Question 1

Which of the following is not a typical property of a discount function $d ( t )$ ?
A) $d ( t )$
Is non-decreasing in the maturity
 $t$
)
B) $d ( t )$
Is non-increasing in the maturity
 $t$
)
C) $d ( t )$
Decreases as the zero-coupon rate
 $z ( t )$
For maturity
 $t$
Increases.
D) $d ( t )$
Is independent of compounding convention.

Accepted Answer

A discount function $d(t)$ is typically non-increasing as the maturity $t$ increases, not non-decreasing. This is because the value of money today is generally worth more than the same amount in the future due to potential earnings from interest rates. Therefore, as the time to maturity increases, the present value (discounted value) of a future payment typically decreases.

Question 2

Assuming annual compounding,the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99,respectively.Assume coupons are paid annually.The fair price of a two-year 6% coupon bond will be
A)$99.86
B)$100.11
C)$100.57
D)$100.87

Accepted Answer

D

Question 3

Under a semi-annual compounding convention,the present value of a $n$ -period cashflow using its ytm $y$ is given by $P = C / ( 1 + y / 2 ) ^ { n }$ .Which of the following is an equivalent way of expressing the same present value?
A) $P = C \cdot d _ { n } \cdot ( n / 2 )$
,where
 $d _ { n }$
Is the discount function for
 $n$
Periods,and
 $n / 2$
Is the time to maturity of the cash-flow.
B) $P = C e ^ { - z _ { n } \times n / 2 }$
,where
 $Z _ { n }$
Is the semi-annual basis zero-coupon rate for
 $n$
Periods.
C) $P = C / \prod _ { j = 1 } ^ { n } \left( 1 + f _ { j } \right)$
,where
 $f _ { j }$
Is the forward rate for period
 $j$
)
D)None of the above.

Accepted Answer

The given formula for present value under a semi-annual compounding convention does not match any of the provided alternatives exactly. Option A introduces a discount function and time to maturity in a manner not equivalent to the original formula. Option B uses continuous compounding with an exponential function, which differs from semi-annual compounding. Option C involves a product of terms involving forward rates, which also does not match the structure of the original semi-annual compounding formula.

Question 4

Assume that the risk-free zero rates are increasing with maturity (That is,the 6-months zero rate is lower than the one-year zero rate,which is lower than the two-year zero rate,etc).It must be that:
A)The yield-to-maturity curve for coupon bonds lies below the yield-to-maturity curve for zeros
B)The yield-to-maturity curve for coupon bonds equals the yield-to-maturity curve for zeros
C)The yield-to-maturity curve for coupon bonds lies above the yield-to-maturity curve for zeros
D)The yield-to-maturity curve for coupon bonds lies above the yield-to-maturity curve for zeros for maturities less than one year,but lies below the yield-to-maturity curve for zeros for maturities greater than one year.

Accepted Answer

When the risk-free zero rates are increasing with maturity, the yield-to-maturity curve for coupon bonds will lie below the yield-to-maturity curve for zeros. This is because the present value of the coupon payments on a coupon bond will be discounted at a higher rate than the present value of the face value on a zero-coupon bond.

Question 5

The price of a three-year 5% coupon Treasury bond in the Wall Street Journal is quoted at 101-20.The yield-to-maturity of the bond is
A)5.19%
B)5.34%
C)5.67%
D)5.88%

Accepted Answer

The quoted price of 101-20 translates to 101 20/32 or 101.625% of the face value. The yield-to-maturity (YTM) is a function of the coupon rate, the face value, the price, and the time to maturity. Without performing the exact calculation, which requires financial formulas or a financial calculator, the correct answer cannot be determined precisely from the information given. However, since the bond is trading above par (100%), its yield to maturity will be lower than its coupon rate. Given the options, the only yield lower than the coupon rate of 5% is 5.19%, making A the best choice based on the information provided.

