FINC2011 Tutorial 4 BMA Ch.3 Problems 2, 4, 5, 10, 11, 12, 14, 15, 16, 1, 21, 2!, 2" 2.
The following statements are true. Explain why. a. If a bond's coupon rate rate is higher than than its yield yield to maturity maturity,, then the the bond will will sell for more than face value. b. If a bond's coupon coupo n rate is lower than its yield to maturity, matur ity, then the bond's price will increase over ov er its remaining maturity.
Answer a. If the coupon rate is higher than the yield, then investors must be expecting a decline in the capital value of the bond over its remaining life. Thus, the bond’s price must be greater than its face value. b. Conversely, if the yield is greater than the coupon, the price will be below face value and it will rise over the remaining life of the bond.
4.
A1!year "erman government bond #bund$ has a face value of %1 and a coupon rate of & paid annually. Assume that the interest rate #in euros$ is e(ual to ) per year. *hat is the bond's +-
Answer
With annual coupon payments: 1
PV = 5 ×
0.06
−
1
0.06 × (1.06)
10
100 = €92.64 + 10 (1.06)
onstruct some simple simple examples to illustrate your answers to the following/ 5.
a. If interest rates rise, do bond prices rise or fall b. If the bond yield is greater than the coupon, is the price of the bond greater or less than 1c. If the price of a bond exceeds 1, is the yield greater or less than the coupond. 0o high!coupon bonds sell at higher or lower prices than low!coupon bondse. If interest rates change, does the price of high!coupon bonds change proportionately more than that of low!coupon low!coupo n bonds-
Answer a. Fall. xample! Assume a one"year, #$% bond. If the interest rate is #$%, the bond is worth &##$'#.# ( &#$$. If the interest rate rises to #)%, the bond is worth &##$'#.#) ( &*).+). b. ess -e.g., see )aif the bond yield is #)% but the coupon rate is lower at #$%, the price of the bond is less than &#$$/. c. ess -e.g., with r ( )%, one"year #$% bond is worth &##$'#.$) ( &#$0.1+/. d. 2igher -e.g., if r ( #$%, one"year #$% bond is worth &##$'#.# ( &#$$, while one" year 3% bond is worth &#$3'#.# ( &*3.#3/. e.4o. ow"coupon bonds have longer durations -unless there is only one period to maturity/ and are therefore more volatile -e.g., if r falls from #$% to )%, the value of a two"year #$% bond rises from &#$$ to &#$*.5 -a rise of *.5%/. The value of a two"year )% bond rises from &*#.5 to &#$$ -a rise of *.)%/.
10.
a. An , five!year bond yields ). If the yield remains unchanged, what will be its price one year hence- Assume annual coupon payments. b. *hat is the total return to an investor who held the bond over this yearc. *hat can you deduce about the relationship between the bond return over a particular period and the yields to maturity at the start and end of that period-
Answer
a. + 2 #. 3 41$ 3 ##1 5 .)$ 6 71 5 8.)#1 9 .)$&:;$ 9 41 5 1.)& + 2 41.<= +1 2 #. 3 41$ 3 ##1 5 .)$ 6 71 5 8.)#1 9 .)$<:;$ 9 41 5 1.)< +1 2 41).>?
b. 6eturn ( -3 7 #$+.*5$/'#$3.08) " # ( .$+, or +%. c. If a bond’s yield to maturity is unchanged, the return to the bondholder is e9ual to the yield.
11.
True or false- Explain. a. @onger!maturity bonds necessarily have longer durations. b. The longer a bond's duration, the lower its volatility. c. ther things e(ual, the lower the bond coupon, the higher its volatility.
d. If interest rates rise, bond durations rise also. Answer a. False. :uration depends on the coupon as well as the maturity. b. False. ;iven the yield to maturity, volatility is proportional to duration. c. True. A lower coupon rate means longer duration and therefore higher volatility. d. False. A higher interest rate reduces the relative present value of -distant/ principal repayments.
12.
alculate the durations and volatilities of securities A, B, and . Their cash flows are shown below. The interest rate is .
Answer
14. The two!year interest rate is 1 and the expected annual inflation rate is &. a. *hat is the expected real interest rate b. If the expected rate of inflation suddenly rises to C, what does Disher's theory say about how the real interest rate will change- *hat about the nominal rate Answer
a.
6eal rate ( #.#$'#.$) < # ( .$01+, or 0.1+%.
b.
The real rate does not change. The nominal rate increases to #.$01+ = #.$1 < # ( .#8$*, or #8.$*%.
ere are the prices of three bonds with 1!year maturities/ 15.
If coupons are paid annually, which bond offered the highest yield to maturity*hich had the lowest- *hich bonds had the longest and shortest durations Answer
Bond 1 FTG 2 <.? Bond = FTG 2 <.= Bond ? FTG 2 ?.> Bond 1 0uration 2 >.& Bond = 0uration 2 .<= Bond ? 0uration 2 C.)& >ields to maturity are about 0.5% for the 8% coupon, 0.8% for the 0% coupon, and 5.*% for the 3% coupon. The 3% bond had the shortest duration -1.+) years/, the 8% bond the longest -*.$1 years/.The 0% bond had a duration of 3.08 years.
