Articles by kenli

LO 13.3: Describe uses and benefits o f the arbitrage-free models and assess the issue o f fitting models to market prices.
Broadly speaking, there are two types of models: arbitrage-free models and equilibrium models. The key factor in choosing between these two models is based on the need to match
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market prices. Arbitrage models are often used to quote the prices of securities that are illiquid or customized. For example, an arbitrage-free tree is constructed to properly price on-the-run Treasury securities (i.e., the model price must match the market price). Then, the arbitrage-free tree is used to predict off-the-run Treasury securities and is compared to market prices to determine if the bonds are properly valued. These arbitrage models are also commonly used for pricing derivatives based on observable prices of the underlying security (e.g., options on bonds).
There are two potential detractors of arbitrage-free models. First, calibrating to market prices is still subject to the suitability of the original pricing model. For example, if the parallel shift assumption is not appropriate, then a better fitting model (by adding drift) will still be faulty. Second, arbitrage models assume the underlying prices are accurate. This will not be the case if there is an external, temporary, exogenous shock (e.g., oversupply of securities from forced liquidation, which temporarily depresses market prices).
If the purpose of the model is relative analysis (i.e., comparing the value of one security to another), then using arbitrage-free models, which assume both securities are properly priced, is meaningless. Hence, for relative analysis, equilibrium models would be used rather than arbitrage-free models.
V a s i c e k M o d e l

LO 13.4: Construct a short-term rate tree under the Ho-Lee M odel with time- dependent drift.
The Ho-Lee model further generalizes the drift to incorporate time-dependency. That is, the drift in time 1 may be different than the drift in time 2; additionally, each drift does not have to increase and can even be negative. Thus, the model is more flexible than the constant drift model. Once again, the drift is a combination of the risk premium over the period and the expected rate change. The tree in Figure 4 illustrates the interest rate structure and effect of time-dependent drift.
Figure 4: Interest Rate Tree with Time-Dependent Drift
It is clear that if X1 = X2 then the Ho-Lee model reduces to Model 2. Also, it should not be surprising that X j and X2 are estimated from observed market prices. In other words, the observed one-period spot rate. Xj could then be estimated so that the model rate equals the observed two-period market rate. X2 could be calibrated from using observed market rate for a three-period security, and so on.
and X1 and the
is
A r b i t r a g e -F r e e M o d e l s

LO 13.3: Describe methods for addressing the possibility o f negative short-term rates in term structure models.
Note that the terminal nodes in the two-period model generate three possible ending rates: rQ + 2a V dt, rQ, and rQ – 2a V dt. This discrete, finite set of outcomes does not technically represent a normal distribution. However, our knowledge of probability distributions tells us that as the number of steps increases, the terminal distribution at the nodes will approach a continuous normal distribution.
One obvious drawback to Model 1 is that there is always a positive probability that interest rates could become negative. On the surface, negative interest rates do not make much economic sense (i.e., lending $100 and receiving less than $100 back in the future). However, you could plausibly rationalize a small negative interest rate if the safety and/or inconvenience of holding cash were sufficiently high.
The negative interest rate problem will be exacerbated as the investment horizon gets longer, since it is more likely that forecasted interest rates will drop below zero. As an illustration, assume a ten-year horizon and a standard deviation of terminal interest rates of 1.2% x VlO = 3.79%. It is clear that negative interest rates will be well within a two standard deviation confidence interval when centered around a current rate of 6%. Also note that the problem of negative interest rates is greater when the current level of interest rates is low (e.g., 4% instead of the original 6%).
There are two reasonable solutions for negative interest rates. First, the model could use distributions that are always non-negative, such as lognormal or chi-squared distributions. In this way, the interest rate can never be negative, but this action may introduce other non- desirable characteristics such as skewness or inappropriate volatilities. Second, the interest rate tree can force negative interest rates to take a value of zero. In this way, the original interest rate tree is adjusted to constrain the distribution from being below zero. This method may be preferred over the first method because it forces a change in the original distribution only in a very low interest rate environment whereas changing the entire distribution will impact a much wider range of rates.
