CFA Level 1 (2009) - 1

Scudy Session 2

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

Present Value of a Single Sum

The PV of a single sum is today's value of a cash flow that is to be received at some point in the future. In other words, it is the amount of money that must be invested today,

at a given rate of return over a given period of time, in order to end up with a specified ry. As previously mentioned, the process for finding the PV of a cash flow is known as discoumillg (i.e .. future cash flows are "discounted" back to the present). The interest r~\ll' used in the d.iscounting process is common]:' referred ro as the discount rate bur m3\' also be referred to as the opportunit~·cost, required rate of return. and the cost of capitaL \VhatcvtT you want to call it, it represents the annual compound rate of retlIrn rhar can be earned on an investment.

The relationship berween PV and FV can be secll b:· examining the FV expression stared earlier. Rewriting the FV equation in terms of PV. we get:

Note that for a single future cash flow, PV is always less than the FV whenever the discount rate is positive.

The quantity 1/(1 + I1Y)N in the PV equation is frequently referred to as the present value factor, present value interest factor, or discount factor for a single cash flow at I1Y over N compounding periods.

Example: PV of a single sum

Given a discount rate of 9%, calculate the PV of a $1,000 cash flow that will be received in five years.

Answer:

To solve this problem, input the relevant data and compute pv.

N = 5; I1Y = 9; FV =1,000; CPT - t PV =-$649.93 (ignore the sign)

Seudy Session 2

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

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'.--':.-.'5; ':~~

,This'~~I~ti~1IysimpIeproblem couIdalso?e,s~lved usingthe following PV equation.

;PV~ ,1,000 '=$649.93 '(1+0.09)5

On.theTI,enter 1.09[yX] 5 [=] [l/x] [x] 1,000[=],.

Annuities

An annuity is a ,~tream of equal cashfloU's that occurs at equal ;nten1fll.. over a Si"c:tl period. Receiving S 1,000 per year at the end of each of the next eight years is an example of an annuiey. There are two types of annuities: ordinary annuities and annuities due. The ordinarJI mmui~)1 is the most common type of annuity. It is characterized by cash flows that occur at the end of each compounding period. This is a typical cash flow pattern for many investment and business finance applications. The other type of annuity is called an annuity due, where payments or receipts occur at the beginning of each period (i.e., the first payment is raday at t = 0).

Computing the FV or PV of an annuity with your calculator is no more difficult than it is for a single cash flow, You will know four of the five relevant variables and solve for the fifth (either PV or FV). The difference between single sum and annuity TVM problems is that instead of solving for the PV or FV of a single cash flow, we solve for the PV or FV of a stream of equal periodic cash flows, where the size of the periodic cash flow is defined by the payment (PMT) variable on your calculator.

Example:FV of an ordinary annuity

What is the future value of an ordinary annuity that pays $150 per year at the end of each ofthe next 15 years, given the investment is expected to earn a 7% rate of return?

Srudy Session 2

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

Answer:

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.~h'tim, lin, furth, ~shaoWSi~~ii~p¥~W~~.i"d,pict'djnth,figni,~(~~~~Yj~;'

'pvofan Ordinary Annuity

0 2 3 15

As indicated here, the sum of the compounded values of the individual cash flows in this IS-year ordinary annuity is $3,769.35. Note that the annuity payments

themselves amounted to $2,250 = 15 x $150, and the balance is the interest earned at the rate of 7% per year.

To find the PV of an ordinary annuity, we use the future cash flow stream, PMT, that we used with FY annuity problems, but we discount the cash flows back to the present (time = 0) rather than compounding them forward to the terminal date of the annuity.

Here again, the PMT variable is a single periodic payment, not the total of all the payments (or deposits) in the annuity. The PYAo measures the collective PY of a stream of equal cash flows received at the end of each compounding period over a stated number of periods, N, given a specified rate of return, I/Y. The following examples illustrate how to determine the PV of an ordinary annuity using a financial calculator.

Example: PV of an ordinary annuity

What is the PV of an annuity that pays $200 per year at the end of each of the next ' 13 years given a 6% discount rate?

Study Session 2

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

Answer: . -F -

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. s(jJve-tJ;i:~~~pro1:*ril~;:~~#;the relevantjn:formation.art(p~onipt1~~·pv:·;~~

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';;;:N;;13;I1:y,~';;;;6;:-pMT=-200; FV =-0; CPT ~PV;"SniZ054 -

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Example:_ PV:C?f' ~.ordiOary annuity beginning later than t = 1

Whatis the present-value offour $100 end-of-year payments if the first payment is to be received three years from today and the appropriate rate of return is 9%?

Answer:

The time line for this cash flow stream is shown in the following figure.

PV of an Annuity Begiruiing at t = 3

0 2 3 4 5 6

PYu =$272.68 ~ PY2 =$323,97

Step 1: Find the present value ofthe annuity as ofthe end ofyear 2 (PV2 ).

Input the relevant data and solve for PV2.

