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IPCOR421 Finance Alex Kane 1 CLASS NOTES WEEK III READING ASSIGNMENT BMA 3

Finance Week 3

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Page 1: Finance Week 3

IPCOR421 FinanceAlex Kane 1

CLASS NOTES

WEEK III

READING ASSIGNMENT

BMA 3

Page 2: Finance Week 3

IPCOR421 FinanceAlex Kane 2

PV revisited

• We know that the PV of a future CF in t=1 is:PV=C(1)/(1+r) = DF(1)*C(1)

• We will stick to the notation that C(t) is cash flow at time t and, for now, assume t measures years.

• Thus C(2) denotes CF in year 2 and so on

• What is the PV of C(2)?

• We follow the same procedure as before, and ask: What is the price of a 2-year T-bill?

Page 3: Finance Week 3

IPCOR421 FinanceAlex Kane 3

A 2-year Strip (of a T-bond)

• The U.S. Treasury doesn’t sell a two year claim to $10,000. Beyond one year maturity, T-bonds and notes bear semi-annual interest coupons

• However, financial institutions engage in “stripping.” This is the act of putting T-bonds in escrow and issuing separately claims to Zero Coupon Treasuries called Strips

• Thus, you can purchase the equivalent of a 2-year (and up) T-bill, or simply observe its price

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The 2-year DF and market interest rate

• Suppose we observe a 2-year stripC(2)=$10,000 Price = $9100

• We can compute:DF(2) = 9100/10000 = 0.91 = PV(2-year safe $)

• We can also compute the rater(2) = (10000–9100)/9100 = 0.0989 (9.89%)

• But, r(2) is the total rate for a two-year investment

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The annual rate embedded in r(2) and DF(2)

• Suppose we invest $1 in the 2-year strip. What annual rate will be earned? Denote it by r2

• Compounding r for two years, a $1 investment will yield 1 + r(2) = 1.0989 = (1+r2)*(1+r2)= (1+r2)^2

1+r2 = 1.0989^(1/2) = 1.0483 (r2=4.83%)

• Equivalently: 1+r2 = 1/DF(2)^0.5 = 1/0.91^0.5 = 1.0483

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Why do we see several levels of r?• The strips of various maturities are safe in terms of

default, that is, you are guaranteed to earn the promised rate

• But, there are other risks involved: uncertain future inflation makes the true (real) value of the earned dollars uncertain. Also, if interest rates go up after one year, you will be better off investing in a 1-year claim and then another and so on

• We commonly observe different annual rates in different maturities. We must match the maturity of rates to the maturity of the CF we discount

• For now we assume the rate is same for all maturities

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IPCOR421 FinanceAlex Kane 7

PV and NPV of projects• A project involves investment, C(0), and future CF:

C(1), C(2),…,C(T) ; T is the project horizonSome of the C(t) can be zero or negativeC(T) includes scrap value, termination costs or resalevalue -- the net C(T) also can be positive or negative

(e.g., reclamation of land - environmental obligation)• PV is additive: PV(sum) = sum (PV)

PV[C(1)+C(2)+…+C(T)] = PV[C(1)] + … + PV[C(T)] = PV[C(t)] = [C(t) / (1+r)^t]

• As before: NPV = C(0) + PV = C(0) + [C(t) /(1+r)^t]

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Annuities and perpetuities

Time line

$1 $1 $1

0 1 32 4 5

$1 Perpetuity

1 2 3

$1$1 $1

$1, three-period annuity

Time line

Definition: CF is same each equal-length period

0

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IPCOR421 FinanceAlex Kane 9

Simple CF: Annuities and perpetuities• A cash flow of equal payments spread over equal periods

(not necessarily years) is called “annuity”• An annuity that continues to infinity is called “perpetuity”• If the annuity payments are $C/period, the PV of the annuity

is: PV(annuity) = C* 1/(1+r)^t = C * PVA• PVA = PV($1 annuity) = 1/(1+r)^t • The first term in the summation is 1/(1+r)• Each subsequent term is multiplied by 1/(1+r), called the

“common factor,” • This is a geometric progression• here, the common factor is same as the first term and <1

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IPCOR421 FinanceAlex Kane 10

PV of perpetuities• In a geometric progression, the first term is denoted

by “a” and the common factor by “q”• The sum of T terms of such progression is:

