MGT 326 Ch. 4: Time Value of Money (bdh), Part 2 1 Semiannual & Other Compounding Periods What do...

MGT 326 Ch. 4: Time Value of Money (bdh), Part 2

1

Semiannual & Other Compounding Periods

What do you do for other than annual compounding?Annual Compounding: Not often used in business/finance world; but it's easier to introduce compounding/discounting with this compounding periodSemiannual Compounding: Used most often in bondsQuarterly Compounding: Often used by banks for business loans. Monthly Compounding: Used most often by banks for consumer loans and investments (CD's); also used in short-term bonds (< 1 yr); very common with leasesDaily Compounding: Used by banks to lend/borrow from each other for very short terms (days & weeks)Continuous Compounding: Used in mathematical models of various, really complicated financial concepts (i.e. duration, convexity, pricing an option contract, interest rate options & swaps, etc.)


2

Example: Calculate the FV of $100 invested for 2 years at 8% if interest is compounded annually and semiannually

Nominal Interest Rate ( rnominal ): This is often what people quote as your interest rate for loans and bank accounts and credit cards and bonds. It is also called the quoted rate It must also be accompanied by a statement indicating the compounding frequencyIn the example above the nominal or quoted interest rate is 8%

Annual: rnominal= 8%, compounded annuallySemiannual: rnominal = 8%, compounded semiannually

Periodic Rate: this is the rate charged per compounding period.periodic Rate = rperiodic = rnominal / m (Learn & know this!)m is the number of compounding/payment periods per yearin the example above……

m = 1 for the annual case m = 2 for the semiannual case

In the above example, the periodic rates are:Annual: 8%Semiannual: 4%

Example: A $1,000 face value bond has a coupon rate of 6.0000% per year but it pays interest semiannually. How much interest is earned each interest paying period?1) Compute the periodic rate:

rperiodic = rnominal / m = 6.0000%/2 = 3.0000%2) Compute the interest payment:

PMT = Principle x Interest rate = $1000 (0.03) = $30.00These two steps can be combined:

PMT = Principle(Interest Rate / m)

Semiannual & Other Compounding Periods (continued)


3


Using FV and PV formulas with other-than-annual compounding:

FV = PV(1 + rnominal/m)n PV = FV / (1 + rnominal/m)n

Example(repeated): Calculate the FV of $100 invested for 2 years at 8% if interest is compounded annually and semiannually

0 1 2

FV = ?

100Formula:FV = PV(1 + rnominal/m)n = $100(1 + 0.08/1)2 = $116.64

Financial Calculator:

Enter parameters:Clear TVM registers [2nd, CLEAR TVM]Set payments per year = 1; [2nd, P/Y, 1, ENTER, CE/C]Enter number of periods [2, N]Enter PERIODIC interest rate [8, I/Y] Enter PV [-100, PV] Find FV [CPT, FV] and voila! FV = $116.64

Annual Case:r = 8%

T = # of years = 2m = # of periods per year = 1n = total # of periods = m x T = 1 x 2 = 2


4


0

1 2

FV = ?

100

Semiannual Case:

1 2 3 4

years

compounding periods

r = 8%

T = # of years = 2m = # of discounting per year = 2n = total # of periods = m x T = 2 x 2 = 4

Formula:FV = PV(1 + rnominal/m)n = $100(1 + 0.08/2)4 = $116.99


Option 1 for Semiannual Case: Enter parameters:1) Find rperiodic: rnomianl/m = 8%/2 = 4%2) Enter parameters

Clear TVM registers [2nd, CLEAR TVM]Leave payments per year = 1;[2nd, P/Y, 1, ENTER, CE/C]Enter number of periods [4, N]Enter PERIODIC interest rate [4, I/Y] Enter PV [-100, PV] Find FV [CPT, FV] and voila! FV = $116.99

Option 2 for Semiannual Case: 1) Set payments per year = 2; [2nd, P/Y, 2, ENTER, CE/C]2) Enter parameters:

Enter number of periods [ 4 , N]Enter NOMINAL interest rate [8, I/Y] Enter PV [-100, PV] Find FV [CPT, FV] and voila! FV = $116.99


5


Example:(extended) Calculate the FV of $100 invested for 2 years at 8% if interest is compounded quarterly and monthly

Quarterly Case:

$100

7 8

1 2

0 4 5 61 2 3

r = 8%

FV = ?

years

compounding periods



Formula:


6

Monthly Case:

0r = 8%

FV = ?