As a final note, it is ultimately up to the user to decide on the appropriateness of the model. For example, if the purpose of the term structure model is to price coupon-paying bonds, then the valuation is closely tied to the average interest rate over the life of the bond and the possible effect of negative interest rates (small probability of occurring or staying negative for long) is less important. On the other hand, option valuation models that have asymmetric payoffs will be more affected by the negative interest rate problem.
M odel 1 Effectiveness
Given the no-drift assumption of Model 1, we can draw several conclusions regarding the effectiveness of this model for predicting the shape of the term structure:
The no-drift assumption does not give enough flexibility to accurately model basic
term structure shapes. The result is a downward-sloping predicted term structure due to a larger convexity effect. Recall that the convexity effect is the difference between the model par yield using its assumed volatility and the par yield in the structural model with assumed zero volatility.
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Model 1 predicts a flat term structure of volatility, whereas the observed volatility term
structure is hump-shaped, rising and then falling.
Model 1 only has one factor, the short-term rate. Other models that incorporate
additional factors (e.g., drift, time-dependent volatility) form a richer set of predictions. Model 1 implies that any change in the short-term rate would lead to a parallel shift in the yield curve, again, a finding incongruous with observed (non-parallel) yield curve shifts.
T e r m S t r u c t u r e M o d e l w i t h D r i f t (M o d e l 2)
Casual term structure observation typically reveals an upward-sloping yield curve, which is at odds with Model 1, which does not incorporate drift. A natural extension to Model 1 is to add a positive drift term that can be economically interpreted as a positive risk premium associated with longer time horizons. We can augment Model 1 with a constant drift term, which yields Model 2:
dr = \d t + odw
Lets continue with a new example assuming a current short-term interest rate, drift, \ , of 0.24%, and standard deviation, a, of 1.30%. As before, the div realization drawn from a normal distribution (with mean = 0 and standard deviation = 0.2887) is 0.2. Thus, the change in the short-term rate in one month is calculated as:
of 3%,
dr = 0.24% x (1/12) + 1.5% x 0.2 = 0.32%
Hence, the new rate, rv is computed as: 5% + 0.32% = 5.32%. T he monthly drift is 0.24% x 1/12 = 0.02% and the standard deviation of the rate is 1.5% x Vl /12 = 0.43% (i.e., 43 basis points per month). The 2bps drift per month (0.02%) represents any combination of expected changes in the short-term rate (i.e., true drift) and a risk premium. For example, the 2bps observed drift could result from a 1.5bp change in rates coupled with a 0.5bp risk premium.
The interest rate tree for Model 2 will look very similar to Model 1, but the drift term, \dt, will increase by \d t in the next period, 2 \d t in the second period, and so on. This is visually represented in Figure 3. Note that the tree recombines at time 2, but the value at time 2, rQ + 2\dt, is greater than the original rate, rQ, due to the positive drift.
Figure 3: Interest Rate Tree with Constant Drift
0.5
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M odel 2 Effectiveness
As you would expect, Model 2 is more effective than Model 1. Intuitively, the drift term can accommodate the typically observed upward-sloping nature of the term structure. In practice, a researcher is likely to choose Hence, the term structure will fit better. The downside of this approach is that the estimated value of drift could be relatively high, especially if considered as a risk premium only. On the other hand, if the drift is viewed as a combination of the risk premium and the expected rate change, the model suggests that the expected rates in year 10 will be higher than year 9, for example. This view is more appropriate in the short run, since it is more difficult to justify increases in expected rates in the long run.
and X based on the calibration of observed rates.
H o -L e e M o d e l

LO 13.2: Calculate the short-term rate change and standard deviation o f the rate change using a model with normally distributed rates and no drift.
In Model 1, since the expected value of dw is zero [i.e., E(dw) = 0], the drift will be zero. Also, since the standard deviation of dw = V dt, the volatility of the rate change = a V dt. This expression is also referred to as the standard deviation of the rate.
In the preceding example, the standard deviation of the rate is calculated as:
1.2% x >/l /12 = 0.346% = 34.6 basis points
Returning to our previous discussion, we are now ready to construct an interest rate tree using Model 1. A generic interest rate tree over two periods is presented in Figure 1. Note that this tree is recombining and the ending rate at time 2 for the middle node is the same as the initial rate, rQ. Hence, the model has no drift.
