Semiannual

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 period
• Semiannual Compounding Used most often in bonds
• Quarterly 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 (lt 1 yr) very common
  with leases
• Daily 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
Semiannual Other Compounding Periods (continued)

• 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 frequency
• In the example above the nominal or quoted
  interest rate is 8
• Annual rnominal 8, compounded annually
• Semiannual 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 year
• in 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

3

• Semiannual Other Compounding Periods
  (continued)
• 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

FV ?
Annual Case
r 8
0
1
2
T of years 2 m of periods per year
1 n total of periods m x T 1 x 2 2
100

• Formula
• 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

4
Semiannual Other Compounding Periods (continued)
Semiannual Case
FV ?
1
2
years
r 8
0
1
2
3
4
compounding periods
T of years 2 m of discounting per year
2 n total of periods m x T 2 x 2 4
100

• Formula
• FV PV(1 rnominal/m)n 100(1 0.08/2)4
  116.99
• Financial Calculator
• 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 12nd, 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

5

• Semiannual Other Compounding Periods
  (continued)
• Example(extended) Calculate the FV of 100
  invested for 2 years at 8 if interest is
  compounded quarterly and monthly
• Quarterly Case

Formula
Financial Calculator
6
Semiannual Other Compounding Periods (continued)
Example (continued)
Monthly Case
Formula
Financial Calculator
7
Semiannual Other Compounding Periods
(continued) 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?
50
50
50
50
rsimple 4
rperiodic ?
PV ?
50
50
50
50
CF1/(1 r/m)1
CF2/(1 r/m)2
CF3/(1 r/m)3
CF4/(1 r/m)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/(10.04/2)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
8
Semiannual Other Compounding Periods
(continued) Example (continued)
50
50
50
50
rsimple 4
rperiodic ?
PV ?
T of years 2 m of discounting per year
2 n total of periods m x T 2 x 2 4
Financial Calculator
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?
100
100
100
100
100
100
100
100
r ?
m 4 n 8
7
8
0
4
5
6
1
2
3
T of years 2 m of discounting per year
4 n total of periods m x T 4 x 2 8
765.17

• 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 / m
• rsimple 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

Note One of the two cash inputs must be negative
Note One of the two cash inputs must be negative
10

• 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 them
• As 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 _at_ 8 compounded annually, 2 yrs
  116.64
• FV of 100 _at_ 8 compounded semiannually, 2 yrs
  116.98
• FV of 100 _at_ 8 compounded quarterly, 2 yrs
  117.17
• FV of 100 _at_ 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 - 1
• m number of compounding/discounting periods
  per year

11

• 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 _at_ 8 compounded annually, 2 yrs
  116.64
• EAR 8
• Note EAR for annual compounding rnominal
• FV of 100 _at_ 8 compounded semiannually, 2 yrs
  116.98
• EAR 8.16
• FV of 100 _at_ 8 compounded quarterly, 2 yrs
  117.17
• EAR 8.24
• FV of 100 _at_ 8 compounded monthly, 2 yrs
  117.29
• EAR 8.30
• In the last 3 cases, the effective annual rate of
  return is greater than the quoted/nominal rate of
  return.

12

• Effective Annual Rate (EAR) or (EFF) (continued)
• 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 Solution
• 1) 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

13

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

14

• Note You can use EAR (EFF) to solve Annuities
  but you must annualize the payments
• Example 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?

100
100
100
100
100
100
100
100
r 5
7
8
0
4
5
6
1
2
3
T of years 2 m of discounting per year
4 n total of periods m x T 4 x 2 8
PV?

• 1) Annualize the payments Find the FV of 4
  payments of 100 _at_ 5
• a) Find rperiodic rnominal / m 5/4 1.25
• b) Find FV
• Enter 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 _at_ 5
  equal one annual payment of 407.56

The CF diagram now looks like this
407.56
407.56
1
2 years
rnominal 5 reffective ?
7
8 qtrs
0
4
5
6
1
2
3
PV?

• Example Using Ear (continued)
• 2) Find EAR
• Access 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 PV
• Set 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

15
Why should you care about EAR? Answer Its used
as a basis of comparison to choose the best
rsimple/nominal/quoted between investment/loan
options that have different payment frequencies
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.
16
Find the Annual ROR, given a Total Return
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

• 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 numbers
• But for all practical purposes, APR rquoted
  since it is the rate that the financial
  institution will quote you
• APR ? EAR

17
Uneven Cash Flows
500
300
m 1
250
200
r 4
PV ?
FV ?
(-) 150
(-) 450

• General Equations
• PV 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 Solution
• PV -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
• 517.47
• FV Left as an exercise for the student

18
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?
700
400
300
1 yr
3
0
2
4
1
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
500
19

• Uneven Cash Flows (continued)
• FV Calculator Solution
• If your calculator has a NFV (Net Future Value)
  key, youre in luck! (TI BA II Plus Professional
  has this function)
• If theres no NFV key, you have to compound each
  CF to the last (terminal) time period

• Fractional Time Periods
• Example Calculate the FV of 100 invested for 18
  months in a bank account that pays a quoted rate
  of 10, compounded annually.

FV ?
rnominal 10
rperiodic ?
m 1 n ?
2
1
0
18 mos
PV 100
T of years 18mos/12mos per yr 1.5 m
of discounting per year 1 n total of
periods m x T 1 x 1.5 1.5

• Numerical Solution
• 1) Find n 18 mos / 12 mos per period 1.5
  periods
• 2) 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
  periods
• 3) 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

20
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 years
• Example 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

21

• Continuous Compounding (continued)
• EAR with continuous compounding
• Example 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 annuity
• The uniform payments go on indefinitely

PMT
r ?
0
8
1
2
3
4
5
PV

PV


PVperpetuity PMT / (r/m) PMT /
rperiodic Example What is the PV of a
perpetuity that pays 500 per year _at_ 8 APR? PV
PMT / rperiodic 500/0.08 6,250.00
22

• Perpetuities (continued)
• Example An endowment is established with an
  initial deposit of 1m. How much can be drawn
  out each month _at_ 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 perpetuities
• We 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?

23

• 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 term
• most 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 _at_ 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?

PV 16,785
isimple 8 iperiodic ?
1
3
years
0
T 3 m 4 n 3x4 12
compounding periods
12
11
10
1
2
3
4
PMT ?

• 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

24
Payments Dont 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?
FV ?
r 3.7
2
years
0
5
0
1
2
3
4
months
22
23
24
PMT 400
T of years 2 m of payments per year
12 n 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

25
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.
FV ?
r 5.00
2
years
0
5
0
1
2
3
4
compounding periods
22
23
24
PMT ?
T of years 2 m of compounding per year
12 n 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)
  12.50
• 2) 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

26
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?
50k
50k
50k
50k
r1-2 7 APR r3-4 5 APR
PV ?
m 1 n 4
Solution Approach divide the problem into two
parts
PV2 ?
PV0 ?
m 1 n 4
50k
50k
50k
50k
r1-2 7 APR r3-4 5 APR
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