1 2 3 4 5

2 years

compounding periods22 23 24

$100

0


Semiannual & Other Compounding Periods (continued)Example: (continued)

Formula:



7


Find PV of an Annuity

Example: An ordinary annuity pays $50 semiannually for two years. If the current market interest rate for this annuity is 4%, what is it worth today?

0 1 2 3 4

5050 50

PV = ?

rperiodic = ?

rsimple = 4%

Formula:

PV = CF1/(1 + r/m)1 + CF2/(1 + r/m)2 + CF3/(1 + r/m)3 + CF4(1 + r/m)4

= 50/(1 + 0.04/2)1 + 50/(1 + 0.04/2)2 + 50/(1 + 0.04/2)3 + 50/(1+0.04/2)4 = 50/(1.02)1 + 50/(1.02)2 + 50/(1.02)3 + 50(1.02)4

= 50/1.02 + 50/1.0404 + 50/1.0612 + 50/1.0824 = 49.0196 + 48.0584 + 47.1161 + 46.1923 = $190.39

50

0 1 2 3 4

5050 5050

CF1/(1 + r/m)1

CF2/(1 + r/m)2

CF3/(1 + r/m)3 CF4/(1 + r/m)4


8


Example (continued):

0 1 2 3 4

5050 50

PV = ?

rperiodic = ?

rsimple = 4% 50




9

Semiannual & Other Compounding Periods (continued)Find the Yield (r, k, i, ROR, etc.) of an Annuity

Example: An ordinary annuity paying $100 every quarter for 2 years is currently selling for $765.17. What return is this security yielding?

7 80 4 5 61 2 3

$765.17

r = ?

100 100100100100100100100

Option 1: 1) Clear your calculator: [2nd, CLEAR TVM]2) Leave payments per year to 1: [2nd, P/Y, 1, ENTER, CE/C]3) Enter parameters:Enter N [8, N]Enter PV [- 765.17, PV]Enter Pmt [100, PMT] Find I/Y, [CPT, I/Y] and voila! I/Y = 1% This is rperiodic!

4) Find iquoted/nominal: rperiodic = rsimple / mrsimple = rperiodic x m = 1% x 4 = 4%

Option 2:1) Set payments per year to 4: [2nd, P/Y, 4, ENTER, CE/C]2) Enter parameters:Enter N [8, N]Enter PV [- 765.17, PV]Enter Pmt [100, PMT] Find I/Y, [CPT, I/Y] and voila! I/Y = 3.9997% = 4%

m = 4n = 8


Note: One of the two cash inputs must be negative

Note: One of the two cash inputs must be negative


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Effective Annual Rate (EAR) or EFF% (Calculator Symbol) (From Ch 5)

Financial institutions have to tell us the interest rate they charge for loans or the interest rate they pay when you invest with themAs stated before, the rate they often tell you is called the nominal interest rate or the quoted interest rate

This is an annual rate (i.e. 12% per year)They must also tell you the compounding rate (i.e. daily, weekly, monthly, semiannually, annually, bi-annually, etc.)

The examples we previously covered showed us that an investment earns more money when the compounding rate is more frequent

FV of $100 @ 8% compounded annually, 2 yrs: $116.64FV of $100 @ 8% compounded semiannually, 2 yrs : $116.98FV of $100 @ 8% compounded quarterly, 2 yrs : $117.17FV of $100 @ 8% compounded monthly, 2 yrs : $117.29

The nominal rate is no help in mathematically expressing the power of compounding

The EAR expresses an interest rate that compounds more than once per year. This is the actual rate of return being earned or paid per year, when compounding is factored in.

EAR (EFF%) = ( 1 + rnominal / m )n - 1m = number of compounding/discounting periods per year

n = the total number of compounding periods in question; in most cases, this will be the same as “m” since you’re usually dealing with an annual return

EAR is also the annual ROR; the profit is recognized once per year at the end of the year


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Effective Annual Rate (EAR) or (EFF%) (continued)

The examples we previously covered showed us that an investment earns more money when the compounding rate is more frequent. This means that the effective rate of return or effective annual rate (EAR) is greater than the nominal rate when there is more than one interest payment (compounding period) per year.