Figure 1: Interest Rate Tree with No Drift
The interest rate tree using the previous numerical example is shown in Figure 2. One period from now, the observed interest rate will either increase with 30% probability to: 6% + 0.346% = 6.346% or decrease with 30% probability to: 6% – 0.346% = 5.654%. Extending to two periods completes the tree with upper node: 6% + 2(0.346%) = 6.692%, middle node: 6% (unchanged), and lower node: 6% – 2(0.346%) = 5.308%.
Figure 2: Numerical Example of Interest Rate Tree with No Drift
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LO 13.1: Construct and describe the effectiveness o f a short term interest rate tree assum ing normally distributed rates, both with and without drift.
T e r m S t r u c t u r e M o d e l w i t h N o D r i f t (M o d e l I)
This topic begins with the simplest model for predicting the evolution of short rates (Model 1), which is used in cases where there is no drift and interest rates are normally distributed. The continuously compounded instantaneous rate, denoted rt, will change (over time) according to the following relationship:
dr = crdw
where: dr = change in interest rates over small time interval, dt dt = small time interval (measured in years) (e.g., one month = 1/12) a = annual basis-point volatility of rate changes dw = normally distributed random variable with mean 0 and standard deviation Vdt
Given this definition, we can build an interest rate tree using a binomial model. The probability of up and down movements will be the same from period to period (30% up and 30% down) and the tree will be recombining. Since the tree is recombining, the up- down path ends up at the same place as the down-up path in the second time period.
For example, consider the evolution of interest rates on a monthly basis. Assume the current short-term interest rate is 6% and annual volatility is 120bps. Using the above notation, rQ = 6%, a = 1.20%, and dt = 1/12. Therefore, dw has a mean of 0 and standard deviation of Vl
112 = 0.2887.
After one month passes, assume the random variable dw takes on a value of 0.2 (drawn from a normal distribution with mean = 0 and standard deviation = 0.2887). Therefore, the
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change in interest rates over one month is calculated as: dr = 1.20% x 0.2 = 0.24% = 24 basis points. Since the initial rate was 6% and interest rates changed by 0.24%, the new spot rate in one month will be: 6% + 0.24% = 6.24%.

LO 12.5: Calculate the price and return o f a zero coupon bond incorporating a risk premium.
Suppose an investor expects 1-year rates to resemble those in Figure 7. In this example, there is volatility of 400 basis point of rates per year where 1-year rates in one year range from 4% to 12% in the second year.
Figure 7: Decision Tree Illustrating Expected 1-Year Rates for Two Years
Next year, the 1-year return will be either 10% or 6%. A risk-neutral investor calculates the price of a 2-year zero-coupon bond with a face value of $1 as follows:
x0.5 $1 $1 $1 | 1.10 ^ 1.06 1.08
[$0.90909 +$0.94340] x 0,5 =
1.08
In this example, the price of $0.85763 implies a 1-year expected return of 8%. Flowever, this is only the average return. The actual return will be either 6% or 10%. Risk-averse investors would require a higher rate of return for this investment than an investment that has a certain 8% return with no variability. Thus, risk-averse investors require a risk premium for bearing this interest rate risk, and demand a return greater than 8% for buying a 2-year zero-coupon bond and holding it for the next year.
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Example: Incorporating a risk premium
Calculate the price and return for the zero-coupon bond using the expected returns in Figure 7 and assuming a risk premium of 30 basis points for each year of interest rate risk.
Answer:
The price of a 2-year zero-coupon bond with a 30 basis point risk premium included is calculated as:
|
$1 $1 1.103 ^ 1.063 1.08
x 0 -5
[$0.90662 + $0.940731×0.5
1.08
$0.85525
Notice that this price is less than the $0.83763 price calculated previously for the risk- neutral investor. Next year, the price of the 2-year zero-coupon bond will either be $0.90909 or $0.94340, depending on whether the 1-year rate is either 10% or 6%, respectively. Thus, the expected return for the next year of the 2-year zero-coupon bond is 8.3%, calculated as follows:
($0.90909 + $0.94340) x 0.3 – $0.83323
$0.85525
0.083
Therefore, risk-averse investors require a 30 basis point premium or 8.3% return to compensate for one year of interest rate risk. For a 3-year zero-coupon bond, risk-averse investors will require a 60 basis point premium or 8.6% return given two years of interest rate risk.
