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ACC 471
Practice Problem Set #2
Fall 2002
Suggested Solutions
1. Text Problems:
11-6
a.
i. Current yield: 70 960 7 29%.
ii. Yield to maturity: solving
960
1 1 y y
11 000
y
10
35
10
for y gives a yield to maturity of 4% semi-annually (or 8% annual bond equivalent yield).
iii. Start by calculating the future value of receiving $35 every 6 months for the next three years:
FV
1 03 06 1 6
35
$226 39
Three years from now, the bond will be selling at par (since the coupon is equal to the forecasted yield to maturity). Therefore total proceeds in three years will be $1,226.39. The realized
compound yield (on a semi-annual basis) is therefore:
$960
b.
1 y 6
$1 226 39
y
4 166% or 8.332% annual bond equivalent yield.
i. Current yield does not account for capital gains or losses on bonds bought at prices other than par
value. It also does not account for reinvestment income on coupon payments.
ii. Yield to maturity assumes that the bond is held until maturity and that all coupon income can be
reinvested at a rate equal to the yield to maturity.
iii. Realized compound yield is affected by the forecast of reinvestment rates, holding period, and
yield of the bond at the end of the investor’s holding period.
11-13 The reported bond price is 100.02 percent of par, or $1,000.20. However, 15 days have passed since the
last semi-annual coupon was paid, so accrued interest equals $35 15 182
$2 885. The invoice price
is the reported price plus accrued interest, or $1,0003.085.
11-17 The price schedule is:
Year
0
1
2
...
19
20
Remaining Maturity (T )
20
19
18
...
1
0
Constant yield value (1000 1 08T )
214.55
231.71
250.25
...
925.93
1,000.00
Imputed interest
n.a.
17.16
18.54
...
n.a.
74.07
Note that the imputed interest is simply the increase in the constant yield value.
11-19
a. The bond price is:
P0
$40
035 60 $1 000 1 1 035
1 035
$40
1 1 y y 1$1 100
y Yield to call:
$1 124 72
$1 124 72
60
10
10
1
y
3 368% b. In this case yield to call is:
$1 124 72
1 1 y y 1$1 050
y 10
$40
10
y
2 976% c. Now yield to call is:
$1 124 72
1 1y y 1$1 100
y 4
$40
y
4
3 031% 11-20 The promised yield to maturity is:
$900
1 1 y y 1$1 y000
y
y
8 526% 10
$140
10
16 075%
However, the expected yield to maturity is:
$900
1 1 y y 1$1 y000
10
$70
10
11-22 Zero coupon bonds provide no coupons to be reinvested. Therefore, the investor’s proceeds from the
bond are independent of the rate at which coupons could be reinvested (if they were paid). There is no
reinvestment rate uncertainty with zeros.
11-25 Factors which might make the ABC debt more attractive to investors, therefore justifying a lower coupon
rate and yield to maturity are:
The ABC debt is a larger issue and thus may sell with more liquidity.
An option to extend the term from 10 years to 20 years is favorable if interest rates in 10 years are
lower than today. In contrast, if interest rates increase, the investor can present the bond for payment
and reinvest the money for better returns.
In the event of financial distress, the ABC debt is a more senior claim. It has more underlying security
in the form of a first claim against real property.
The call feature on the XYZ bonds makes the ABC bonds relatively more attractive since ABC bonds
cannot be called from the investor.
The XYZ bond has a sinking fund requiring XYZ to retire part of the issue each year. Since most
sinking funds give the firm the option to retire this amount at the lower of par or market value, the
sinking fund can work to the detriment of bondholders.
11-30
a. The call provision requires the firm to offer a higher coupon (or higher promised yield to maturity) on
the bond to compensate the investor for the firm’s option to call back the bond at a specified price if
interest rates fall sufficiently. Investors are willing to grant this option to issuers, but only for a price
that reflects the possibility that the bond will be called. That price is the higher promised yield at
which they are willing to buy the bond.
b. The call option will reduce the expected life of the bond. If interest rates fall substantially and the
likelihood of call increases, investors will begin to treat the bond as if it will “mature” and be paid off
at the call date, not at the stated maturity date. On the other hand, if interest rates rise, the bond must
be paid off at the maturity date, not later. This asymmetry means that the expected life of the bond
will be less than the stated maturity.
c. The advantage of a callable bond is the higher coupon (and a higher promised yield to maturity)
when the bond is issued. If the bond turns out not to be called, then one will earn a higher realized
compound yield on a callable bond than an otherwise identical but non-callable bond issued on the
same date. The disadvantage of the callable bond is the risk of call. If rates fall and the bond is called,
the investor will receive the call price and will have to reinvest the proceeds at rates lower than the
yield to maturity at which the bond was originally issued. In this event, the firm’s savings in interest
payments is the investor’s loss.
