in BGN mode as an annuity due. To do this, put your calculator in BGN mode [2 nd] [BGN] [2nd] [SET] [2nd] [QUIT] on the TI or [g] [BEG] on the HP. Then enter:

N = 30; PMT = -25,000; I/Y = 10; CPT --4 PV = 259,240.14

If you do it this way, make certain you reset your calculator to the END mode.

Step 2: The annuity payment that must be made to accumuLate the required amount over 25 years can be determined by entering the reLevant data and computing

PMT.

N = 25; I1Y = 12.5; FV =-259,240; CPT --4 PMT = $1,800.02

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

Thus, the in~estormust deposit $1,800 per year at the end of each of the next 25 years in order to accumulate $259,240. With this amount she will be able to withdraw $25,000 per year for the following 30 years.

Note that all thesecalculations assume that the investor will earn 12.5% on the payments prior to retirement and 10% on the funds held in the retirement account thereafter.

The Connection Between Present Values, Future Values, and Series of Cash

Flows

As we have explained in the discussion of annuities and series of uneven cash flows, the sum of the present values of the cash flows is the present value of the series. The sum of the future values (at some furure time = Il) of a series of cash flows is the furure value of rhar series of cash flows.

One inrerpretation of the presenr value of a series of cash flows is how much would have (Q be pur in the bank raday in order (Q make these furure withdrawals and exhaust the accounr with the final withdrawal? Let's illusrrate this with cash flows of $1 00 in year 1. $200 in year 2, $300 in year 3. and an assumed interest rate of 10%.

Calculate the present value of these three cash flows as:

If we pur $481.59 in an account yielding 10%, at the end of the year we would have 481.59 x 1.1 = $529.75. Withdrawing $100 would leave $429.75.

Over the second year, the $429.75 would grow (Q 429.75 x 1.1 = $472.73. Withdrawing $200 would leave $272.73.

Over the third year, $272.73 would grow (Q 272.73 x 1.1 = $300, so that the last withdrawal of $300 would empty the account.

The interpretation of the fumre value of a series of cash flows is srraightforward as well. The FY answers the question, how much would be in an account when the last of a series of deposits is made? Using the same three cash flows-$l 00, $200, and

$300-and the same interest rate of 10%, we can calculate the furure value of the series as:

100 0.1)2 + 200 (1.1) + 300 = $641

This is simply the sum of the t = 3 value of each of the cash flows. Note that the t = 3 value and the t '" 0 (present) value of the series are related by the interest rate, 481.59

0.1)3 = 641.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

The $100 cash flow (deposit) comes at t = 1, so it will earn interest of] 0% compounded for two periods (until t = 3). The $200 cash flow (deposit) will earn 10% between t = 2 and t = 3, and the final cash flow (deposit) of $300 is made at t = 3, so $300 is the future (t = 3) value of that cash flow.

We can also look at the future value in terms of how the account grows over time. At t = 1 we deposit $100, so at t = 2 it has grown to $110 and the $200 deposit at t = 2 makes the account balance $310. Over the next period the $310 grows to 310 x 1.1

=$341 at t = 3, and the addition of the final $300 deposit puts the account balance at $641. This is, of course, the future value we calculated initially.

G Professor's Note: This last view ofthe future value ofa series ofcash flows suggests a quick way to calwlate the future value ofan uneven cash flow series. The process described previous0' for the future value ofa series ofend-ofperiod payments can be writtm mathematicall)' as [( 100 x 1.]) + 200J x ].1 + 300 = 641. aile! tI,i.' migh{ be ({ qltick way to do some fitfure pl'o!l/CIII.i.

NOLl' that questions 0\\ the future \'Jlue of an flJll/lli~l' due refer [() the amOUlH in the accounr one period :lfrcr the last deposit is made. If the three deposits considered here were made at the beginning of each period (at t = 0, 1, 2) the amount in the ;lCCOUI1t :It

of cash flows equals the sum of the present values of the cash flows. There are different applications of this principle in time value of money problems. If we have two series of cash flows, the sum of the present values of the rwo series is the same as the present values of the rwo series taken together, adding cash flows that will be paid at the same point in time. We can also divide up a series of cash flows any way we like, and the present value of the "pieces" ,,,,,ill equal the present value of the original series.