FV of $100 @ 8% compounded annually, 2 yrs: $116.64EAR = 8%

Note: EAR for annual compounding = rnominal

FV of $100 @ 8% compounded semiannually, 2 yrs : $116.98EAR = 8.16%

FV of $100 @ 8% compounded quarterly, 2 yrs : $117.17EAR = 8.24%

FV of $100 @ 8% compounded monthly, 2 yrs : $117.29EAR = 8.30%

In the last 3 cases, the effective annual rate of return is greater than the quoted/nominal rate of return.

The Effective Annual Rate is also referred to as the Annual Percentage Yield (APY) or Effective Annual Yield (EAY) by banks and some investment companies


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Example: $400 dollars is deposited in a checking account that pays 5% interest compounded monthly. What is the effective annual rate?Option 1: Formula Solution

EAR (EFF%) = ( 1 + rnominal / m )n - 1= ( 1 + 0.05/12)12 - 1= ( 1.004167)12 - 1

= 1.051162 - 1 = 5.1162%Option 2: Calculator Financial Function Solution1) Access interest rate conversion worksheet [2nd, ICONV]2) Enter rnominal [5, ENTER]3) Enter # of payments/compounding periods [↓, ↓, 12, ENTER]4) Find EFF [↓, ↓, CPT] and viola! EFF% = 5.1162%

Another way to look at EAR, an Empirical Demonstration:

Example: $400 dollars is deposited in a checking account that pays 5% interest compounded monthly. Find FV after 1 year using iperiodic and EAR.a. Find FV using rperiodic:

1) Find rperiodic: 5% / 12 = 0.4167%2) Enter parameters:

Enter number of periods [ 12, N]Enter periodic interest rate [ 0. 4167 , I/YR] Enter PV [-400, PV] Find FV, [CPT,FV] and voila! FV = $420.47

b. Find ROR:Profit = $420.47 - $400 = $20.47ROR = $20.47 / $400 = 0.0512 = 5.1162% = EAR


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Empirical Demonstration of EAR: (continued)Find FV using EAR:

1) Find EAR (EFF%): 5.1162% (as per above example)2) Enter parameters:

Enter number of periods [2nd, P/Y, 1, ENTER, CE/C]Enter N [1, N]Enter periodic interest rate [5.1162, I/Y] Enter PV [-400, PV] Find FV, [CPT,FV] and voila! FV = $420.46

Another Example: If a security earns 6% p.a. with monthly compounding, what would be the total ROR if the security is held for 2 years? EAR (EFF%) = ( 1 + rnominal / m )n – 1 (n = the total # of compounding periods = 2 yrs x 12 per/yr = 24 per.)

= ( 1 + 0.06/12)24 - 1= ( 1.0050)24 - 1

= 1.127160 - 1 = 12.7160%


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Example Using Ear (continued)2) Find EARAccess interest rate conversion worksheet [2nd, ICONV]Enter isimple [5, ENTER]Enter # of payments/compounding periods [↓, ↓, 4, ENTER]Find EAR (EFF%): [↓, ↓, CPT] and viola! EFF% = 5.0945%

3) Find PVSet payments per year to 1: [2nd, P/Y, 1, ENTER, CE/C]Enter number of periods [2, N]Enter EFF% [5.0945, I/Y] Enter Annualized PMT [407.56, PV] Find PV, [CPT,PV] and voila! FV = -$756.81

7 8 qtrs0 4 5 61 2 3

PV?

rnominal = 5%reffective = ?

407.56

1 2 years

407.56The CF diagram now looks like this:

1) Annualize the payments: Find the FV of 4 payments of $100 @ 5% a) Find rperiodic: rnominal / m = 5%/4 = 1.25% b) Find FVEnter number of periods [ 4, N]Enter periodic interest rate [ 1.25 , I/Y] Enter PMT [100, PMT] Press FV, [CPT,FV] and voila! FV = $407.56, Interpretation: 4 quarterly pymts of $100 @ 5% equal one annual payment of $407.56

7 80 4 5 61 2 3

PV?

r = 5%

100 100100100100100100100


Note: You can use EAR (EFF%) to solve Annuities but you must “annualize” the paymentsExample Using EAR: An ordinary annuity paying $100 every quarter for 2 years is currently yielding 5%. What is the fair market value of this security yielding?