2
12-1
i. Expectations hypothesis: The yields on long-term bonds are geometric averages of present and expected future short rates. An upward-sloping yield curve is explained by expected future short rates
being higher than the current short rate. A downward-sloping curve implies expected future short
rates are lower than the current short rate. Thus bonds of different maturities have different yields if
expectations of future short rates are different from the current short rate.
ii. Liquidity preference hypothesis: Yields on long-term bonds are greater than the expected return from
rolling over short-term bonds in order to compensate investors in long-term bonds for bearing interest
rate risk. Thus bonds of different maturities can have different yields even if expected future short
rates are all equal to the current short rate. An upward-sloping yield curve can be consistent even with
expectations of falling short rates if liquidity premiums are high enough. If, however, the yield curve
is downward-sloping and liquidity premiums are assumed to be positive, then we can conclude that
future short rates are expected to be lower than the current short rate.
iii. Segmentation hypothesis: This hypothesis would explain a sloping yield curve as an imbalance between supply and demand for bonds of different maturities. An upward-sloping curve is evidence of
supply pressure in the long-term market and demand pressure in the short-term market. According to
the segmentation hypothesis, expectations of future rates have little to do with the shape of the yield
curve.
12-3 True. Under the expectations hypothesis, there are no risk premia built into bond prices. The only reason
for long-term yields to exceed short-term yields is an expectation of higher short-term rates in the future.
12-4 Uncertain. Lower inflation will usually lead to lower nominal interest rates, but if the liquidity premium is
sufficiently high, long-term yields may exceed short-term yields despite expectations of falling short rates.
12-5
Maturity
1
2
3
4
Price
$943.20
$898.47
$847.62
$792.16
YTM
Forward Rates
1 000 943 20 1 6 00%
n.a.
1 000 898 47 1 5 50% 1 0550 1 0600 1 5 00%
1 000 847 62 1 5 67% 1 0567 1 0550 1 6 000%
1 000 792 16 1 6 000% 1 06000 1 0567 1 7 000%
1 2
2
3
1 3
1 4
2
4
3
12-6 The expected price path of the 4 year zero coupon bond is as follows. Note that we discount the face value
by the appropriate sequence of forward rates implied by the current yield curve.
Start of Year
1
2
3
4
Expected Price
$792.16
$839.69 (1 000 1 05 1 06 1 07 )
$881.68 (1 000 1 06 1 07 )
$934.58 (1 000 1 07)
Expected Rate of Return
6.00% (839 69 792 16 1)
5.00% (881 68 938 69 1 )
6.00% (934 58 881 68 1 )
7.00% (1 000 934 58 1 )
12-8 You should expect it to lie about the curve since the bond must offer a premium to investors to compensate
them for the option granted to the issuer.
12-12
a. We have:
P
1 907 1109
08 2
$101 86
b. We have:
1 9 y 1109
y y 7 958% c. The forward rate for next year is f 1 08 1 07 1 9 01%. Thus the forecasted bond price is
P 109 1 0901 $99 99.
d. If the liquidity premium is 1%, then the forecasted interest rate is f 01 8 01%. Thus P 109 1 0801 $100 92.
$101 86
2
2
2
2
3
12-16
a. To calculate the 5 year spot rate:
1 70y 1 70y 1 70y 1 70y 11 070
y 70
70
70
70
1 070
1 05 1 0521 1 0605 1 0716 1 y 758 32 11 070
y y 7 13% Then the 5 year forward rate is 1 0713 1 0716 1 7 01%.
$1 000
1
2
2
2
5
3
3
3
4
4
4
5
5
5
5
5
5
5
4
b. Yield to maturity is the single discount rate that equates the present value of a series of cash flows to
a current price. It is the internal rate of return.