Professors Note: In the previous example, it is assumed that rates can change only once a year, so in the first year there is no uncertainty o f interest rates. There is only uncertainty in what the 1-year rate will be one and two years from today.
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K e y C o n c e p t s
LO 12.1
If expected 1-year spot rates for the next three years are rate, r (2), is computed as r (2) = ^ (l + q )(l + r2) 1, and the 3-year spot rate, r (3), is
rv and ry then the 2-year spot
computed asr(3) = ^(l + q )(l + r2)(l + r3) 1.
LO 12.2
The volatility of expected rates creates convexity, which lowers future spot rates.
LO 12.3
The convexity effect can be measured by using Jensens inequality: E
1
(l + r)
>
1 E[l +
LO 12.4
Convexity lowers bond yields due to volatility. This reduction in yields is equal to the value of convexity. Thus, we can say that the value of convexity increases with volatility. The value of convexity will also increase with maturity, because the price-yield relationship will become more convex over time.
LO 12.3
Risk-averse investors will price bonds with a risk premium to compensate them for taking on interest rate risk.
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C o n c e p t C h e c k e r s
1.
An investor expects the current 1-year rate for a zero-coupon bond to remain at 6%, the 1-year rate next year to be 8%, and the 1-year rate in two years to be 10%. What is the 3-year spot rate for a zero-coupon bond with a face value of $ 1, assuming all investors have the same expectations of future 1-year rates for zero-coupon bonds? A. 7.888%. B. 7.988%. C. 8.000%. D. 8.088%.
Suppose investors have interest rate expectations as illustrated in the decision tree below where the 1-year rate is expected to be 8%, 6%, or 4% in the second year and either 7% or 3% in the first year for a zero-coupon bond.
If investors are risk-neutral, what is the price of a $ 1 face value 2-year zero-coupon bond today? A. $0.88113. B. $0.88634. C. $0.89007. D. $0.89032.
If investors are risk-neutral and the price of a 2-year zero-coupon bond is $0.88033 today, what is the implied 2-year spot rate? A. 4.339%. B. 5.230%. C. 5.827%. D. 6.579%.
4.
less concave. What is the impact on the bond price-yield curve if, all other factors held constant, the maturity of a zero-coupon bond increases? The pricing curve becomes: A. B. more concave. C. D. more convex.
less convex.
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Suppose an investor expects that the 1-year rate will remain at 6% for the first year for a 2-year zero-coupon bond. The investor also projects a 50% probability that the 1-year spot rate will be 8% in one year and a 50% probability that the 1-year spot rate will be 4% in one year. Which of the following inequalities most accurately reflects the convexity effect for this 2-year bond using Jensens inequality formula? A. $0.89031 > $0.89000. B. $0.89000 > $0.80000. C. $0.94340 >$0.89031. D. $0.94373 > $0.94340.
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C o n c e p t C h e c k e r An s w e r s
1. B The 3-year spot rate can be solved for using the following equation:
$1
$1
(1.06) (1.08) (1.10)
(l + ;(3 )f
Solving for r(3) = ^/(l.06)(l.08)(l.10) 1 = 7.988%
2. C
Assuming investors are risk-neutral, the following decision tree illustrates the calculation of the price of a 2-year zero-coupon bond using the expected rates given. The expected price in one year for the upper node is $0.93458, calculated as $1 / 1.07. The expected price in one year for the lower node is $0.95238, calculated as $1 / 1.05. Thus, the current price is $0.89007, calculated as:
[0.5 x ($0.93458 / 1.06)] + [0.5 x ($0.95238 / 1.06)] = $0.89007
$0.89007
$1
$1
$1
3. D The implied 2-year spot rate is calculated by solving for r (2) in the following equation:
$0.88035 =
$1
(l + r(2))
i (2) =
= 0-06579 or 6.579%
Alternatively, this can also be computed using a financial calculator as follows:
P V = -0.88035; FV = 1; PMT = 0; N = 2; CPT
I/Y = 6.579%.