Question 6

Today's forward rate $f \left( t _ { 1 } , t _ { 2 } \right)$ for a period $\left( t _ { 1 } , t _ { 2 } \right)$ in the future ( $t _ { 1 } < t _ { 2 }$ )is
A)Unknown until $t _ { 1 }$
)
B)Settled when we get to $t _ { 2 }$
)
C)Depends on the ytm of coupon bonds of maturities $t _ { 1 }$
And
 $t _ { 2 }$
)
D)Depends on the ytm of zero-coupon bonds of maturities $t _ { 1 }$
And
 $t _ { 2 }$
)

Accepted Answer

The forward rate is the rate that is implied by the current prices of zero-coupon bonds of maturities $t _ { 1 }$ and $t _ { 2 }$.

Question 7

The yield-to-maturity (ytm)is the
A)Return on the bond at maturity.
B)The coupon rate on the bond.
C)The internal rate of return of the bond.
D)The coupon rate divided by the current price of the bond.

Accepted Answer

Yield-to-maturity (YTM) is defined as the internal rate of return of the bond, meaning the rate that equates the sum of present value of all cash flows from the bond to its current market price. It takes into account the bond's current price, interest rate, coupon rate, time to maturity, and the face value of the bond. Therefore, option C is the correct answer. Options A and B are not correct because they do not consider the current market price of the bond, which is an important component of YTM. Option D is incorrect because it refers to the yield-to-price ratio, which is not the same as YTM.

Question 8

If zero rates (i.e. ,discount rates)are the same for all maturities and remain the same over the next year,the price of a zero-coupon bond that matures ten years from today will:
A)Decrease over the next year
B)Remain the same over the next year
C)Increase over the next year
D)Cannot tell with the given information

Accepted Answer

The price of a zero-coupon bond increases as it approaches maturity if the discount rates remain constant. This is because the present value of its single future cash flow (the face value at maturity) becomes greater as the time to receive that cash flow decreases.

Question 9

The prices of a one-year 4% coupon bond,a two-year 5% coupon bond,and a three-year 6% coupon bond are $101,$100 and $99,respectively.Coupons are paid annually.What is the price of a bond that pays $37 each year?
A)$99.00
B)$100.08
C)$101.21
D)$111.43

Accepted Answer

The price of a bond that pays $37 each year can be calculated using the concept of arbitrage. Arbitrage involves combining the given bonds in such a way that their coupon payments match the desired $37 payment. By doing so, we can determine the price of the desired bond without directly calculating its yield to maturity or using any bond pricing formula.Given:- 1-year 4% coupon bond priced at $101 (pays $4 annually)- 2-year 5% coupon bond priced at $100 (pays $5 annually)- 3-year 6% coupon bond priced at $99 (pays $6 annually)To create a bond that pays $37 annually, we can combine these bonds as follows:- Buy 7 of the 6% coupon bonds: 7 * $6 = $42- Sell 1 of the 5% coupon bonds: -$5- Sell 0 of the 4% coupon bonds: $0This combination results in a net payment of $37 annually ($42 - $5). The cost of this combination is:- Cost of buying 7 of the 6% bonds: 7 * $99 = $693- Proceeds from selling 1 of the 5% bonds: -$100- Net cost: $693 - $100 = $593However, to accurately reflect the price of a bond paying exactly $37 annually, we need to adjust this combination to exactly match the $37 payment. Since the initial combination gives us $42 in coupons and we need to reduce this to $37, we adjust by considering the sale of the 5% bond (which reduces the coupon by $5). The net payment is then exactly $37, and the net cost calculated above ($593) is for a combination that effectively pays $37 annually over the period considered.Given the complexity of directly calculating the exact price without knowing the total number of bonds or the specific combination that would yield exactly $37 in annual payments without any remainder, the approach simplifies to understanding that arbitrage and combination of existing bonds can give us an insight into the pricing of a new bond with desired characteristics. The correct answer, therefore, involves recognizing the principle of arbitrage and the need for a detailed calculation or combination strategy, which was not fully explored due to the initial misinterpretation of the question's requirements.

Question 10

If zero rates (also known as discount rates)are positive for any maturity,the discount function $d ( t )$ ,which gives the present value of a dollar receivable at time $t$ in the future,
A)Must be between 0 and 1
B)Must be greater than 0 and can be greater than 1
C)Can be a negative number
D)is usually 0.99

Accepted Answer

The discount function $d(t)$ reflects the present value of $1 receivable at time $t$ in the future. If zero rates are positive, it means that the future value is being discounted to present value, which will always result in a number between 0 and 1, but never reaching 0 or exceeding 1, as the value today of a future dollar is less than a dollar.

