FINANCE MBI
Session 1/2
 Time Value of Money
( chapter 4 )
 Bonds
( chapter 5 )
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
1
VALUATION OF CASH FLOWS
1.
2.
3.
4.
5.
6.
7.
Future value
Present value
Perpetuity value
Annuity values
Inflation
Discrete interest rates
Continuous rate of return
-----------------------------------------------You start with € 1000 at t = 0
Annual interest rate
6%
After 1 year this is:
1000  (1 + 0,06) = 1060
After t years
1000  (1  0,06)
t
Assumption: no intermediate cash
withdrawals.
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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So in the next period you also earn interest
on previous interest payments
This is called Compound Interest. Applying
compound interest generates Future Values.
Example
We have arranged a loan
 amount € 4 mln (principal)
 to be paid back after 6 years
 interest rate 7 % annually
 no intermediate payments of interest
Final payment will be:
4,000.000* (1  0.07)6
= € 6.002,921
 Side comment: Do you think a bank
would like such an arrangement?
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Future Value and Present Value factors
can be found in:
Tables
A.1
A.2
(p. 684)
(p. 686)
Better: just calculate it yourself
Present value
= a future cash flow calculated backwards
to get a value today ( at t = 0 ).
Example:
Worth today
A cash flow
as:
Annual interest rate 6 %
Future cash flow € 1000
Timing one year hence
1000
(1  0.06)
CF
t
= € 943.40
will be valued today
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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CFt
PV = (1  r ) t
Note: r will be presented in a decimal
notation. So 6 % interest per period is
r = 0.06 and t the number of periods in the
formula.
In 2013 we expect to have a cash flow of
€ 3 mln. Today (6 years earlier) this will be
valued with 6 % interest rate as:
3
PV = (1  0.06) 6 = € 2.114,882
1
The expression (1  r) t
is called the
discount factor for interest rate r and time t.
book: example 4.3 (page 80)
We have an IOU  zero coupon bond
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Suppose r = 8.53 %,
25 years
It pays $ 1000 after 25 years. Today´s
value will be, rounded to the dollar:
1000
PV = (1.0853) 25 = $ 129
The other way around:
Assume you pay today $ 150 in stead.
After 25 years you will receive $ 1000.
What is the implied interest rate?
1000
150 = (1  r ) 25
(1  r ) 25 = 6.6667
(1 + r) = (6.6667)
1
25
or 6.66670.04
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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hence 1 + r = 1.0788…
Annual interest rate is 7.88 %
For a series of cash flows we have a similar
approach:
CF2
CFt
CF1
PV = 1  r + (1  r ) 2 + … + (1  r ) t
A perpetual bond (or perpetuity) is a loan
with a principal that never will be paid back.
Interest payments will continue forever.
 Assume the annual interest payment
equals € 80
 What is a fair price for this bond?
Just suppose the bond is traded on the
market for € 900 (So this is the present
value!)
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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The return on investment in such a case will
be
r
80
900

 0.0889
So, the market is happy with a (permanent)
interest rate of 8.89%.
Now let´s think the market rate of return
changes to 10%
We then get:
PV

80
0.10
 € 800
This implies that the market price for the
bond will decrease with
€ 900  € 800 = € 100 (loss)
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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General conclusion:
Changes is the rate of return, required in a
financial market, directly influences the
value of the assets in that market:
Required rate up: value down, and
Required rate down: value up.
(… but of course, the market interest rate is
NOT the only factor that determines value )
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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ANNUITIES.
You negotiate today a loan with a principal
A.
There will be t annual payments C, starting
one year hence.
After t years everything wil be payed back,
including all interest due. This implies that
the following must be valid:
C
C
C
A

 ... 
2
1  r (1  r )
(1  r ) t
just work with the formula….
C
(1  r ) A  C 
1 r
C
 ... 
(1  r ) (t 1)
tric: Take line 2 minus line 1
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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C
C
(1  r ) A  C 
 ... 
1 r
(1  r ) t 1
A
C
C
C
 ... 

t 1
1 r
(1  r )
(1  r )t
The difference will be:
C
rAC
(1  r )t

1
r A  C 1 
t
(
1

r
)




1
1 

A  C  
t 
 r r (1  r ) 
A: total amount borrowed, present value.
C: annual payment
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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What you read between the brackets is
sometimes called the annuity present value
factor.
This relates the annual payments to the
current value of the principal.
( It is a bit of mathematics, but you can read the formula as:
the present value of a perpetuity C minus the present value
of a perpetuity C, starting at t+1 )
Example
Let us assume that you borrow € 4 mln to be
paid back in 6 equal annual payments.
Interest rate = 7 % per year.
Let us try to find the annuity factor
1
1
1
1




t
6
r r (1  r )
0.07 0.07(1.07)
14.2857 - 9.5192 = 4.7665
hence A = C x 4.7665…
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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A
4
C