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Why should you care about EAR? Answer: It’s used as a basis of comparison to choose the best rsimple/nominal/quoted between investment/loan options that have different payment frequencies

r (APR) EFF%

6.350% 1 6.350%

6.238% 2 6.335%

6.223% 4 6.370%

6.187% 12 6.365%

6.177% 52 6.368%

# of Compounding

Periods per Year

Example: You are considering two different stock mutual funds in which to invest. Fund A offers 8.5808% APR rate of return with quarterly reinvestment of profits. Fund B offers a 8.5410% APR rate of return with monthly reinvestment of profits. Which fund is more profitable? Fund A: 2nd, INCONV, 8.5808%, ENTER, ↓, ↓, 4, ENTER, ↓, ↓, CPT: 8.8609%Fund B: 2nd, INCONV, 8.5410%, ENTER, ↓, ↓, 12, ENTER, ↓, ↓, CPT: 8.8834% Fund B is more profitable

Example: Your company needs to borrow $100,000.00 for a warehouse modification. You have received five different quoted rates (rates and compounding periods per year are shown below). Which one should you choose?Answer: Compute the EAR for each quoted rate. The one with the lowest EAR is the lowest annual rate of cost.


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Example: Your broker proposes an investment scheme that will pay you $1000 two years from now for an initial cost of $900 today. The investment promises a total return of 11.11%. What is the annual rate of return on this investment?

ROR (per annum) = (1 + RORtotal)1/n – 1 = (1 + 0.1111)1/2 – 1 = (1.1111)1/2 – 1 = 1.054093 – 1 = 0.054093 = 5.4093%

Find the Annual ROR, given a Total Return

Annual Percentage Rate (APR)

This is the rate reported (as required by law) to borrowers. (Look at your mortgage or auto loan paper work)There are several different formulas to compute APR and they result in different numbersBut for all practical purposes, APR = rquoted since it is the rate that the financial institution will quote youAPR EAR


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Uneven Cash Flows

General EquationsPV = CF0 + CF1/(1 + r/m)1 + CF2/(1 + r/m)2…+ CFn/(1 + r/m)n FV = CF0(1 + r/m)n + CF1(1 + r/m)n-1 + CF2(1 + r/m)n-2 ….+ CFn

Formula SolutionPV = -450 + 300/(1.04) + 250/(1.04)2 + -150/(1.04)3 + 200/(1.04)4 + 500/(1.04)5

= -450 + 300/(1.04) + 250/(1.08499) + -150/(1.1248) +200/(1.1698) + 500/(1.2166) = -450 + 288.4615 + 230.4169 - 133.3570 + 170.9694 + 410.9814 = $517.47FV = Left as an exercise for the student

0 1 2 53 4

$300$250 $200

(-) $150

$500

(-) $450

r = 4%FV = ?PV = ?

m = 1

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Present Value of Uneven Cash Flows Example: You are tasked with estimating the fair market value of a security that promises uneven future payments. The table below shows the quarterly payment schedule (each cash flow occurs at the end of the quarter). You consider 7.2000% APR to be the appropriate opportunity cost. What is the theoretical value of this security?

0 2 3 41

300400

500

700

1 yr

Formula Solution:PV = CF1/(1 + r/m)1 + CF2/(1 + r/m)2 + CF3/(1 + r/m)3 + CF4(1 + r/m)4

= 300/(1 + 0.072/4)1 + 400/(1 + 0.072/4)2 - 500/(1 + 0.072/4)3 + 700/(1 +0.072/4)4 = 300/(1.018)1 + 400/(1.018)2 + 500/(1.018)3 + 700(1.018)4

= 300/1.018 0+ 400/1.03632 - 500/1.05498 + 700/1.07397 = 294.6955 + 385.9797 - 473.9426 + 651.7889 = $858.52 Calculator Solution: 1) Compute periodic rate: rperiodic = rnominal/m = 7.2000%/4 = 1.8000%2) Access Cash Flow Worksheet [CF]3) Clear registers: [2nd, CLR WORK]2) Enter parameters:

Enter CF0; [Default is 0]Enter CF1; [↓, 300, ENTER]Enter CF2; [↓, ↓, 400, ENTER]Enter CF3; [↓, ↓, -500, ENTER]Enter CF4; [↓, ↓, 700, ENTER]Enter discount rate: [NPV, 1.8, ENTER]Find NPV [↓, CPT] and viola! PV = $858.52 Net Present Value



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Uneven Cash Flows (continued)

FV Calculator SolutionIf your calculator has a “NFV” (Net Future Value) key, you’re in luck! (TI BA II Plus Professional has this function)If there’s no NFV key, you have to compound each CF to the last (terminal) time period

Fractional Time PeriodsExample: Calculate the FV of $100 invested for 18 months in a bank account that pays a quoted rate of 10%, compounded annually.