Question 11

The one-year discount factor today is 0.95.You buy a one-year zero-coupon bond today and hold it until maturity.Suppose that at maturity,the one-year discount factor is 0.92.The return you realize,expressed in simple terms,is:
A)3.43%
B)5.26%
C)8.69%
D)9.50%

Accepted Answer

The return realized on the zero-coupon bond can be calculated using the formula:

Return = (Ending Discount Factor / Beginning Discount Factor) - 1

Plugging in the values given in the question, we get:

Return = (0.92 / 0.95) - 1 = -0.0316 = -3.16%

However, since the question asks for the return in simple terms, we need to convert this annualized return into a simple return. To do this, we divide the annualized return by the number of periods held, which in this case is one year. Therefore, the simple return is:

Simple Return = -3.16% / 1 = -3.16%

However, since the return is negative, we need to consider the absolute value of the return. Therefore, the answer is:

Answer: B (5.26%)

Question 12

You are to receive a cash-flow of $40 after three years.If the ytm of this cash-flow is 6%,what is its present value? (Assume a semi-annual basis for compounding and discounting. )
A)$28.20
B)$33.50
C)$33.58
D)$37.74

Accepted Answer

The answer of You are to receive a cash-flow of...

Question 13

Find the yield-to-maturity of a 5% two-year bond that has a price of $102.Assume coupons are paid quarterly.
A)4.33%
B)4.44%
C)4.55%
D)4.66%

Accepted Answer

The answer of Find the yield-to-maturity of a 5% two-year...

Question 14

The zero-coupon rate (zcr)is
A)The rate of return each period on a bond that pays no coupons.
B)The yield-to-maturity of a zero-coupon bond.
C)The return on the bond's appreciation excluding coupon payments.
D)Zero.

Accepted Answer

The zero-coupon rate (zcr) is the annual interest rate that would be earned on an investment if all interest income was reinvested at the same rate until final maturity, and is equivalent to the yield-to-maturity of a zero-coupon bond.

Question 15

Assuming annual compounding and annual coupon payments,the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99,respectively.The zero-coupon rate for two years is:
A)4.5%
B)5.1%
C)5.6%
D)6.2%

Accepted Answer

The zero-coupon rate for two years can be found by solving for the yield that equates the present value of the cash flows from the two-year bond to its price. The two-year bond has a 5% annual coupon and a face value of $100 (assuming par value), so it pays $5 in one year and $105 in two years. Given the price of $99, we can set up the equation: $99 = $5/(1+r) + $105/(1+r)^2. Solving for r gives us a rate of approximately 5.6%.

Question 16

The 6-months risk-free zero rate is 2.84%,and the one-year zero rate is 3.17%.Assuming no-arbitrage,the yield-to-maturity on a $1,000 par of a one-year,12% Treasury bond,that pays $60 after 6 months and $1060 after one-year,must be
A)Greater than 0% and less than 2.84%
B)Greater than 2.84% and less than 3.17%
C)Greater than 3.17% and less than 6.01%
D)6.01%

Accepted Answer

The yield-to-maturity must be greater than the 6-month risk-free zero rate (2.84%) and less than the one-year zero rate (3.17%) to avoid arbitrage opportunities.

Question 17

If the one year rate expressed with semi-annual compounding is 6%,what is the equivalent rate with quarterly compounding.
A)3.00%
B)5.87%
C)5.96%
D)6.00%

Accepted Answer

To find the equivalent rate with quarterly compounding when given a semi-annual compounding rate, we use the formula to convert interest rates: $ (1 + \frac{r_{m}}{m})^{m} = (1 + \frac{r_{n}}{n})^{n} $, where $r_{m}$ is the rate with compounding frequency $m$ and $r_{n}$ is the rate with compounding frequency $n$. For semi-annual compounding at 6%, $r_{m} = 0.06$ and $m = 2$. We want to find $r_{n}$ with $n = 4$ (quarterly compounding). Rearranging the formula and solving for $r_{n}$, we find that the equivalent quarterly rate is approximately 5.96%.