ExampJe: Additivity principle

A security will make the following paymen tS at (he end of the next four years: $100, $100, $400, and $100. Calculate the present value of these cash flows using the concept of the present value of an annuity when the appropriate discount rate is 10%.

Answer:

We can divide the cash flows so that we have:

Srudy Session 2

Cross-Reference to CPA Institute Assigned Reading #5 - The Time Value of Money

The additivity principle tells us that to get the present value of the original series, We

For the annuity, N =4, PMT = 100,.FV = 0, I/Y = 10, CPT ---+ PV = -$316.99 .

For the single payment: N = 3, PMT = 0, FV = 300, I/Y = 10,

CPT ---+PV =-$225.39

The sum of these two values is 316.99 + 225.39 = $542.38.

The sum of these two (present) values is identical (except for rounding) to the sum of the present values of the payments of the original series: .

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

KEy CONCEPTS

LOS 5.a

An interesr rare can be interprered as rhe rerum required in equilibrium, rhe discount rare for calcularing rhe presenr value of future cash flows, or as rhe opportunity cosr of consuming now, rarher rhan saving and invesring.

LOS 5.b

The required rare of return on a securiry = real risk-free rare + expecred inflarion + default risk premium + liquidity premium + maturity risk premium.

LOS 5.c

Furure value: FY = PY(l + I/Y)]\;; presenr value: PY = FY I (l + I/y)N.

The effecrivc annual rare when rhere are 11/ compounding periods =

For non-annual rime value of money problems, divide rhe srared annual interesr rarc by rhe numbcr of compounding periods per year, 111, and mulriply rhe number of ycars by rhe number of compounding periods per year.

LOS 5.d

An annuiry is a series of equal cash flows rhar occurs ar evenly spaced inrervals over rime. Ordinary annuity cash flows occur ar rhe end of each rime period. Annuity due cash flows occur ar rhe beginning of each rime period.

Perperuiries are annuiries wirh infinire lives (perpetual annuiries);

PMT

PYpcrpetuity = IN .

LOS '5.c

The present (future) valuc of any series of cash flows is equal to rhe sum of rhe present (future) values of rhe individual cash flows.

Construc'ring a rime line showing furure cash flows will help in solving many rypes of

TYM problems.

A mortgage is an amortizing loan, repaid in a series of equal payments (an annuiry), where each payment consisrs of rhe interesr for rhe period (which decreases wirh each payment) and rhe amount of principal repaid (which increases wirh each payment).

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

CONCEPT CHECKERS

1.The amount an investor will have in 15 years if $1 ,000 is invesred wday ar an annual interesr rare of 9% will be closest w:

A.$1,350.

B.$3,518.

C.$3,642.

A.$4,550. B. $45,502.

C.$91,004.

3.How much musr bt" invesred toda~' at ()l}() to have $100 in thrcl' ,'cars?

A. S""7.7').

B. $100.00.

C. $126.30.

4.How much must be invested roday, at 8~'o interesr, to accumulate enough ro retire a $] 0,000 debt duc seVl:n years from raday? The amount dlat must be invested today is closest [0:

A. $5,835. B. $6,123. C. $8.794.

5.An analyst esrimates that XYZ's earnings will grow from $3.00 a share to $4.50

per share over the nexr eight years. The rate of growth in >..ryZ's earnings is closest to:

A.4.9%.

B.5.2%.

C.6.7%.

6.If $5,000 is invested in a fund offering a rare of return of 12% per year, approximately how many years will it take for rhe investment [Q reach $1 O,OOO?

A.4 years.

B.5 years.

C.6 years.

7.An investment is expected to produce the cash flows of $500, $200, and $800 at the end of the next three years. If the required rate of return is 12%, rhe present value of this investment is closest to:

A.$835.

B.$1,175.

C.$1,235.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

8.Given an 8.5% discounr rare, an asser rhar generares cash flows of$10 in year 1,

-$20 in year 2, $10 in year 3, and is rhen sold for $150 ar rhe end of year 4, has

a presenr value of:

A.$108.29.

B.$135.58.

C.$163.42.

9.An investor has juSt won the lottery and will receive $50,000 per year at the end of each of the next 20 years. At a 10% interest rate, the present value of the winnings is closest to:

A.$425,678.

B.$637,241.

C.$2,863.750.