N = 4; I/Y = 9; PMT =-100; FV = 0; CPT ---t PV = PV2 = $323.97

Step 2: Find the present valueofPV2 .

Input the relevant data and solve for PVo'

N = 2; I/Y = 9; PMT = O;FV =-323.97; CPT ~ PV =PVo = $272.68

In this solution, the annuity was treated as an ordinary annuity. ThePVwas computed one period before the first payment, and we discountedPV2 = $323.97 over two years. We need to stress this important point.ThePV annuity function on your ca1culatorset ill "END" mode gives you the value one period 'before the annuity begins. Although the annuity begins at t = 3, we discounted the result for only tWo periods to get thepresent (t = 0) value.

Study Session 2

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

Example: PV of a bond's cash flows

A bond will make coupon interest payments of70 euros (7% of its face value) at the end of eachyear and will also pay its face value of l;OOOeuros at maturity in five years. Ifthe appropriate discount rate is 8%, whads the present value of the bond's promised cash flows? '

Answer:

The five annual.coupon payments of 70 euros each can be viewed as an ordinary

,annuity. The maturity value of 1,000euros is the future va.hie ofthe bond at the time the last coupon payment is made.' On a rimeline, the promised payment stream is as shown below. "

Cash flows for a 5-year, 7%, 1,000 euro bond

70 70 70 70 +-- payments an, all ordinan' annUil\'

+ 1.000 +-- the maturiry pa~'me\1( is a single future value

The PV of the bond's cash flows can be broken down into the PV of a 5-payment ordinary annuity, plus the PV of a 1,000 euro lump sum to be received five years from now.

The calculator solution is:

N", 5; PMT = 70; I/Y = 8; FV = 1,000; CPT PV = -960

With a yield to maturity of 8%, the value of the bond is 960euros.

Note that the PMT and FV must have the same sign, since both are cash flows paid to the investor (paid by the bond issuer). The calculated PV will have the opposite sign from PMT and FY.

Future Value of an Annuity Due

Sometimes it is necessary to find the FV ~fan annuity due (FVAo), an annuity where the annuity payments (or deposits) occur at the beginning of each compounding period. Fortunately, our financial calculators can be used to do this, but with one slight modification-the calculator must be set to the beginning-of-period (BGN) mode. To switch berween the BGN and END modes on rhe TI, press [2nd] [BGN] [2nd] [SET]. When this is done, "BGN" will appear ill the upper right corner of the display window. If the display indicates the desired mode, press [2nd] [QUIT]. You will normally want your calculator to be in the ordinary annuity (END) mode, so remember to switch out of BGN mode after working annuity due problems. Note that nothing appears in the

upper right corner of the display window when the TI is set to the END mode. It should

Study Session 2

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

be mentioned that while annuity due payments are made or received at the beginning of each period, the FV of an annuity due is calculated as of the end of the last period.

Another way to compute the FV of an annuity due is to calculate the FV of an ordinary annuity, and simply multiply the resulting FV by [1 + periodic compounding rate (I1Y)]. Symbolically, this can be expressed as:

FVAo = FVAo x (l + I/Y)

The following examples illustrate how to compute the FV of an annuity due .

. Example: FV of an annuity due

•What is the future value of an annuity that pays $100 per year at the beginning of each of the next three years, commencing today, if the cash Rows can be invested at an annual rate of 10%? Note in the time line in the figure below that the FV is computed as of the end of the last year in the life of the annuity, year 3, even though the final payment occurs at the beginning of year 3 (end of year 2).

Answer:

To solve this problem, put your calculator in the BGN mode ([2nd] [BGN] [2nd]

Alternatively, we could calculate the FV for an ordinary annuity and multiply it by (l + IIY). Leaving your calculator in the END mode, enter the following inputs:

N = 3; IIY = 10; PMT = -100; CPT ~ FVAo = $331.00

FVAD = FVAo x (l + IIY) = 331.00 x 1.10 = $364.10

Study Session 2

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

Example: FV of an annuity due

If you deposit $1,000 in the bank today and at the beginning of each of the next three years, how much will you have six years from today at 6% interest? Thetimdine for this problem is shown in the following figure.

Answer:

FV for an Annuity Due

Step 1: Compute the FV ofthe annuity due at the end ofyear 4 (FV4 ).

Set your calculator to the annuity due (BGN) mode, emer the relevant data, and compute FV4•

N = 4; I1Y =6; PMT =-1,000; CPT - 4 FV = $4,637.09

Step 2: Find the future value ofFV4 two years from year 4. Enter the relevant data and compute FV6.

N = 2; IN = 6; PV =-4,637.09; CPT - 4 FV = $5,210.23

or

4,637.09(1.06)2 = $5,210.23

Present Value of an Annuity Due

While less common than those for ordinary annuities, there may be problems on the exam where you have to find the PV ofan annui~J' due (PVA o )' Using a financial

calculator, this really shouldn't be much of a problem. With an annuity due, there is one less discounting period since the first cash flow occurs at t = 0 and thus is already its PY. This implies that, all else equal, the PV of an annuity due will be greater than the PV of an ordinary annuity.