16. A 1!year H.. Treasury bond with a face value of 41, pays a coupon of &.& #=.C& of face value every six months$. The semiannually compounded interest rate is &.= #a six!month discount rate of &.=5= 2 =.)$. a. *hat is the present value of the bond b. "enerate a graph or table showing how the bond's present value changes for semiannually compounded interest rates between 1 and 1&.
Answer
a. + 2 #.=C& 3 41,$ 3 ##1 5 .=)$ 6 71 5 8.=)#1 9 .=)$13= :;$ 9 41, 5 #1 9 . =)$13= + 2 41,=?.1)
b. #iel$ to Maturit%
P& o' Bo($
41,<=C.== 1,?1&. 1,=1<.) 1,1==.)< 1,?.>C >)=.1 >?.<1 ?.1= CC=.?) C1>.) )C1.?) )=C.=? &).1 &<>.C& &1&.C)
1 = ? < & ) C > 1 11 1= 1? 1< 1&
1. A six!year government bond maJes annual coupon payments of & and offers a yield of ? annually compounded. uppose that one year later the bond still yields ?. *hat return has the bondholder earned over the 1=!month period Kow suppose that the bond yields = at the end of the year. *hat return would the bondholder earn in this case-
Answer ?urchase price for a six"year government bond with )% annual coupon! 1
PV = 50 ×
0.03
−
1,000 = 41,108.34 + 0.03 × (1.03) 6 (1.03) 6 1
The price one year later is e9ual to the present value of the remaining five years of the bond! 1
PV = 50 ×
0.03
−
1,000 = 41,091.59 + 0.03 × (1.03)5 (1.03)5 1
6ate of return ( @&)$ 7 -&#,$*#.)* < &#,#$3.50/'&#,#$3.50 ( 5.$
%$ ?rice one year later -yield ( 8%/! 1
PV = 50 ×
0.02
−
1,000 = 41,141.40 + 0.02 × (1.02) (1.02) 5 1
5
6ate of return ( @&)$ 7 -&#,#0#.0$ < &#,#$3.50/'&#,#$3.50 ( 1.0*%. 21. alculate durations and modified durations for the ? bonds in Table?.=. Fou can follow the procedure set out in Table?.< for the > coupon bonds. onfirm that modified duration predicts the impact of a 1 change in interest rates on the bond prices. Answer To calculate the duration, consider the following table similar to Table 5.0!
Year Payment ($) PV(C ) t at 4% ($) Fraction of total value [PV(C )/PV] t Year × fraction of total value Duration (Years)
# 5$ 83.30+
8 5$ 81.151
5 5$ 8+.+1$
0 5$ 8).+00
) 5$ 80.+)3
+ 5$ 85.1$*
1 #,$5$ 138.1#)
$.$5#
$.$5$
$.$83
$.$81
$.$8+
$.$8)
$.355
$.$5#
$.$)*
$.$3)
$.#$*
$.#5#
$.#)#
).38*
Totals *5*.*3$ #.$$$
!"#
The duration is the sum of the year = fraction of total value column, or +.5*) years. The modified duration, or volatility, is +.5*)'-# 7 .$0/ ( +.#). The price of the 5% coupon bond at 5.)%, and 0.)% e9uals &*+*.05 and &*##.+#, respectively. This price difference -&)1.38/ is +.#)% of the original price, which is very close to the modified duration.
uppose that you buy a two!year bond at its face value. 2!.
a. *hat will be your nominal return over the two years if inflation is ? in the first year and & in the second- *hat will be your real return b. Kow suppose that the bond is a TI+. *hat will be your real and n ominal returns Answer
a.
Four nominal return will be 1.= !1 2 1).)< over the two years. Four real return is #1.51.?$ 3 #1.51.&$ ! 1 2 C.&.
b.
*ith the TI+, the real return will remain at per year, or 1).)< over two years. The nominal return on the TI+ will e(ual #1. 3 1.?$ 3 #1. 3 1.&$ 6 1 2 =).1&.
2". A bond's credit rating provides a guide to its price. As we write this in early =1&, Aaa bonds yield ?.< and Baa bonds yield <.<. If some bad news causes a 1 five!year bond to be unexpectedly downrated from Aaa to Baa, what would be the effect on the bond price- #Assume annual coupons.$
Answer The bond price at a 5.0% yield is!
1 1 1,000 − + = 41,298.84 5 5 × 0.034 0.034 (1.034) (1.034)
PV = 100 ×
If the yield increase to 0.0%, the price would decrease to!
PV = 100 ×
1
0.044
−
1,000 + = 41,246.53 0.044 × (1.044) (1.044) 5 1
5