4. D As the maturity of a bond increases, the price-yield relationship becomes more convex.
5. A The left-hand side of Jensens inequality is the expected price in one year using the 1-year
spot rates of 8% and 4%.
$1 ( i + 0
= 0.5 X
$1 (1.08)
+ 0.5 X
$1
(1.04)
= 0.5 X 0.92593 + 0.5 x $0.96154 = $0.94373
The expected price in one year using an expected rate of 6% computes the right-hand side of the inequality as:
$1
0.5×1.08 + 0.5×1.04
$1 1.06
= 0.94340
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Next, divide each side of the equation by 1.06 to discount the expected 1-year zero-coupon bond price for one more year at 6%. The price of the 2-year zero-coupon bond equals $0.89031 (calculated as $0.94373 / 1.06), which is greater than $0.89000 (the price of a 2-year zero-coupon bond discounted for two years at the expected rate of 6%). Thus, Jensens inequality reveals that $0.89031 > $0.89000.
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The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
T h e A r t o f T e r m S t r u c t u r e M o d e l s : D r i f t
Topic 13
E x a m F o c u s
This topic introduces different term structure models for estimating short-term interest rates. Specifically, we will discuss models that have no drift (Model 1), constant drift (Model 2), time-deterministic drift (Ho-Lee), and mean-reverting drift (Vasicek). For the exam, understand the differences between these short rate models, and know how to construct a two-period interest rate tree using model predictions. Also, know how the limitations of each model impact model effectiveness. For the Vasicek model, understand how to convert a nonrecombining tree into a combining tree.

LO 12.4: Evaluate the im pact o f changes in maturity, yield, and volatility on the convexity o f a security.
The convexity effect can be measured by applying a special case of Jensens inequality as follows:
Example: Applying Jensens inequality
Assume that next year there is a 50% probability that 1-year spot rates will be 10% and a 50% probability that 1-year spot rates will be 6%. Demonstrate Jensens inequality for a 2-year zero-coupon bond with a face value of $ 1 assuming the previous interest rate expectations shown in Figure 1.
Answer:
The left-hand side of Jensens inequality is the expected price in one year using the 1-year spot rates of 10% and 6%.
0.5 x $1
(1.10) T 0.5 x $1
(1.06)
= $0.92624
The expected price in one year using an expected rate of 8% computes the right-hand side of the inequality as:
____________________________$1____________________________ 0.5×1.10 + 0.5×1.06
$1 1.08
0.92593
Thus, the left-hand side is greater than the right-hand side, $0.92624 > $0.92593.
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If the current 1-year rate is 8%, then the price of a 2-year zero-coupon bond is found by simply dividing each side of the equation by 1.08. In other words, discount the expected 1-year zero-coupon bond price for one more year at 8% to find the 2-year price. The price of the 2-year zero-coupon bond on the left-hand side of Jensens inequality equals $0.85763 (calculated as $0.92624 / 1.08). The right-hand side is calculated as the price of a 2-year zero-coupon bond discounted for two years at the expected rate of 8%, which equals $0.85734 (calculated as $1 / 1.082).
The left-hand side is again greater than the right-hand side, $0.85763 > $0.85734.
This demonstrates that the price of the 2-year zero-coupon bond is greater than the price obtained by discounting the $1 face amount by 8% over the first period and by 8% over the second period. Therefore, we know that since the 2-year zero-coupon price is higher than the price achieved through discounting, its implied rate must be lower than 8%.
Extending the above example out for one more year illustrates that convexity increases with maturity. Suppose an investor expects the spot rates to be 14%, 10%, 6%, or 2% in three years. Assuming each expected return has an equal probability of occurring results in the decision tree shown in Figure 3.
Figure 3: Risk-Neutral Decision Tree Illustrating Expected 1-Year Rates for Three Years
The decision tree in Figure 4 uses the expected spot rates from the decision tree in Figure 3 to calculate the price of a 3-year zero-coupon bond.