0

PV = $100

rperiodic = ?

rnominal = 10%

2

FV = ?

18 mos

m = 1n = ?

Numerical Solution1) Find n: 18 mos / 12 mos per period = 1.5 periods2) FV0.75 = PV(1 + r/m)n = 100(1 + 0.10/1)1.5 = 100(1.10)1.5

= $115.37

Calculator Solution:1) Find rperiodic: 10% / 1 = 10%2) Find n: 18 mos / 12 mos per period = 1.5 periods3) Enter parameters:

Enter number of periods [ 1.5, N]Enter periodic interest rate [ 10 , I/YR] Enter PV [100, PV]Find FV, [CPT,FV] and viola! FV = (-)$115.37

T = # of years = 18mos/12mos per yr = 1.5m = # of discounting per year = 1n = total # of periods = m x T = 1 x 1.5= 1.5

1


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Fractional Time Periods (continued)

Example: Today you deposit $2000 in a bank account that pays 3.6% APR compounded quarterly. How much money would you have in that account 20 months from now.

Continuous Compounding

Used in mathematical models of various more complicated financial concepts (i.e. duration, convexity, pricing an option contract, interest rate options & swaps, etc.)Formula: FV = PVerT where is an annual rate and T is time in yearsExample: If today you deposit $1,000 in to an account that pays 7.2000% per annum with continuous compounding, how much will you have in the account three years from now?

FV = PVerT = $1,000e(0.072)(3) = $1,000e(0.216) = $1,000(1.2411) = $1,241.10


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Continuous Compounding (continued)

EAR with continuous compoundingExample: If rnominal is 6% what is the EAR with continuous compounding?

EARcontinuous = er – 1 = e(0.06) – 1 = 1.061837 = 0.061837 = 6.1837%

Perpetuities

A type of annuityThe uniform payments go on indefinitely

0r = ?

PV1 2 3 4 5

∞

PMT

PVperpetuity = PMT / (r/m) = PMT / rperiodic

Example: What is the PV of a perpetuity that pays $500 per year @ 8 % APR?

PV = PMT / rperiodic = 500/0.08 = $6,250.00

PMT(1 + r/m)1

PMT(1 + r/m)2

PMT

(1 + r/m)∞+ + ……+PV =

= ∞n = 1

PMT(1 + r/m)n


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Perpetuities (continued)Example: An endowment is established with an initial deposit of $1m. How much can be drawn out each month @ 6% APR?

1) Find rperiodic: 6% / 12 = 0.5%2) Find PMT: PMT = PV(rperiodic) = $1m(0.005) = $5,000

Why worry about Perpetuities?Answer:Many pensions are perpetuitiesWe will use the perpetuity model to find stock values

and:Capitalize (Capitalization)Example: What is the value of a firm that earns $100m per year and its cost of debt is 10%? (Assume this firm is totally financed by debt.) VFirm = $100m / 0.10 = $1 billion

Growing Perpetuity: A perpetuity in which the cash flows are not constant; they grow at a particular rate indefinitely

Example: A wealthy businessman wishes to establish a scholarship endowment for a local university business school. The donator wants to initially provide $6,000 per semester but he wants that amount to grow to compensate for inflation. He estimates that inflation is likely to be 3.5% per year. The endowment account pays 6.5% p.a. How large must the endowment be?

PV(Growing Perp.) = Initial PMT / (r/m – g/m) (g is the growth rate) = $6,000 / (0.065/2 – 0.035/2) = $6,000 / (0.03/2) = $6,000 / 0.015 = $400,000.00


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Amortized Loan:

Definition: a loan in which portions of the principle are combined with periodic interest payments to form a series of uniform payments

the entire principle is paid back to the lender by the end of the loan termmost consumer loans (mortgages, auto loans, etc.) are amortized

Each successive payment contains a little less interest and a little more balance but the total amount of each payment is the same

Example: You finance the entire $16,785 cost of a new car @ 8% APR for 3 years. You have managed to convince your bank to allow you to make quarterly payments. What is the amount of each quarterly payment?