The spot rate for a given period is the yield to maturity on a zero coupon bond which matures at the
end of the period. A spot rate is the discount rate for each period. Spot rates are used to discount
each cash flow of a coupon bond to calculate a current price. Spot rates are the rates appropriate for
discounting future cash flows of different maturities.
A forward rate is the implicit rate that links any two spot rates. Forward rates are directly related
to spot rates, and therefore yield to maturity. Some would argue (as in the expectations theory) that
forward rates are market expectations of future interest rates. Regardless, a forward rate represents a
break-even rate that links two spot rates. It is important to note that forward rates link spot rates, not
yields to maturity.
Yield to maturity is not unique for any particular maturity. In other words, two different bonds with
the same maturity but different coupon rates may have different yields to maturity. In contrast, spot
and forward rates for each date are unique.
c. The 4 year spot rate is 7.16%. Thus, 7.16% is the expected yield to maturity for the zero coupon bond.
The price of the zero is $1 000 1 07164 $758 35.
12-17 The price of the coupon bond is:
120
1 120
$1 113 99
1 058 1 0582
If the coupons were stripped and sold as zeros, they could be sold separately for:
1 120
1 06 120
1 05
2
$1 111 08
The arbitrage strategy is to buy zeros with face values of $120 and $1,120 and respective maturities of one
and two years, and simultaneously sell the coupon bond. The profit equals $2.91 on each bond.
12-19
a. We have:
Maturity (Years)
Price
Yield to Maturity (Spot Rate)
Forward Rate
1
925.93
1 000 925 93 1 8 00%
n.a.
1
2
2
2
853.39
1 000 853 39
1 8 25%
1 0825 1 0800 1 8 50%
3
782.92
1 000 782 92 1 3 1 8 50% 1 08503 1 08252 1 9 00%
4
715.00
1 000 715 00 1 4 1 8 75% 1 08754 1 08503 1 9 50%
5
650.00
1 000 650 00 1 5 1 9 00% 1 09005 1 08754 1 10 00%
b. You can create the loan by selling 3 year zeros today and buying 4 year zeros. For each 3 year zero
sold, you can use the proceeds to buy 782 92 715 1 095 4 year zeros. Your cash flows are:
Time Cash Flow
0
0
Your purchase of 4 year zeros is financed by the
sale of 3 year zeros.
3
-$1,000
The 3 year zero that you sell matures and you pay
out the face value.
4
+$1,095
The 4 year zeros that you buy mature and you collect
the face value on each one.
4
This is a synthetic loan starting at time 3, with a rate of 9.5%, precisely the forward rate for year 4.
c. For each 4 year zero sold, you can buy 715 650 1 10 5 year zeros. Your cash flows are:
Time Cash Flow
0
0
Your purchase of 5 year zeros is financed by the
sale of 4 year zeros.
4
-$1,000
The 4 year zero that you sell matures and you pay
out the face value.
5
+$1,100
The 5 year zeros that you buy mature and you collect
the face value on each one.
This is a synthetic loan starting at time 4, with a rate of 10%, precisely the forward rate for year 5.
12-20 a. For each 3 year zero that you buy today, you need to sell 782 92 650 1 2045 5 year zeros to make
your initial cash flow equal to zero.
b. Your cash flows are:
Time Cash Flow
0
0
Your purchase of 3 year zeros is financed by the
sale of 5 year zeros.
3
+$1,000
The 3 year zero that you buy matures and you collect
the face value.
5
-$1,204.50 The 5 year zeros that you sell mature and you pay
out the face value on each one.
This is a synthetic two year loan, starting at time 3.
c. The two year forward rate on the forward loan is 1 204 50 1 000 1 20 45%.
d. The one year forward rates for years 4 and 5 and 9.5% and 10% respectively. Notice that 1 095
1 10 1 2045, which equals the two year forward rate on the three year ahead forward loan.