S(T) = a*(1–q^T)/(1–q)When q<1, then the sum of an infinite progression is

S() = a/(1–q), because q^T goes to zeroWith an annuity, the common factor is: q = 1/(1+r) <1Also, the first term is the PV of the first payment which is:

1/(1+r) < 1Therefore: a/(1–q) = 1/(1+r) / {1– [1/(1+r)] } = 1/rIn sum: PV(perpetuity of $C/year) = C/r

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IPCOR421 FinanceAlex Kane 11

PV of annuities (1)

Time line

$1 $1 $1

$1 $1 $1

0 1 32

4 5 6

Perpetuity as of t = 0

Perpetuity as of t = 3

3Time line

1 2 3

$1$1 $1A three-period annuity

Notice:PV(annuity) = difference between PV(perpetuities)Time line0

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IPCOR421 FinanceAlex Kane 12

PV of annuities (II)

• PV(perpetuity as of t=0) = C/r• PV(perpetuity as of t=T) = (C/r)/(1+r)^T• PV(annuity from 0 to T) = C/r – (C/r)/(1+r)^T

= (C/r ) *[1–1/(1+r)^T]• The value of an annuity is less than the value of a

perpetuity, and gets closer to it as T (its length) grows• Example:

(1) at 10%, PV( $1 perpetuity) = 1/0.1 = $10

(2) PVA(r=0.1,T=3) = (1/0.1)*(1–1/1.1^3) = 10*0.2487=$2.49

In Excel, T is denoted by NPER

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Example of annuity vs. perpetuity• Suppose the annuity is of $1000 and the required

rate is either 4% or 16%– Using the formula we have T (years) PV (4%) PV(16%) 1 961.54 862.07 20 13590.33 5928.84

80 23915.39 6249.96perpetuity 25000.00 6250.00

• Notice the effect of the rate on the value of the annuity and its convergence to the value of a perpetuity

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IPCOR421 FinanceAlex Kane 14

Growth annuities and perpetuities• In economics, time series where numbers grow at a

constant (or close to it) rate are common• For CF, instead of a constant C, we have:

C(1), C(2)=C(1)*(1+g), C(3)=C(1)*(1+g)^2, …,C(T)=C(1)*(1+g)^(T–1)

• The PV of a growth annuity is of a similar form to an annuity. We replace C by C*=C(1)/(1+g) and the rate r by r*=(r–g)/(1+g) [ useful in Excel ]

• With this substitution we can use the formulas for annuity/perpetuity for growth as well

• Then PV(growth perpetuity) = C*/r* = C(1)/(r-g)

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IPCOR421 FinanceAlex Kane 15

Example of a growth annuity• Suppose shareholders (SH) expect next year’s

dividend at C(1)= $3/share, with growth of g = 7%/year. The required rate is r = 0.12

• To use annuity formula we substitute:C* = C(1)/(1+g) = 3/1.07 = 2.8037r* = (r–g)/(1+g) = 0.05/1.07 = 0.04673

• When this goes on forever, the value isPV = C(1)/(r–g) = 3 / (0.12–0.07) = $60

• If the stream ends at T=7, then:PV = (C*/r*)*[1–1/(1+r*)^T] = (2.8037/0.04673)*(1–1/1.04673^7) = $16.42

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IPCOR421 FinanceAlex Kane 16

Annual compounding

• With annual compounding you do not earn interest on CF within the year

• Your interest is calculated (and credited) at the end of each year, based on investment value at the beginning of the year

• Most contracts allow for compounding over shorter periods

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Compounding period in general

• When a compounding period is specified in a contract (for example, a mortgage contract calls for a compounding period of one month), interest is calculated and credited at the end of each such period (a month for mortgages) based on the investment value at the beginning of that period

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IPCOR421 FinanceAlex Kane 18

Compounding convention

• A compounding period is specified

• Accordingly, there are n compounding periods per year. For example: with monthly compounding, n=12

• An annual rate is specified and called: APR (Annual Percentage Rate), example: r = 0.07

• The compounding period rate is: APR/n = (monthly) 0.07/12 = 0.05833

• APR/n is used for each compounding period

Page 19: Finance Week 3

IPCOR421 FinanceAlex Kane 19

The Effective Annual Rate (EAR)• EAR = the annual rate actually earned on $1 with the

contract APR, over the compounding periods (n)• By definition

1+EAR = (1+r/n)^n

With the previous example: EAR = (1+0.07/12)^12 – 1 = 0.0723 (7.23%)

• It is obvious that, given an annual rate (APR), EAR is greater the shorter the compounding period (and a larger n). You earn interest on interest more frequently

• Is there a limit to this increase? Put differently, when n grows, does EAR grows without bound?