1 30

PMT = ?

isimple = 8%iperiodic = ?

T = 3m = 4n = 3x4 =12

PV = $16,785

1 2 3 4 121110

interest

principle

Calculator Solution:1) Find rperiodic: 8% / 4 = 2%2) Enter parameters:Enter number of periods [ 12, N]Enter periodic interest rate [ 2 , I/Y] Enter PV [16785, PV]Find PMT, [CPT,PMT] and viola! PMT = -$1,587.18

years

compounding periods


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Payments Don’t Coincide with Compounding Periods (not covered in your text)

Example: Today you open a new savings account that pays 3.7% compounded weekly. You plan to deposit $400 into this account at the end of every month, starting at the end of this month. How much will you have in this account 2 years from now?

0r = 3.7%

FV = ?

1 2 3 4 5

2 years

months22 23 24

PMT = $400

0

T = # of years = 2m = # of payments per year = 12n = total # of payments = m x T = 12 x 2 = 24

Enter Parameters:Set payments per year to 12; [2nd, P/Y, 12, ENTER]Set compounding periods per year to 52; [↓, 52, ENTER, CE/E]Enter number of payments [24, N] (Note: in this case, N is the number of payments, not number of compounding periods)Enter interest rate; [3.7, I/Y]Enter payments; [400, PMT]Find FV, [CPT,FV] and viola! FV = $9,948.65


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You Can Invest Interest Payments at a Different Rate Than You Are Currently Receiving (not covered in your text)

Example: You currently have $5,000 in a savings account that pays 3.00% APR, compounded monthly. For the next two years you plan to reinvest each savings account interest payment in a mutual fund that guarantees 5.00% APR, compounded monthly. How much money would you have in the mutual fund after two years? Assume the current balance in the mutual fund account is 0.

0r = 5.00%

FV = ?

1 2 3 4 5

2 years

compounding periods22 23 24

PMT = ?

0

T = # of years = 2m = # of compounding per year = 12n = total # of periods = m x T = 12 x 2 = 24

1) Find the interest payment from the savings account: PMT = $5,000(rnominal/m) = $5,000(0.03/12) = $12.502) Enter Parameters:Set payments per year to 12; [2nd, P/Y, 12, ENTER]Set compounding periods per year to 12; [↓, 12, ENTER, CE/E]Enter number of compounding periods [24, N] Enter interest rate; [5, I/Y]Enter payments; [12.50, PMT]Find FV, [CPT,FV] and viola! FV = $314.82


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More than One Interest Rate (not covered in your text)

What do you do if the interest rate changes?

Example: You are negotiating a loan with a bank in order to raise funds for the start of your new business. You estimate that your business will earn $50k NI each year for the next 4 years. You manage to convince the bank that your company will become progressively less risky as time goes on. You and the bank agree that your cost of debt for the first two years should be 7% and that it should fall to 5% for the next 2 years. What is the value of your company?

0 1 2 3 4

50k50k 50kPV = ? r1-2 = 7% APR

r3-4 = 5% APR

50k

m = 1n = 4

Solution:Approach: divide the problem into two parts

0 1 2 3 4

50k50k 50kr1-2 = 7% APRr3-4 = 5% APR

50km = 1n = 4

PV2 = ?PV0 = ?

PV = PV0(CFs 1, 2) + PV0(PV2 of CFs 3, 4) = 50/(1.07)1 + 50/(1.07)2 + [ 50/(1.05)1 + 50/(1.05)2] / (1.07)2

= 50/1.07 + 50/1.1449 + (50/1.05 + 50/1.1025)/1.1449 = 46.729 + 43.6719 + (47.619 + 45.3515)/1.1449 = 46.729 + 43.6719 + 81.204 = $90.40k + $81.20k = $171,604.91

MGT 326 Ch. 4: Time Value of Money (bdh), Part 2 1 Semiannual & Other Compounding Periods What do...

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Transcript of MGT 326 Ch. 4: Time Value of Money (bdh), Part 2 1 Semiannual & Other Compounding Periods What do...