13-2 Recall that the duration of a coupon bond is:
1
y 1 y T c y y
c 1 y 1 y
T
where y is the bond’s yield per payment period, c is its coupon rate per payment period, and T is the
number of payment periods. Moreover, this can be simplified to
1
y
y
1
1 1y
T
for bonds selling at par. Therefore:
06
y 10
y
D
D
1 06
1
1
2 833 years
06
1 063
1 10 1 10 3 06 10
2 824 years
10
06 1 103 1
10
Note that we can also calculate duration directly, as follows. The price of the bond is its par value when
y 06, and it is
60
60
1 060
$900 53
2
1 10 1 10
1 103
when y 10. Then
y
06
y
10
1 21 000 00 3 2 833 years 1
2
3
D
900 53
2 824 years D
60
1 06
60
1 062
1060
1 063
60
1 10
60
1 102
1060
1 103
5
06, we have:
13-3 When y
03 1 1
1 03
1 03
5 58 half-years
2 79 years D
while when y
10, we have:
D
6
05 1 05 6 03 05
1 05
03 1 05 1 05
5 55 half-years
2 775 years 6
13-4
a. Bond B has a higher yield to maturity than Bond A since its coupons and time to maturity are the
same as for A, while its price is lower. Therefore, its duration must be shorter.
b. Bond A has a lower yield and a lower coupon, both of which cause it to have a longer duration than
Bond B. Morever, A cannot be called, which makes its maturity at least as long as that of B, which
generally increases duration.
13-5
a. The present value of the firm’s liabilities is:
10 000 000
11
000 4 000
1 1
5
$11 574 594 38
Therefore the duration of the liabilities is:
D
1 10 000 000
11
5 4 000 000
1 15
11 574 594 38
1 858 years Thus the required maturity of the zero coupon bond is 1.858 years.
b. The market value of the zero must be $11,574,594.38, the same as the market value of the obligations.
$13 817 413 73.
Therefore, the face value must be 11 574 594 38 1 11 858
13-6
a. The call feature provides a valuable option to the issuer, since it can buy back the bond at a given
call price even if the present value of the scheduled remaining payments exceeds the call price. The
investor will demand, and the issuer will be willing to pay, a higher yield on the issue as compensation
for this feature.
b. The call feature will reduce both the duration (interest rate sensitivity) and the convexity of the bond.
The bond will not experience as large a price increase if interest rates fall. Moreover, the usual
curvature that would characterize a straight bond will be reduced by a call feature. The price-yield
curve flattens out (see text Figure 13.7) as the interest rate falls and the option to call the bond becomes
more attractive. In fact, at very low interest rates, the bond exhibits “negative convexity”.
13-10
a. D
D 1 y
10 1 08 9 26 years.
b. For option-free coupon bonds, modified duration is better than maturity as a measure of the bond’s
sensitivity to changes in interest rates. Maturity considers only the final cash flow, while modified
duration includes other factors such as the size and timing of coupon payments and the level of interest
rates (yield to maturity). Modified duration, unlike maturity, tells us the approximate percentage
change in bond price for a given change in yield to maturity.
c. i. Modified duration increases as the coupon decreases.
ii. Modified duration decreases as the maturity decreases.
d. Convexity measures the curvature of the bond’s price-yield curve. Such curvature means that the duration rule for bond price change (which is based only on the slope of the yield curve at the original
6
yield) is only an approximation. Adding a term to account for the convexity of the bond will increase the accuracy of the approximation. That convexity adjustment is the last term in the following
equation:
1
∆P
D ∆y
Convexity
∆y 2
P
2
a. In an interest rate swap, one firm exchanges or “swaps” a fixed payment for another payment that is
tied to the level of interest rates. One party in the swap agreement must pay a fixed interest rate on
the notional principal of the swap. The other party pays the floating interest rate (typically based on
LIBOR) on the same notional principal.
b. There are several applications of interest rate swaps. For example, a portfolio manager who is holding
a portfolio of long-term bonds, but is worried that interest rates might increase, causing a capital loss
on the portfolio, can enter a swap to pay a fixed rate and receive a floating rate, thereby converting
the holdings into a synthetic floating rate portfolio. Or, a pension fund manager might identify some
money market securities that are paying excellent yields compared to other comparable risk short term
securities. However, the manager might believe that such short-term assets are inappropriate for the
portfolio. The fund can hold these securities and enter a swap where it receives a fixed rate and pays
a floating rate. It thus captures the benefit of the advantageous relative yields on these securities, but
still establishes a portfolio with characteristics more like those of long-term bonds.
13-12
13-14 The firm should enter a swap where it pays a fixed rate of 7% and receives LIBOR on $10 million of
notional principal. The firm’s combined position will be LIBOR 01
07 LIBOR
08, i.e. it
will be paying an interest rate of 8%.