The price of a 1-year zero-coupon bond in two years with a face value of $1 for the upper node is $0.89286 (calculated as $1 / 1.12). The price of a 1-year zero coupon bond in two years for the middle node is $0.92593 (calculated as $1 / 1.08). The price of a 1-year zero coupon bond in two years for the bottom node is $0.96154 (calculated as $1 / 1.04).
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The price of a 2-year zero-coupon bond in one year using the upper node expected spot rates is calculated as:
[0.5 x ($0.89286 / 1.10)] + [0.5 x ($0.92593 / 1.10)] = $0.82672
The price of a 2-year zero-coupon bond in one year using the bottom node expected spot rates is calculated as:
[0.5 x ($0.92593 / 1.06)] + [0.5 x ($0.96154 / 1.06)] = $0.89032
Lastly, the price of a 3-year zero-coupon bond today is calculated as:
[0.5 x ($0.82672 / 1.08)] + [0.5 x ($0.89032 / 1.08)] = $0.79493
Figure 4: Risk-Neutral Decision Tree for a 3-Year Zero-Coupon Bond
$0.79493
$1
$1
$1
$1
To measure the convexity effect, the implied 3-year spot rate is calculated by solving for r(3)in the following equation:
0.79493
1
(1 + r(3))
r (3) = 3/o j5493 1 = 0-0795 or 7.95%
Notice that convexity lowers bond yields and that this reduction in yields is equal to the value of convexity. For the 3-year zero-coupon bond, the value of convexity is 8% 7.95% = 0.05% or 5 basis points. Recall that the value of convexity for the 2-year zero-coupon bond was only 1.84 basis points. Therefore, all else held equal, the value of convexity
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increases with maturity. In other words, as the maturity of a bond increases, the price-yield relationship becomes more convex.
This convexity occurs due to volatility. Thus, we can also say that the value of convexity increases with volatility. The following decision trees in Figures 5 and 6 illustrate the impact of increasing the volatility of interest rates. In this example, the 1-year spot rate in one year in Figure 5 ranges from 2% to 14% instead of 4% to 12% as was shown in Figure 1.
Figure 5: Risk-Neutral Decision Tree Illustrating Volatility Effect on Convexity
(25% joint probability)
(50% joint probability)
(25% joint probability)
Using the same methodology as before, the price of a 2-year zero-coupon bond with the listed expected interest rates in Figure 5 is $0,858.
Figure 6: Price of a 2-Year Zero-Coupon Bond with Increased Volatility
This price results in a 2-year implied spot rate of 7.958%. Thus, the value of convexity is 8% – 7.958% = 0.042% or 4.2 basis points. This is higher than the previous 2-year example where the value of convexity was 1.84 basis points when expected spot rates ranged from 4% to 12%, instead of 2% to 14%. Therefore, the value of convexity increases with both volatility and time to maturity.
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R i s k P r e m i u m

LO 12.3: Estim ate the convexity effect using Jensens inequality.

LO 11.10: Evaluate the appropriateness o f the Black-Scholes-Merton model when valuing derivatives on fixed income securities. * 1
The Black-Scholes-Merton model is the most well-known equity option-pricing model. Unfortunately, the model is based on three assumptions that do not apply to fixed-income securities:
1. The models main shortcoming is that it assumes there is no upper limit to the price of the underlying asset. However, bond prices do have a maximum value. This upper limit occurs when interest rates equal zero so that zero-coupon bonds are priced at par and coupon bonds are priced at the sum of the coupon payments plus par.
2.
3.
It assumes the risk-free rate is constant. However, changes in short-term rates do occur, and these changes cause rates along the yield curve and bond prices to change.
It assumes bond price volatility is constant. With bonds, however, price volatility decreases as the bond approaches maturity.
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B o n d s W i t h E m b e d d e d O p t i o n s

LO 11.9: Evaluate the advantages and disadvantages o f reducing the size o f the time steps on the pricing o f derivatives on fixed income securities.
For the sake of simplicity, the previous example assumed periods of six months. Flowever, in reality, the time between steps should be much smaller. As you can imagine, the smaller the time between steps, the more complicated the tree and calculations become. Using daily time steps will greatly enhance the accuracy of any model but at the expense of additional computational complexity.
F i x e d -In c o m e S e c u r i t i e s a n d B l a c k -S c h o l e s -M e r t o n