克罗恩病最常见的症状CHAPTER 4
Interest Rates
Practice Questions
Problem 4.1.
A bank quotes you an interest rate of 14% per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?
(a) The rate with continuous compounding is 0144ln 1013764.⎛⎫+=. ⎪⎝
⎭ or 13.76% per annum.
(b) The rate with annual compounding is 4
01411014754.⎛⎫+-=. ⎪⎝⎭ or 14.75% per annum.
Problem 4.2.
佛手是什么What is meant by LIBOR and LIBID. Which is higher?
LIBOR is the London InterBank Offered Rate. It is calculated daily by the British Bankers Association and is the rate a AA-rated bank requires on deposits it places with other banks. LIBID is the London InterBank Bid rate. It is the rate a bank is prepared to pay on deposits from other AA-rated banks. LIBOR is greater than LIBID.
Problem 4.3.
The six-month and one-year zero rates are both 10% per annum. For a bond that has a life of 18 months and pays a coupon of 8% per annum (with miannual payments and one having just been made), the y ield is 10.4% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quoted with miannual compounding.
Suppo the bond has a face value of $100. Its price is obtained by discounting the cash flows at 10.4%. The price is 23
441049674105210521052++=.... If the 18-month zero rate is R , we must have 23
441049674105105(12)R ++=...+/ which gives 1042R =.%.
Problem 4.4.
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An investor receives $1,100 in one year in return for an investment of $1,000 now. Calculate the percentage return per annum with a) annual compounding, b) miannual compounding, c) monthly compounding and d) continuous compounding.
(a) With annual compounding the return is 11001011000
-=. or 10% per annum.
(b) With mi-annual compounding the return is R where 2
1000111002R ⎛⎫+= ⎪⎝⎭ i.e.,
1104882
R +
==. so that 00976R =.. The percentage return is therefore 9.76% per annum.
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(c) With monthly compounding the return is R where 12
10001110012R ⎛⎫+= ⎪⎝⎭ i.e.
1110079712R ⎛⎫+==. ⎪⎝⎭
so that 00957R =.. The percentage return is therefore 9.57% per annum.
(d) With continuous compounding the return is R where:
10001100R e =
<,
11R e =.
so that ln1100953R =.=.. The percentage return is therefore 9.53% per annum.
Problem 4.5.
Suppo that zero interest rates with continuous compounding are as follows:
Calculate forward interest rates for the cond, third, fourth, fifth, and sixth quarters.
The forward rates with continuous compounding are as follows to
Problem 4.6.
Assume that a bank can borrow or lend at the rates in Problem 4.5. what is the value of an FRA where it will earn 9.5% for a three-month period starting in one year on a principal of $1,000,000? The interest rate is expresd with quarterly compounding.
The forward rate is 9.0% with continuous compounding or 9.102% with quarterly compounding. From equation (4.9), the value of the FRA is therefore
0086125[1000000025(0095009102)]89356e -.⨯.,,⨯.⨯.-.=.
or $893.56.
Problem 4.7.
The term structure of interest rates is upward sloping. Put the following in order of magnitude:
(a) The five-year zero rate
(b) The yield on a five-year coupon-bearing bond
(c) The forward rate corresponding to the period between 4.75 and 5 years in the future
What is the answer to this question when the term structure of interest rates is downward sloping?
When the term structure is upward sloping, c a b >>. When it is downward sloping, b a c >>.
Problem 4.8.
What does duration tell you about the nsitivity of a bond portfolio to interest rates? What are the limitations of the duration measure?
Duration provides information about the effect of a small parallel shift in the yield curve on the value of a bond portfolio. The percentage decrea in the value of the portfolio equals the duration of the portfolio multiplied by the amount by which interest rates are incread in the small parallel shift. The duration measure has the following limitation. It applies only to parallel shifts in the yield curve that are small.
Problem 4.9.
What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding?
The rate of interest is R where:
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015112R e .⎛⎫=+ ⎪⎝⎭ i.e.,
01512ln 112R .⎛⎫=+ ⎪⎝⎭
01491=.
The rate of interest is therefore 14.91% per annum.
Problem 4.10.
A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
The equivalent rate of interest with quarterly compounding is R where 401214R e .⎛⎫=+ ⎪⎝
⎭ or
0034(1)01218R e .=-=.
The amount of interest paid each quarter is therefore: 0121810000304554
.,⨯=. or $304.55.
Problem 4.11.
Suppo that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%,
4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum miannually.
The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price is 004050042100044150046200482522221029804e e e e e -.⨯.-.⨯.-.⨯.-.⨯-.⨯.++++=.
Problem 4.12.
A three-year bond provides a coupon of 8% miannually and has a cash price of 104. What is the bond’s yield?
The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yield is the value of y that solves
05101520253044444104104y y y y y y e e e e e e -.-.-.-.-.-.+++++=
Using the Solver or Goal Seek tool in Excel, 006407y =. or 6.407%.
Problem 4.13.
Suppo that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%, and 7% respectively. What is the two-year par yield?
Using the notation in the text, 2m =, 007208694d e -.⨯==.. Also
老年人健康知识00505006100065150072036935A e e e e -.⨯.-.⨯.-.⨯.-.⨯.=+++=.
The formula in the text gives the par yield as
(10010008694)2707236935
-⨯.⨯=.. To verify that this is correct we calculate the value of a bond that pays a coupon of 7.072% per year (that is 3.5365 every six months). The value is
0050500610006515007203536353653536103536100e e e e -.⨯.-.⨯.-.⨯.-.⨯..+.+.+.= verifying that 7.072% is the par yield.
Problem 4.14.
Suppo that zero interest rates with continuous compounding are as follows:
Calculate forward interest rates for the cond, third, fourth, and fifth years.
The forward rates with continuous compounding are as follows:
Year 2: 4.0%
Year 3: 5.1%
Year 4: 5.7%动态感
Year 5: 5.7%
Problem 4.15.
Suppo that the 9-month and 12-month LIBOR rates are 2% and 2.3%, respectively. What is the forward LIBOR rate for the period between 9 months and 12 months? What is the value of an FRA w多人爱爱
here 3% is received and LIBOR is paid on $10 million for the period? All rates are quarterly compounded. Assume that LIBOR is ud as the risk-free discount rate.
The 9 month and 12 month rates are 0.5% per quarter and 0.575% per quarter. If the forward LIBOR rate is R with quarterly compounding we must have (1.0053)×(1+R /4)=1.005754 so that R =3.201%. We value the FRA by assuming that the forward LIBOR will be realized. The value of the FRA is
10,000,000×(0.03−0.03201)×0.25/(1.00575)4 = −$4,919.47
Problem 4.16.
A 10-year, 8% coupon bond currently lls for $90. A 10-year, 4% coupon bond currently lls for $80. What is the 10-year zero rate? (Hint: Consider taking a long position in two of the 4% coupon bonds and a short position in one of the 8% coupon bonds.)
Taking a long position in two of the 4% coupon bonds and a short position in one of the 8% coupon bonds leads to the following cash flows
Year 09028070Year 10200100100
:-⨯=-:-= becau the coupons cancel out. $100 in 10 years time is equivalent to $70 today. The 10-year rate,R , (continuously compounded) is therefore given by
1010070R e =