13-15 a. The present value of the obligation is 2 16 $12 5 million. The duration of the obligation is
1 16 16 7 25 years. Let w be the weight on the 5 year bond (which has a duration of 4 years).
Then:
w 4 1 w 11 7 25
w
5357
Therefore, 5357 12 5 $6 696 million should be invested in the 5 year bonds and 4643
$5 804 million in the 20 year bonds.
b. The price of the 20 year bond is:
60
1
1 16
16
11 16000 20
20
12 5
$407 12
Therefore, the bond sells for .40712 times its par value, and so the par value needed for the position
is 5 804 40712 $14 256 million.
a. The duration of the perpetuity is 1 05 05 21 years. Let w be the weight of the zero coupon bond.
Then:
w 5 1 w 21 10
w
6875
13-16
so you would have 68.75% invested in the zero and 31.25% in the perpetuity.
b. The zero coupon bond will now have a duration of 4 years, while the perpetuity’s duration will remain
at 21 years. Then:
w 4 1 w 21 9
w
7059
13-17
so the percentage invested in the zero increases to 70.59%, while that in the perpetuity falls to 29.41%.
a. Recall that the duration of an annuity is:
1
y
T
y
1 y 1 T
Therefore, if the annuity were to start in 1 year, its duration would be
10 4 7255 years 1 1 1
1 10
10
10
Because the payment stream starts in 5 years, instead of 1 year, we must add 4 years to the duration,
resulting in duration of 8.7255 years.
7
b. The present value of the deferred annuity is:
1 10
1 1 1 10
10
4
$10 000
$41 968
Let w be the weight of the portfolio in the 5 year zero. Then:
8 7255 w 7516 Therefore the investment in the 5 year zero is 7516 $41 968 $31 543 and the investment in
the 20 year zero is 2484 $41 968 $10 425. These are the present values of each investment.
The face value of the 5 year zeros is $31 543 1 10 $50 800, while that of the 20 year zeros is
$10 425 1 10 $70 134.
w
5 1 w 20
5
20
13-18 We can calculate the exact prices of the bonds as follows:
y
7%
1
y
y
1 07 07
1 1 08 120 08
1 1 09 120 09
120
8%
8%
30
1 000 1 07 $1 620 45
1 000 1 08 $1 450 31
1 000 1 09 $1 308 21
30
30
30
30
30
Using the duration rule, if the yield falls to 7%, we would predict a price change of:
111 08 54 01 $1 450 31 $154 97 so our predicted new price would be 1 450 31 $154 97 $1 605 28. The actual price is $1,620.45,
implying a percentage error of 1 605 28 1 620 45 1 620 45 0 94%.
Using the duration rule, if the yield rises to 9%, we would predict a price change of:
111 08 54 01 $1 450 31 $154 97 so our predicted new price would be 1 450 31 $154 97 $1 295 34. The actual price is $1,308.21,
implying a percentage error of 1 295 34 1 308 21 1 308 21 0 98%.
Using the duration-with-convexity rule, if the yield drops to 7% we would predict a price change of:
54 01 0 5 111 08
so our estimated new price would be 1 450 31 01 $1 450 31 $168 92 $168 92 $1 619 23, with a percentage error of -0.075%.
2
192 4
Using the duration-with-convexity rule, if the yield increases to 9% we would predict a price change of:
54 01 0 5 192 4 01 $1 450 31 $141 02 111 08
so our estimated new price would be 1 450 31 $141 02 $1 309 29, with a percentage error of 0.083%.
2
The duration-with-convexity rule provides more accurate approximations to the true change in price. In
this example, the percentage error using convexity with duration is less than one-tenth the error using only
duration to estimate the price change.
13-23 The economic climate is one of impending rate increases (i.e. falling bond prices), so we will want to
shorten portfolio duration.
a. Choose the short maturity (2001) bond.
b. Choose the Arizona bond since it likely has lower duration (it has slightly lower coupons, but substantially higher yield).
8
c. Choose the 15 3/8 coupon bond—the maturities are about the same, but it has a much higher coupon,
implying a lower duration.
d. Choose the Shell bond. Although it has a little lower coupon, it has a higher yield to maturity and
earlier start to the sinking fund repayments.
e. Choose the floating rate bond. It has a duration approximately equal to the adjustment period of 6
months.