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Continuous Compounding• 1+EAR = (1+r/n)^n, will grows as n grows, but cannot

exceed the limit:Limit(n–>∞) of (1+r/n)^n = exp(r)

• Example: with APR=0.07, and continuous compounding, EAR= exp(0.07)–1 = 0.0725 (7.25%)

• It appears that continuous compounding is an artificial construct, but is isn’t

• For example, since information flows to the market continuously, values of financial assets compound continuously, although we observes them from transactions only infrequently

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Compounding frequency and the EAR• Example: semi-annual monthly weekly daily Cont.

n = 2 12 52 365 ∞

EAR(r=5%)= 5.0625 5.1162 5.1246 5.1268 5.1271

EAR(r=15%)=15.5625 16.0754 16.1583 16.1798 16.1834

• Obviously, one cannot profit from setting up a contract with more frequent compounding

• Market conditions determine the appropriate EAR and the compounding interval is chosen for convenience

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Note on terminology

• Consumer law requires that the EAR be clearly shown in any contract

• Over time, some confusion took over and now frequently APR is listed as the EAR

• You will find from your own experience that ignorance about compounding still is wide-spread

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Mortgages: monthly compounding• Consider a $100,000, 30-year mortgage

payments are to be made every month and your obligation is compounded n=12 times a year

Suppose APR is 9%

the monthly rate is then

r(month) = APR/n = 0.09/12 = 0.0075 (0.75%)

• A dollar you owe in the beginning of the year becomes: (1+APR/n)^n = (1+0.0075)^12 = 1. 0938 = 1+EAR

EAR = 0.0938 (9.38%) the Annual Effective Rate

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Mortgage payments

• It is required that PV of payments equal the mortgage value. Using Excel’s function, PV: PV(rate=0.075, nper=360, PMT, FV=0,0) = 100,000PMT is the annuity amount needed to get the right PVExcel allows for a “balloon” payment (FV) at the end, here

FV=zero. [The last argument, when set to 1, allows for PMT at beginning of month]

PV = (C/r)*(1–1/(1+r)^T) = (C/0.075)*(1–1/1.075^360)Excel’s function, PMT, easily produces the monthly

payment:PMT(rate=0.0075,nper=360,PV= –100000,0) = $804.62

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Inflation and real rates• Suppose a one-year T-bill yields 7%, so a $10,000 bill will

cost 10000/1.07 = $9345.79• Suppose inflation over the next year is: i = 0.04 (4%)• An average $1-item will cost you $1.04 next year• Therefore, when you get your money, the purchasing power

will be only 10000/1.04, or: Amount/(1+inflation rate)• Hence, your real rate (rr) is taken from

1+rr = (10000/9345.79)/(1+i) = (1+r)/(1+i)

rr = (r – i)/(1+i). r is now referred to as the nominal rate

• The approximation rr = r –i is good only when inflation is low. With continous compounding: ccrr=ccr – cci (exactly)

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Real and nominal CF• The real value of a CF at t, RC(t), is found by discounting it

at the rate of inflationRC(t) = C(t)/(1+i)^t

• Conversely, the nominal CF, C(t), with purchasing power of C(0), is obtained by compounding at the rate of inflationC(t) = C(0)*(1+i)^t

• Examples: With i=4%, looking t=10 years forward.(1) The purchasing power of C(t) = $10,000 will be only

RC(t) = $6,755.64(2) To maintain a $50,000 standard of living we’ll need

C(t) = $74,012.21This type of calculations is critical for personal finance!

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PV of projects with inflation

• The PV of a project can be computed in two equivalent ways:(1) PV(nominal CF) discounted at the nominal rate

(2) PV(real CF) discounted at the real rate

• It is necessary to be consistent in the calculation. You choose one of the identical versions according to convenience and to what fits the presentation you may have to make