Temp_store

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

LO 11.6: Define option-adjusted spread (OAS) and apply it to security pricing.
The option-adjusted spread (OAS) is the spread that makes the model value (calculated by the present value of projected cash flows) equal to the current market price. In the previous CM T example, the model price was equal to $1,466.63. Now assume that the market price of the CM T swap was instead $1,464.40, which is $2.23 less than the model price. In this case, the OAS to be added to each discounted risk-neutral rate in the CM T swap binomial tree turns out to be 20 basis points. In six months, the rates to be adjusted are 7.25% in the up node and 6.75% in the down node. Incorporating the OAS into the six-month rates generates the following new swap values:
($2,500 x 0.6) + ($0 x0.4)
+ 0.0745 1
+ $1,250
$2,696.13
($ 0 x 0 .6 )+ (-$ 2 ,5 0 0 x 0 .4 )
t | 0.0695
2
$1,250
-$ 2 ,2 1 6 .4 2
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Notice that the only rates adjusted by the OAS spread are the rates used for discounting values. The OAS does not impact the rates used for estimating cash flows. The final step in this CM T swap valuation is to adjust the interest rate used to discount the price back to today. In this example, the discounted rate of 7% is adjusted by 20 basis points to 7.2%. The updated initial CM T swap value is:
($2,696.13 x 0.76) + (-$2,216.42 x 0.24)
+ 0.072 1
$1,464.40
Now we can see that adding the OAS to the discounted risk-neutral rates in the binomial tree generates a model price ($1,464.40) that is equal to the market price ($1,464.40). In this example, the market price was initially less than the model price. This means that the security was trading cheap. If the market price were instead higher than the model price we would say that the security was trading rich.
T i m e S t e p s

LO 11.8: Calculate the value o f a constant maturity Treasury swap, given an interest rate tree and the risk-neutral probabilities.
In addition to valuing options with binomial interest rate trees, we can also value other derivatives such as swaps. The following example calculates the price of a constant maturity Treasury (CMT) swap. A CM T swap is an agreement to swap a floating rate for a Treasury rate such as the 10-year rate.
Example: CM T swap
Assume that you want to value a constant maturity Treasury (CMT) swap. The swap pays the following every six months until maturity:
/
$ 1,000,000
2
X (y
V C M T
Tc m t ls a semiannually compounded yield, of a predetermined maturity, at the time of payment (y( is equivalent to 6-month spot rates). Assume there is a 76% risk-neutral probability of an increase in the 6-month spot rate and a 60% risk-neutral probability of an increase in the 1-year spot rate.
Fill in the missing data in the binomial tree, and calculate the value of the swap.
Figure 5: Incomplete Binomial Tree for CM T Swap
Int. rate = 7.25% Swap price = ?
0.76
Int. rate = 6.75% Swap price = ?
Int. rate = 7.50% Swap price = ?
0.60
Int. rate = 7.00% Swap price = ?
Int. rate = 6.50% Swap price = ?
0.40
Today
Six Months
One Year
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Answer:
In six months, the top node and bottom node payoffs are, respectively:
payoff^ = ———— x (7.25% 7.00%) = $1,250
$ 1,000,000
payoff^ = ————-x (6.75% 7.00%) = $1,250
$ 1,000,000
rr
rr
2
2
^
^
Similarly in one year, the top, middle, and bottom payoffs are, respectively:
r r
$ 1, 000,000
/ r r i r A f t /
payoff2ju = ————-x (7.50% 7.00%) = $2,500
^
^ payoff2jM = ————-x (7.00% 7.00%) = $0
/ r r / w w w
r
r
$ 1,000,000
2
rr
$ 1,000,000
payoff2 L = ————-x (6.50% 7.00%) = $2,500
2
The possible prices in six months are given by the expected discounted value of the 1-year payoffs under the risk-neutral probabilities, plus the 6-month payoffs ($1,250 and
$1,250). Hence, the 6-month values for the top and bottom node are, respectively:
l,u Vi
V
1>L
($2,500×0.6)+ ($0x0.4)
1+ 0.0725
+ $1,250 = $2,697.53
($ 0 x 0 .6 )+ (-$2,500×0.4)
+ 0.0675 1
-$1,250 = -$2,217.35
Today the price is $1,466.63, calculated as follows:
w ^ V Q — —————————ri'”r\~v———————- $1,466.63
($2,697.53 x 0.76) + ($2,217.35 x 0.24)
.
1+ 0.07
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Figure 6 shows the binomial tree with all values included.
Figure 6: Completed Binomial Tree for CM T Swap
0.60
Int. rate = 7.50%
Swap price = $2,500.00
Int. rate = 7.00%
Swap price = $1,466.63
0.76
0.24
Int. rate = 7.25%
Swap price = $2,697.53
Int. rate = 6.75%
Swap price = -$2,217.35
Int. rate = 7.00%
Swap price =
$0
Int. rate = 6.50%
Swap price = -$2,500.00
0.40
Today
Six Months
One Year
O p t i o n -Ad j u s t e d S p r e a d

LO 11.7: Describe the rationale behind the use o f recombining trees in option pricing.
In the previous example, the interest rate in the middle node of period two was the same (i.e., 6.34%) regardless of the path being up then down or down then up. This is known as a recombining tree. It may be the case, in a practical setting, that the up then down scenario produces a different rate than the down then up scenario. An example of this type of tree may result when any interest rate above a certain level (e.g., 3%) causes rates to move a fixed number of basis points, but any interest rate below that level causes rates to move at a pace that is below the up states fixed amount. When rates move in this fashion, the movement process is known as state-dependent volatility, and it results in nonrecombining trees. From an economic standpoint, nonrecombining trees are appropriate; however, prices can be very difficult to calculate when the binomial tree is extended to multiple periods.
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C o n s t a n t M a t u r i t y T r e a s u r y S w a p