MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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 (< 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.)
1
MGT 326 Ch. 4: Time Value of Money (bdh), Part 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%
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) = $30.00
These two steps can be combined:
PMT = Principle(Interest2 Rate / m)
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
3
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
Semiannual & Other Compounding Periods (continued)
Semiannual Case:
1
r = 8%
0
FV = ?
1
100
2
3
2
years
4
compounding periods
T = # of years = 2
m = # of discounting per year = 2
n = 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
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 = 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
4
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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:
FV = ?
1
r = 8%
0
1
2
3
4
5
$100
6
7
2
years
8
compounding periods
T = # of years = 2
m = # of discounting per year = 4
n = total # of periods = m x T = 4 x 2 = 8
Formula:
Financial Calculator:
5
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
Semiannual & Other Compounding Periods (continued)
Example: (continued)
Monthly Case:
FV = ?
r = 8%
2
0
0
1
2
3
$100
4
5
years
compounding periods
24
22
23
T = # of years = 2
m = # of discounting per year = 12
n = total # of periods = m x T = 12 x 2 = 24
Formula:
Financial Calculator:
6
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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?
rsimple = 4%
rperiodic = ?
0
50
50
50
50
1
2
3
4
50
50
50
50
1
2
3
4
PV = ?
0
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/(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
7
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
Semiannual & Other Compounding Periods (continued)
Example (continued):
rsimple = 4%
rperiodic = ?
0
50
50
50
50
1
2
3
4
PV = ?
T = # of years = 2
m = # of discounting per year = 2
n = total # of periods = m x T = 2 x 2 = 4
Financial Calculator:
8
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
m=4
n=8
100
100
100
100
100
100
100
2
3
4
5
6
7
8
r=?
0
1
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:
Note: One of the two cash inputs
Enter N [8, N]
Enter PV [- 765.17, PV] must be negative
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:
Note: One of the two cash inputs
Enter N [8, N]
Enter PV [- 765.17, PV] must be negative
Enter Pmt [100, PMT]
Find I/Y, [CPT, I/Y] and voila! I/Y = 3.9997% = 4%
9
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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 @ 8% compounded annually, 2 yrs: $116.64
FV of $100 @ 8% compounded semiannually, 2 yrs : $116.98
FV of $100 @ 8% compounded quarterly, 2 yrs : $117.17
FV 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 - 1
m = 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
10
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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: $116.64
EAR = 8%
Note: EAR for annual compounding = rnominal
FV of $100 @ 8% compounded semiannually, 2 yrs : $116.98
EAR = 8.16%
FV of $100 @ 8% compounded quarterly, 2 yrs : $117.17
EAR = 8.24%
FV of $100 @ 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.
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
11
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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%
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 = $20.47
ROR = $20.47 / $400 = 0.0512 = 5.1162% = EAR
12
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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%
13
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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%
0
1
2
3
4
5
6
7
8
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 @ 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 @ 5% equal one annual payment of $407.56
407.56
407.56
The CF diagram now looks like this:
1
rnominal = 5%
0
reffective = ?
1
2
3
4
2 years
5
6
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! FV14= -$756.81
7
8 qtrs
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
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.
r (APR)
# of
Com pounding
Periods per
Year
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%
15
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
16
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
Uneven Cash Flows
$500
m=1
$300
$250
0
r = 4%
1
$200
2
3
PV = ?
4
5
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
17
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
300
400
1 yr
0
1
2
3
4
Formula Solution:
500
1
2
PV = CF1/(1 + r/m) + CF2/(1 + r/m) + 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
18
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
Uneven Cash Flows (continued)
FV Calculator Solution
If 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 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=?
0
PV = $100
1
2
18 mos
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
19
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
20
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
1
2
3
4
5
∞
PV
PV =
PMT
+
(1 + r/m)1
S
∞
=
PMT
PMT
+
……+
(1 + r/m)2
(1 + r/m)∞
PMT
(1 + r/m)n
n=1
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
21
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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 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?
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
22
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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 @ 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
0
T=3
m=4
n = 3x4 =12
interest
principle
isimple = 8%
iperiodic = ?
1
1
2
3
4
3
10
11
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
23
years
12 compounding
periods
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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?
FV = ?
r = 3.7%
0
0
1
2
3
4
5
22
PMT = $400
23
2
years
24
months
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
24
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
0
0
1
2
3
4
5
22
PMT = ?
23
24
years
compounding periods
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
25
MGT 326 Ch. 4: Time Value of Money (bdh), Part 2
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
1
2
3
4
r1-2 = 7% APR
r3-4 = 5% APR
PV = ?
0
m=1
n=4
Solution:
Approach: divide the problem into two parts
PV0 = ?
m=1
n=4
PV2 = ?
50k
50k
50k
50k
1
2
3
4
r1-2 = 7% APR
r3-4 = 5% APR
0
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
26