Chapter 4
Understanding Interest Rates
1
Present Value
One lira paid to you one year from now
is less valuable than one lira paid to
you today. Even if we assume inflation
zero, people prefer immediate money.
People are impatient.
 This is called time value of money.

2
Present Value



If the government promises to pay you 100 liras
one year later, or alternatively pay you 100 liras
right now, which one would you prefer?
What If the government promises to pay you 115
TL one year later, or alternatively pay you 100 TL
right now, which one would you prefer?
What is the present value of 115 TL one year
later for you? In other words, how many liras you
receive today would make you equally happy with
115 TL received one year later? (assume infl. is
zero)
3
Present Value

If your answer is 100 TL today, then your
discount rate is 15%:
115
PresentValue 
 100TL
(1  0.15)

The present value of 115 liras one year later is
equal to 100 liras now. Notice that the present
value decreases as the discount (interest) rate
increases. If discount rate was 20%, then PV of
115 liras next year would be _____liras now.
4
Present Value

If we think of a longer time period such as two
years, how much money two years later would
be equivalent to 100 liras now? With 15%
annual discount=interest rate,
FutureCash Flow
100 
(1  0.15) 2

Cash Flow is the two-years-later value (or the
repayment amount) of 100 TL (loan) today. We
find
CF = _132.25___ liras.
5
Present Value
Today
Year 1
Year 2
Year n
100 x 1  0.15
n
100
115
132.25
6
Simple Present Value
CF
PV 
where
n
1  i 
PV  present value
CF  fut ure cash flow payment
i  annualint erestrat e
n  number of years
7
Simple Present Value
İf there is more than one payment in the
future, then we add the present value of
each payment.
 For example, if there are two payments:
1000 TL one year later and 1500 TL three
years later:

PV=(1000/(1+i))+(1500/(1+i)3)
8
Four Types
of Credit Market Instruments

Simple Loan
 Lender provides the borrower the loan amount
at the present. Borrower must repay the
principal + interest payment when the
maturity date comes.

Ex: 100 TL loan at 50% simple interest rate,
1 yr. maturity. Principal: 100 TL, interest
payment: 50 TL.
9
Four Types
of Credit Market Instruments

Fixed Payment Loan

Repayment of a certain loan is divided into
equal payments. Ex: You borrowed 100,000
TL now to buy a house. If the interest rate is
10%, you will repay this loan by paying a fixed
amount of 11,017 TL every year for the
maturity of 25 years. Ex: Mortgages, car loans.
11,017 11,017
11,017
100,000 

 .. 
2
(1  i) (1  i)
(1  i) 25
10
Four Types
of Credit Market Instruments

Coupon Bond

Issuer of this bond makes fixed interest (coupon)
payments to the holder every year until the
maturity date. For example, assume you bought a
coupon bond with 1000 TL face value, 50%
coupon rate, annual coupon payments and 5
years maturity issued by the Treasury. The
treasury will pay you 500 TL every year and will
pay you the 1000 TL five years later.
500
500
500
500
1500
PV 




2
3
4
(1  i) (1  i) (1  i) (1  i) (1  i)5
11
Four Types
of Credit Market Instruments
Discount Bond(zero-coupon bond)



The issuer pays the holder the face value at
the maturity date. No coupon payments. The
current price of the bond is smaller than the
face value.
Ex: Treasury discount bond with 1000 TL
face value and one year maturity is currently
sold for 900 TL. then the interest rate is
(1000-900) / 900 = 11.1 %
12
Yield to Maturity
To compare different instruments, we need
a standard measure of the interest rate.
 The interest rate that equates the
present value of future cash flow payments
with the value of the loan today is called
yield to maturity (YTM).
 Assumption is that the security is held until
maturity and payments are done on time.
We know future cash flows and present
value; we calculate the annual interest
rate which is YTM.

13
Simple Loan—Yield to Maturity
If one borrows 100 TL credit for one year,
and repays 118 TL at the end of the year,
YTM of this credit is
= (118-100)/100 = 0.18, i.e. 18%

14
Fixed Payment Loan—
Yield to Maturity
The same cash flow payment every period throughout
the life of the loan
LV = loan value
FP = fixed yearly payment
n = number of years until maturity
FP
FP
FP
FP
LV =


 ...+
2
3
n
1 + i (1 + i) (1 + i)
(1 + i)
15
Fixed Payment Loan—
Yield to Maturity


You take out a 100,000 TL house mortgage with
an annual interest rate (YTM) of 20% to be paid
in 20 years. What is your payment every year?
Using a financial calculator, we find annual
payments (FP) as 20,536 TL. Total amount paid
in 20 years: 410,720 TL. If the interest rate
(YTM) increases, annual payments also increase
(keeping maturity and loan value constant). For
example, if interest rate was 10%, then we would
find annual payments as FP = 11,746 TL.
16
Coupon Bond—Yield to Maturity
Using the same strategy used for the fixed-payment loan:
P = price of coupon bond
C = yearly coupon payment
F = face value of the bond
n = years to maturity date
C
C
C
C
F
P=


. . . +

2
3
n
n
1+i (1+i) (1+i)
(1+i) (1+i)
17
Coupon Bond—Yield to Maturity
For a treasury coupon bond with 1000 YTL
face value, ten years maturity and 100 TL
coupon payments every year: If the yield
to maturity is 10%, what is the current
price of the bond?
 The current price =present value is 1000
TL.

