Post on 24-Dec-2015
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Lecture 7: Measuring interest rate
Mishkin chapter 4 – part A
Page 67-78
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Future value Deposit $1 in bank, annual interest rate
i=0.1, how much you would get after 1 year? 2 years? … n years?
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1 (1 0.1) $1.1
1 (1 0.1) (1 0.1) 1 (1 0.1) $1.21
$1 has a future value of $[1 (1 ) ] in year nni
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Present value – discounting $1
1.33
1 2 3 n
1.211.1 1*(1 + .1)n
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(1 .1)n
1
2 3
1.1 1.21 1.331 ...
1 0.1 (1 0.1) (1 0.1)
$1 received after n years has a present value of:
1
(1 )ni
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Present value - meaning
A dollar paid to you one year from now is less valuable than a dollar paid to you today.
Present value of $1 is the minimum number of dollars that you would have to give up today in return for receiving $1 in year n.
Why? impatient forgone interest
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Present value – discounting a future cash flow
n
PV = today's (present) value
CF = future cash flow (payment)
= the interest rate
CFPV =
(1 + )
i
i
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Present value – discounting multiple future cash flows
2 3
100 150 200
(1 .1) (1 .1) (1 .1)PV
$100 $150 $200
Start date Maturity date
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Example
Your bank offers you a CD with 3% interest rate for five year investment. You wish to invest $1500 for five years.
How much your investment will be worth then? Known current value and need to calculate
future value.
5$1500 (1 0.03) $1738.91
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Example
You have been offered $40,000 to sell your printing business, payable in two years. Suppose the market interest rate is 8%. How much is the offer worth to you today?
(In other words, what is the offer’s present value?)
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$40,000$34,293.6
(1 0.08)
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Compare debt instruments
1. Simple loan2. Fixed payment loan3. Coupon bond4. Discount bond
Difference: repayment schemes. Calculate yield to maturity for each.
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Yield to maturity (YTM)
Yield to maturity (YTM) is the interest rate (i) that equates the present value of cash flow payments received from a debt instrument with its value today.
the most accurate measure of interest rates.
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Simple loan Payment scheme:
pay the loan value (LV) together with an interest payment (I) on the maturity date.
Time line:
Loan value (LV)
Loan value (LV)
+ Interest payment (I)
borrower receives
Lender receives
maturity date
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Simple loan - YTM
LV = loan value
I = Interest payment
n = years to mature(1 )n
LV ILV
i
Example:
borrow a simple loan of $100, interest rate is 0.1, or, need to pay interest of $10, mature in one year. What’s the yield to maturity?
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Example – Cont’d
1
1
$100
$10
1
(1 )
100 10100
(1 )
100 (1 ) 110
.1
LV
I
n
LV ILV
i
i
i
i
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Fixed payment loan Payment scheme:
makes periodic fixed payments to the lender until a specified maturity date.
These periodic fixed payments include both principal (loan value) and interest, so at maturity there is no lump-sum repayment of principal.
example: home mortgage
Loan value (LV)
Fixed payments (FP) FP FP
borrower receives
Lender receives
maturity date
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Fixed payment loan - YTM
2 3
The same cash flow payment every period throughout
the life of the loan
LV = loan value
FP = fixed yearly payment
= number of years until maturity
FP FP FP FPLV = . . . +
1 + (1 + ) (1 + ) (1 + )n
n
i i i i
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Example Consider a particular fixed-payment loan contract
with a loan value $5000, annual fixed payments $660.72, and a maturity of 20 years. What is the yield to maturity for this loan contract?
Answer: Yield to maturity for this fixed-payment loan contract is 0.12 or 12 percent.
2 20
2 20
FP FP FPPV = + +...+
(1+i) (1+i) (1+i)
600.72 600.72 600.725000= + +...+
(1+i) (1+i) (1+i)
0.12i
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Coupon bond Payment scheme:
(1)pay a fixed amount of funds (the coupon
payment) periodically; (2) pay the face value
(or par value) of the bond on maturity date. purchase price may not equal face value
Purchase price (P)
Coupon payments (C) C
Coupon payments (C)
+ Face value (F)
borrower receives
Lender receives
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Coupon bond - YTM
2 3
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
= years to maturity date
C C C C FP = . . . +
1+ (1+ ) (1+ ) (1+ ) (1n
n
i i i i
+ )ni
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Example Consider a coupon bond whose purchase price
is $94, face value is $100, whose coupon payment is $10 and maturity is 10 years. What’s yield to maturity i?
Cash flow is: ( $10, $10, $10, $10, $10, $10, $10, $10, $10, [$10 + $100] ).
2 9 10
10 10 10 (10+100)94 = + + ... + +
(1+i) (1+i) (1+i) (1+i)
0.11i
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1.When bond is at par (sold at face value), yield to maturity equals coupon rate.
2.Price and yield to maturity are negatively related.
Bond price and YTM
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Coupon rate and YTM
2 3...
(1 ) (1 ) (1 ) (1 ) (1 )n n
C C C C FP
i i i i i
Other things unchanged:
1. P increase i decrease.
2. C increase i increase. Coupon rate or coupon payment is positively related to yield to maturity.
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Discount bond Payment scheme:
on maturity date, pay the face value (F) Sold at a discount: price < face value Time line:
Purchase price (P)
Face Value (F)
borrower receives
Lender receives
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Discount bond - YTM
For any one year discount bond
i = F - P
PF = 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.
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Example
A discount bond selling for $15,000 with a face value of $20,000 in one year has a yield to maturity of _____?
2000015000 =
(1+i)
0.333i
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Recap
Present value Yield to maturity
definition
Calculate yield to maturity for: Simple Loan Fixed Payment Loan Coupon Bond Discount Bond