Question 18

If the forward rate curve is downward sloping,then
A)The zero-coupon curve will lie above the yield curve.
B)The zero-coupon curve will lie below the yield curve.
C)The zero-coupon curve will lie above the yield curve in the near maturities and then lie below the yield curve for later maturities.
D)Not enough information to be able to answer the question.

Accepted Answer

The falling forward curve means that the zcr curve will also be falling and will lie below the ytm curve.

Question 19

If the price of a two-year semi-annual pay bond is par (say,$100),and the coupon on the bond is 6%,the yield-to-maturity expressed with semi-annual compounding is
A)5.87%
B)5.96%
C)6.00%
D)6.12%

Accepted Answer

Since the bond is at par, the coupon rate is equal to the yield to maturity. The coupon rate is 6% which means the yield to maturity is also 6%. With semi-annual compounding, the yield is divided by 2 to get a semi-annual yield of 3%. Therefore, the yield-to-maturity expressed with semi-annual compounding is 2* 3%=6%. Therefore, the best choice is C) 6.00%.

Question 20

Assuming annual compounding,the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99,respectively.The forward rate between one and two years is:
A)4.5%
B)5.0%
C)6.8%
D)8.3%

Accepted Answer

The answer of Assuming annual compounding,the prices of a one-year...

Quiz 24: Term Structure of Interest Rates: Concepts

Which of the following is not a typical property of a discount function $d ( t )$ ?

Assuming annual compounding,the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99,respectively.Assume coupons are paid annually.The fair price of a two-year 6% coupon bond will be

Under a semi-annual compounding convention,the present value of a $n$ -period cashflow using its ytm $y$ is given by $P = C / ( 1 + y / 2 ) ^ { n }$ .Which of the following is an equivalent way of expressing the same present value?

Assume that the risk-free zero rates are increasing with maturity (That is,the 6-months zero rate is lower than the one-year zero rate,which is lower than the two-year zero rate,etc) .It must be that:

The price of a three-year 5% coupon Treasury bond in the Wall Street Journal is quoted at 101-20.The yield-to-maturity of the bond is

Today's forward rate $f \left( t _ { 1 } , t _ { 2 } \right)$ for a period $\left( t _ { 1 } , t _ { 2 } \right)$ in the future ( $t _ { 1 } < t _ { 2 }$ ) is

The yield-to-maturity (ytm) is the

If zero rates (i.e. ,discount rates) are the same for all maturities and remain the same over the next year,the price of a zero-coupon bond that matures ten years from today will:

The prices of a one-year 4% coupon bond,a two-year 5% coupon bond,and a three-year 6% coupon bond are $101,$100 and $99,respectively.Coupons are paid annually.What is the price of a bond that pays $37 each year?

If zero rates (also known as discount rates) are positive for any maturity,the discount function $d ( t )$ ,which gives the present value of a dollar receivable at time $t$ in the future,

The one-year discount factor today is 0.95.You buy a one-year zero-coupon bond today and hold it until maturity.Suppose that at maturity,the one-year discount factor is 0.92.The return you realize,expressed in simple terms,is:

You are to receive a cash-flow of $40 after three years.If the ytm of this cash-flow is 6%,what is its present value? (Assume a semi-annual basis for compounding and discounting. )

Find the yield-to-maturity of a 5% two-year bond that has a price of $102.Assume coupons are paid quarterly.

The zero-coupon rate (zcr) is

Assuming annual compounding and annual coupon payments,the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99,respectively.The zero-coupon rate for two years is:

The 6-months risk-free zero rate is 2.84%,and the one-year zero rate is 3.17%.Assuming no-arbitrage,the yield-to-maturity on a $1,000 par of a one-year,12% Treasury bond,that pays $60 after 6 months and $1060 after one-year,must be

If the one year rate expressed with semi-annual compounding is 6%,what is the equivalent rate with quarterly compounding.

If the forward rate curve is downward sloping,then

If the price of a two-year semi-annual pay bond is par (say,$100) ,and the coupon on the bond is 6%,the yield-to-maturity expressed with semi-annual compounding is

Assuming annual compounding,the prices of a one-year 4% coupon bond and a two-year 5% coupon bond are $101 and $99,respectively.The forward rate between one and two years is: