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Solutions to week 5 Tutorial Questions
*************** Part I: Duration Models **************
1.
What are the two different general interpretations of the concept of duration, and what is
the technical definition of this term? How does duration differ from maturity?
Duration measures the weightedaverage life of an asset or liability in economic terms. As such,
duration has economic meaning as the interest sensitivity (or interest elasticity) of an asset’s
value to changes in the interest rate. Duration differs from maturity as a measure of interest rate
sensitivity because duration takes into account the time of arrival and the rate of reinvestment of
all cash flows during the assets life. Technically, duration is the weightedaverage time to
maturity using the relative present values of the cash flows as the weights.
2. A oneyear, $100,000 loan carries a coupon rate and a market interest rate of 12 percent.
The
loan requires payment of accrued interest and onehalf of the principal at the end of six months.
The remaining principal and accrued interest are due at the end of the year.
a.
What will be the cash flows at the end of six months and at the end of the year?
CF
1/2
= ($100,000 x .12 x ½) + $50,000 = $56,000 interest and principal.
CF
1
= ($50,000 x .12 x ½) + $50,000 = $53,000 interest and principal.
b. What is the present value of each cash flow discounted at the market rate?
What is the
total present value?
PV of CF
1/2
= $56,000
1.06
=
$52,830.19
PV of CF
1
=
$53,000
(1.06)
2
=
47,169.81
PV Total CF
= $100,000.00
c. What proportion of the total present value of cash flows occurs at the end of 6 months?
What proportion occurs at the end of the year?
X
1/2
= $52,830.19
$100,000 = .5283 = 52.83%
X
1
= $47,169.81
$100,000 = .4717 = 47.17%
d. What is the duration of this loan?
Duration = .5283(1/2) + .4717(1) = .7358
OR
t
CF
PVof CF
PV of CF x t
½
$56,000
$52,830.19
$26,415.09
1
53,000
47,169.81
47,169.81
$100,000.00
$73,584.91
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View Full DocumentDuration = $73,584.91/$100,000.00 = 0.7358 years
3. A sixyear, $10,000 CD pays 6 percent interest annually and has a 6 percent yield to
maturity. What is the duration of the CD? What would be the duration if interest were paid
semiannually? What is the relationship of duration to the relative frequency of interest
payments? [Hint: In order to save time, set up a spread sheet to automate some
computations.]
Sixyear CD:
Par value = $10,000
Coupon rate = 6%
R = 6%
Maturity = 6 years
Annual payments
t
CF
PV of CF
PV of CF x t
1
$600
$566.04
$566.04
2
$600
$534.00
$1,068.00
3
$600
$503.77
$1,511.31
4
$600
$475.26
$1,901.02
5
$600
$448.35
$2,241.77
6
$10,600
$7.472.58
$44,835.49
$10,000.00
$52,123.64
Duration = $52,123.64/$1,000.00 = 5.2124
R = 6%
Maturity = 6 years
Semiannual payments
t
CF
PV of CF
PV of CF x t
0.5
$300
$291.26
$145.63
1
$300
$282.78
$282.78
1.5
$300
$274.54
$411.81
2
$300
$266.55
$533.09
2.5
$300
$258.78
$646.96
3
$300
$251.25
$753.74
3.5
$300
$243.93
$853.75
4
$300
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 Three '09
 YIP

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