10.If $1 0,000 is invesred today in an account rhar earns interest ar a rate of 9.5%,

whar is the value of rhe equal withdrawals rhar can be raken our of rhe account ar rhe: end of each of the nexr five years if rhe invesror plans to dcplerc rhe account :ll the end of the rime period?

A.52.'!').)

B.S2,604.

C.$2,750.

11.An investor is ro receive a IS-year $8,000 annuiry_ rhe firsr pa)'l1lenr to be received roday. Ar an 11 % discounr rare, rhis anlluin'"s worth roday is closest to:

A.555,855.

B.$57,527.

C.$63.855.

12.Given an 11 % rate of return, the amount that musr be put into an invesrment

account ar the end of each of the next ten years in order to accumulate $60,000 to pay for a child's educarion is closest to:

A.$2,500.

B.$3,588.

C.$4,432.

13.An investor will receive an annuity of $4,000 a year for ren years. The first payment is to be received five years from roday. At a 9% discounr rate, this annuity's worth today is closest to:

A.$16,684.

B.$18,186.

C.$25,671.

J4. If $1,000 is invested today and $1,000 is invesred ar rhe beginning of each of the next three years ar 12% inrerest (compounded annually), rhe amount an investor will have at the end of the fourth year will be closest to:

A.$4,779.

B.$5.353.

C.$6,792.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time VaJue of Money

15.An investor is looking ar a $150,000 home. If 20% musr be pur down and rhe balance is financed ar 9% over rhe nexr 30 years, whar is the monthly mortgage paymenr?

A.$799.33.

B.$895.21.

C.$965.55.

16.Given daily compounding, rhe growrh of $5,000 invesred for one year ar 12% interesr will be closest ro:

A.$5,600.

B.$5,628.

C.$5,637.

17.Terry Corporarion preferred stocks are expecred to pay a $9 annual dividend forever. If rhe required rare of rerurn on equivalent invesrments is II %, a share of Terry preferred should be worth:

A.$81.82.

B.$99.00.

C.$122.22.

18.A share of George Co. preferred srock is selling for $65. Ir pays a dividend of $4.50 per year and has a perperual life. The rare of rerum it is offering irs invesrors is closest to:

A.4.5UAI.

B.6.9%.

C.14.4()0.

19.If $1 0,00') is borrowed at 10% interesr ro be paid back over ren years, how much of rhe second year's payment is interesr (assume annual loan payments)?

A.$937.26.

B.$954.25.

C.$1,037.26.

20.Whar is the effecrive annual rare for a credir card rhar charges 18% compounded monthly?

A.] 5.38%.

B.18.81%.

C.19.56%.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

COMPREHENSIVE PROBLEMS

1.The Parks plan to take three cruises, one each year. They will take their first cruise 9 years from today, the second cruise one year after that, and the third

cruise 11 years from today. The type of cruise they will take currently costs $5,000, but they expect inflation will increase this cost by 3.5% per year on average. They will contribute to an account ro save for these cruises that will

earn 8% per year. What equal contributions must they make roday and every year until their first cruise (ten contributions) in order to have saved enough for all three cruises at thar rime? They pay for cruises when taken.

2.A company's dividend in 1995 was $0.88. Over the next eight years, the dividends were $0.91, $0.99, $1.12, $0.95, $1.09, $1.25, $1.42, $1.26. Calculate the annually compounded growth rate of the dividend over the whole period .

your required ratc of rerum on this investment is 4%, how much would you be willing co pay for the bond coday?

4.A bank quotes certificate of deposit (CD) yields both as annual percentage rares (APR) without compounding and as annual percentage yields (APY) that include the effects of monthl)' compounding. A $100,000 CD will pay $110.471.31 at the end of the year. Calculatc the APR and APY the bank is quoting.

5.A client has $202,971.39 in an account that earns 8% per year. compounded monthly. The client's 35th birthday was yesterday and she will retire when the account value is $1 million.

payment made at retirement. What is rhe annual rare the client would need to earn on a retirement investment ro be indifferent between the two choices?

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

ANSWERS - CONCEPT CHECKERS

3.B Since no interest is earned, $100 is needed today to have $100 in three years.

14.B The key to this problem is to recognize that it is a 4-year annuity due, so switch to BGN mode: N = 4; PMT = -1,000; PV = 0; I1Y = 12; CPT -> FV = 5,352.84. Switch back to END mode.