13-26 The minimum terminal value that the manager is willing to accept is determined by the requirement for a
3% annual return on the initial investment. Therefore, the floor is 1 1 03 5 $1 16 million. Three years
after the initial investment, only two years remain until the horizon date, and the interest rate has risen
to 8%. Therefore, at this time, the manager needs a portfolio worth 1 16 1 08 2 $0 9945 million to be
assured that the target value can be attained. This is the trigger point.
2.
a. The promised yield to maturity is:
$1 000
$721 40
1 3
1
11 5%
b. The expected yield to maturity is:
10 3.
$200 90 $1 000
$721 40
1 3
1 8 44% a. Since the bond is selling at par, its yield to maturity is equal to the coupon rate of 12%. Its yield to call is:
120
120
1 120 120
1 y 1 y 1 y 120
1 y
2
3
4
$1 000
y
14 42%
b. The yield to call is higher. This is because the issuer’s call option reduces the value of the bond to the
holder, implying a lower price and therefore a higher yield. The yield to maturity calculation ignores this
option feature, resulting in a lower yield.
4.
a. For bonds A and B we have:
1100
r
50
924 05 1 r 1 011 80
1
1
11 100
r 1 050
1 r 2
2
2
2
Buying two of bond B and selling one of bond A therefore costs $836.30 and returns $1,000 in two years,
so: 836 30 1 000 1 r2 2
r2 9 35%. Then 1 011 80 100 1 r1
1 100 1 09352
r1
2
3
8 85% and (from bond C) 934 48 70 1 0885 70 1 0935 1 070 1 r3
r3 9 65%.
b. Under the expectations hypothesis, forward rates are equal to expected future spot rates. We have:
f2
f3
11 11 1 1 0935 1 0885 1 9 85%
r 1 1 0965 1 0935 1 10 25%
r r2 2
r1
3
2
2
3
3
2
2
Therefore the one year spot rate is expected to be 9.85% in one year and 10.25% in two years. The usual
assumption for the liquidity preference hypothesis is that the market is dominated by investors with a short
term horizon. For such investors, long term bonds are riskier than short term bonds and thus long term
bonds command a risk premium. This implies that forward rates will exceed expected future spot rates.
On the other hand, if the market is dominated by investors with a long term horizon, short term bonds are
riskier and forward rates are less than expected future spot rates.
9
5. Let x be the spread in the fixed rate market (in this case 140 basis points) and let y be the spread in the floating
rate market (50 basis points). The total gain available is x y
90 basis points. The bank will take 10 basis
points, so that leaves 80 basis points to be split equally between A and B. Therefore, A must end up paying
LIBOR - 30 basis points and B must end up paying 13%. One way to accomplish this is as follows:
A
Total
B
Total
Pay 12% to outside lenders
Pay LIBOR - 0.1% to the bank in the swap
Receive 12.2% from the bank in the swap
12% + LIBOR - 0.1% - 12.2% = LIBOR - 0.3%
Pay LIBOR + 0.6% to outside lenders
Pay 12.2% to the bank in the swap
Receive LIBOR - 0.2% from the bank in the swap
LIBOR + 0.6% + 12.2% - (LIBOR - 0.2%) = 13%
Note that the bank’s profits of 10 basis points come from receiving LIBOR - 10 basis points from A and paying
LIBOR - 20 basis points to B.
6. Let x be the spread in the yen market (in this case 150 basis points) and let y be the spread in the dollar market
110 basis points. The bank will take 50 basis points, so
(40 basis points). The total gain available is x y
that leaves 60 basis points to be split equally between X and Y . Therefore, X must end up paying 9.3% in dollars
and B must end up paying 6.2% in yen. One way to accomplish this is as follows:
X
Total
Y
Total
Pay 5% in yen to outside lenders
Pay 9.3% in dollars to the bank in the swap
Receive 5% in yen from the bank in the swap
9.3% in dollars
Pay 10% in dollars to outside lenders
Pay 6.2% in yen to the bank in the swap
Receive 10% in dollars from the bank in the swap
6.2% in yen
Note that the bank’s profits of 50 basis points come from receiving 6.2% and paying 5% in yen (thereby gaining
120 basis points in yen) while receiving 9.3% and paying 10% in dollars (thus losing 70 basis points in dollars).
10