100
100
1100
1000

 ... 
2
10
(1  0.10) (1  0.10)
(1  0.10)
18



If the current price of the coupon bond is equal to its
nominal value, its yield to maturity equals the coupon
rate. In other words, if the YTM is equal to the coupon
rate, then its current price is equal to its nominal value.
The price of a coupon bond and the yield to maturity are
inversely related
If yield to maturity is greater than the coupon rate then
the current price is below its face value. If YTM=20%,
then current price = $581. (sold at a discount)
19
Consol or Perpetuity

A bond with no maturity date. A consol never repays the
principal but pays fixed coupon payments forever. But you
can sell it anytime.
Pc  C / ic
Pc  price of the consol
C  yearly interest payment
ic  yield to maturity of the consol
Can rewrite above equation as ic  C / Pc
For coupon bonds, this equation gives current yield—
an easy-to-calculate approximation of yield to maturity
20
Consols


The same consol formula can be used to
approximately calculate the YTM or FP of a
long-term coupon bond or a fixed payment loan.
For example, consider a coupon bond with 1000
TL nominal value, 10% coupon rate, 10 years
maturity and present value of 1200 TL. Then YTM
is ~ 8.3 % (exact 7.13%).
Or a fixed payment credit of 100,000 YTL with 20
years maturity has annual payments equal to
20,536 YTL. YTM of this credit is approx. 20.5 %
(exact 20%).
21
Discount Bond—Yield to
Maturity
For any one year discount bond
F-P
i=
P
F = Face value of the discount bond
P = current price of the discount bond
The yield to maturity equals the increase
in price over the year divided by the initial price.
As with a coupon bond, the yield to maturity is
negatively related to the current bond price.
22
Discount Bond—Yield to
Maturity


If a discount bond issued by the treasury
has a face value of 1000 TL, maturity of one
year, and is sold at the price of 900 TL, what
is its yield to maturity?
Answer:
1000  900 100
i

 0.111  11.1%
900
900
23
Rate of Return

In case one does not hold an asset until maturity, rate
of return differs from YTM.

Consider a treasury coupon bond with 20% yield to
maturity (interest rate), 1000 TL face value, 200 TL
coupon payments and two years maturity (initial
price=1000) . One year later, a crisis breaks out and
the interest rate rises to 40%. If you need money and
sell the bond, price of the bond will be 857 TL
(=1200/(1.4)). (If the int. rate did not increase, price
would be still 1200/1.2 = 1000 TL)

Your rate of return is :
24
Rate of Return
Your rate of return is :
200/1000 + (857 - 1000)/ 1000 = 20 - 14.3
= 5.7 % much less than 20%. The
bondholder faces interest rate risk.

25
Rate of Return

More generally, rate of return R for holding a
bond from time t to time t+1 is
C Pt 1  Pt
R 
where
Pt
Pt
Pt is theprice of thebond at t imet
Pt 1 is theprice of thebond at t imet  1
C is coupon payment
Pt 1  Pt
C
is current yield and
is capitalgain
Pt
Pt
26
Rate of Return
and Interest Rates

Rate of return is equal to yield to maturity only
if you hold the bond until the maturity date.

A rise in interest rates causes a fall in bond
prices, and results in a capital loss if the asset
is sold before the time to maturity.

If the bond has a longer term (maturity),
there is a greater risk of capital loss because of
a given interest-rate change. See Table 2. So,
usually long term bonds have higher int. Rates
to compensate for the higher interest rate risk.
27
28
Interest-Rate Risk

Prices and returns for longer-term
bonds are more volatile than those for
shorter-term bonds. There is more interest rate
risk for long term bonds. This is why Turkish
Treasury had a hard time selling long term
bonds.

There is no interest-rate risk for any bond if the
owner holds it until the end of maturity. In that
case, YTM= rate of return and YTM is known
when the bond is purchased.
29
Real and Nominal Interest Rates

Nominal interest rate is the rate quoted by
banks. It is not adjusted for inflation.

Real interest rate is adjusted for inflation so it
more accurately reflects the cost of borrowing

“Ex ante” real interest rate is calculated from
expected rate of inflation at the beginning of the
year.

“Ex post” real interest rate is calculated from
realized inflation at the end of the year.
30
Fisher Equation
i  ir   e
i = nominal interest rate
ir = real interest rate
 e = expected inflation rate
When the real interest rate is low,
there are greater incentives to borrow and fewer incentives to lend.
The real interest rate is a better indicator of the incentives to
borrow and lend.
31
Real Interest Rates in Turkey
110
100
90
80
70
60
50
40
30
20
10
0
-10 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
-20
-30
-40
NomIntRate
RealIntRate
32
Real Interest Rates in USA
33