As you will see in the next example, there are two ways to compute the PV of an annuity due. The first is to put the calculator in the BGN mode and then input all the relevant variables (PMT, IN, and N) as you normally would. The second, and far easier way, is

Study Session 2

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

ra treat the cash Row scream as an ordinary annuity over N compounding periods, and simply multiply the resulting PV by [1 + periodic compounding rate (1/Y)].

Symbolically, this can be stated as:

PVAo = PVAo x (1 + l/Y)

The advantage of this second method is that you leave your calcularar in the END mode and won't run the risk of forgening co reset it. Regardless of the procedure used, the computed PV is given as of the beginning of the first period, t = O.

Exainple: PV of an Annuity Due

Given a discount rate of 10%, what is the present value of a 3-year annuity thar makes a series of $100 paymenrs at the beginning of each of the nexr three years, starring raday? The time line for this problem is shown in the following figure.

Answer:

First, let's solve this problem using the calculator's BGN mode. Set your calcularar-ro the BGN mode ([2nd] [BGN] [2nd] [SET] [2nd] [QUIT] on the TI or [g] [BEG] on the HP), enter the relevant data, and compute PY.

PV = $273.55

Alternatively, this problem can be solved by leaving your calculacor in the END mode. First, compute the PV of an ordinary 3-year annuity. Then multiply this PV by

(l + I/Y). To use this approach, enter the relevanrinputs and compute PY.

N = 3; l/Y = 10; PMT = -100; CPT - 4 PVAO = $248.69

PVAo = PVAO x (1 + I/Y) = $248.69 x 1.10 = $273.55

Study Session 2

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

Present Value of a Perpetuity

A perpetuity is a financial instrument that pays a fixed amount of money at set intervals over an infinite period of rime. In essence, a perpetuity is a perpetual annuity. British consul bonds and most preferred srocks are examples of perpetuities since they promise fixed inrerest or dividend payments forever. Withour going into all the excruciating mathematical details, the discount facror for a perpetuity is just one divided by the appropriate rate of return (i.e., lIr). Given this, we can compute the PV of a perpetuity.

PMT

PVperpetuiry = W

The PV of a perpetuity is the fixed periodic cash flow divided by the appropriate periodic rate of return.

As with other TVM applications, it is possible to solve for unknown variabks in the

Example: PV of a perpetuity

Assume the preferred srock of Kodon Corporation pays $4.50 per year in annual dividends and plans to follow this dividend policy forever. Given an 8% rate of return, what is the value of Kodon's preferred stock?

Answer:

Given that the value of the srock is the PV of all future dividends, we have:

PVperpemity = 4.50 = $56.25

0.08

Thus, if an investor requires an 8% rate of return, the investor should be willing to pay for each share of Kodon's preferred Stock.

•Example: Rate of return for a perpetuity

Using the Kodon preferred stock described in the preceding example, determine the rate of return that an investor would realize if she paid $75.00 per share fat the stock.

Answer:

Rearranging the equation for PVperpetuity' we get:

This implies that the return (yield) on a $75 preferred stock that pays a $4.50 annual dividend is 6.0%.

Study Session 2

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

PV and FV of Uneven Cash Flow Series

It is not uncommon to have applications in investments and corporate finance where it is necessary to evaluate a cash flow stream that is not equal from period to period. The time line in Figure 2 depicts such a cash flow stream.

Thi;, 6-year cash Bow series is nOl an anl1uir~' sine<: rhe cash Bows are different every vear. In fad. rhere is one year with zero G\,h flow and rw() mhe:';, wirh negarive cash flow;,. II~ essence, this series of uneven cash flows is norhing more than a stream of annual singk sum cash A.ows. Thus, to find the PV or FV of this cash Bow stream, all we need to do is sum the PYs or FYs of rhe individual cash A.ows.

Example: Computing the FV of an uneven cash flow series

Using a rate of return of 10%, compute the future value ufthe 6-year uneven cash flow stream described above at the end of the sixth year.

Answer:

The FY for the cash flow stream is determined by first computing the FV of each individual cash flow, then summing the FYs of the individual cash Hows. Note that we need to preserve the signs of the cash flows.

FVI : PV = -1,000; I/Y = 10; N = 5; CPT ---* FV = FV I = -1,610.51

FV2: PV = -500; I/Y = 10; N = 4; CPT ---* FV = FV2 = -732.05

FV3: PV = 0; I/Y = 10; N = 3; CPT ---* FV = FV3 = 0.00

FV4: PV = 4,000; I/Y = 10; N = 2; CPT -. FV = FV4 = 4,840.00

FVs: PV = 3,500; I/Y = 10; N = 1; CPT -. FV = FVs = 3,850.00

FV6: PV = 2,000; I/Y = 10;N = 0; CPT ---* FV = FV6 = 2,000.00

FV of cash flow stream = EFVindividual = 8,347.44