 € 839,183
4.7665 4.7665
So: the loan of 4 million will be fully
reimbursed, including interest due, by 6
equal annual payments of 839,183 each,
starting one year hence.
Alternatively, the value of a series of cash
flows could be determined at the end of that
series, at time t.
( mathematically: multiply the annuity-present-value-factor with the
compound interest factor for t periods and simplify.)
The resulting future value factor of an
annuity is
( 1 + r )t - 1
r
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Example.
A – piteous – participant of the “ Legio
Lease investment scheme “ joined in
January 1998, agreeing to pay Dfl 200
monthly for a period of 5 years. Normally
he/she could earn 0.5 % ( half-a percent ! )
per month in a savings account. On January
1st, 2003 his/her Legio investment turned out
to be worthless. Determine his/her loss at
that day.
The proper future value factor is:
{(1.005)60 – 1} / 0.005 = 69.77……
The loss, measured at the start of 2003, is:
200 * 69.77… = Dfl. 13,954
What a shame!
( In practice more has been lost by these participants, as the market value of
their investment in stock at the time of selling off was lower than their debt to the
lease-banker, thus ending negatively as well! )
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Notice that an annuity series actually could
start immediately in stead of at the end of the
first period.
If this is the case, it is indicated as an
Immediate Annuity or Annuity due,
compared with the ordinary or regular
annuity.
Computational rule:
Given a annuity of n cash flows and an
interest rate of r, multiply the annuity future
value factor and the annuity present value
factor with ( 1 + r ) in order to obtain the
factors for an annuity due.
Question:
Can you intuitively explain, why the PV and
FV of an annuity due is higher than these of
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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an ordinary annuity of an equal size and
time ?
Application: value of bonds.
For the debt financing it is common to make
a distinction between two categories:
- Long term
- Short term
(arbitrary: 1 year will determine the
distinction between long and short)
For the long term we focus on bonds.
Characteristics:
* Nominal value (par value or face value)
* Coupon payments
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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NL: Coupons once a year
USA: semi-annual coupons
* There is an end date at which the
borrower will pay back the principal.
(Maturity date)
We focus on publicly traded bonds.
Issued by:
National government
Local goverment
Utility companies
Corporations
Others
How are they traded?
 We start with the issue of the bonds to
the general public. This is usually
organised by banks
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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 Later the secundairy market
 Sometimes OTC-issues
What determines the value of a bond ?
Prices will depend on the market interest
rate and default risk.
At issue the interest rate usually will be
market conform.
Example:
A 6% bond issue is announced. If the market
rate is slightly under 6%, e.g. 5,9%, the loan
will be issued at a price above 100%.
Since the coupon rate remains constant, the
value of the loan will move up and down,
inversely related to the market interest rate.
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Example:
Outstanding a 6% bond with an annual
coupon. Remaining time to maturity 3 years.
Cash flows 60, 60, 1060.
Suppose the market rate today is 4%.
60
60
1060
PV 


2
1.04 (1.04) (1.04)3
PV = 57.69 + 55.47 + 942.34 =1055.50
So, today we have to pay 55.50 more than
the final repayment after three years!
The 4% is called YIELD-TO-MATURITY
(YtM). This yield equals the current market
price to the expected cash flows over time.
We also have Current Yield, that is the short
term return, that can be calculated taking the
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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coupon payment relative to the market price
of the bond.
Coupon payment
CurrentYield 
Bond price
In the example we would get:
60
return 
 0.0568  5.68%
1055.50
This gives us a confusing picture since the
price of the bond will go down….(at a
market rate of 4%).
Total return = Coupon + Capital gains
What will be the bond price one year later?
Assume the market interest rate does not
change:
60 1060
PV 

 57.69  980.03  1037.72
2
1.04 (1.04)
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Hence the Capital gain will be:
1037.72 – 1055.50 = – 17.78 ! A loss.
Just check:
Coupon + Cap. Gain = 60 – 17.78 = 42.22
????
We invested 1055.50.
42.22
 0.04  4%
1055.50
Exactly right!
An interesting issue is the sensitivity of
bond prices to changes of the market interest
rate. Long term maturities tend to show a
higher sensitivity.
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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Next week more on the characteristics of an
interest rate and its relationship with
inflation.
MBI-5 2007-2008. Financial Management, part finance. Lecture 1/2. 09-06/11-2007.
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