# LO 12.5: Calculate the price and return o f a zero coupon bond incorporating a risk

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. 2018 Kaplan, Inc. Page 157 Topic 12 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 8 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
$0.85525 0.083 Therefore, risk-averse investors require a 30 basis point premium or 8.3% return to compensate for one year of interest rate risk. For a 3-year zero-coupon bond, risk-averse investors will require a 60 basis point premium or 8.6% return given two years of interest rate risk. Professors Note: In the previous example, it is assumed that rates can change only once a year, so in the first year there is no uncertainty o f interest rates. There is only uncertainty in what the 1-year rate will be one and two years from today. Page 158 2018 Kaplan, Inc. Topic 12 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 8 K e y C o n c e p t s LO 12.1 If expected 1-year spot rates for the next three years are rate, r (2), is computed as r (2) = ^ (l + q )(l + r2) 1, and the 3-year spot rate, r (3), is rv and ry then the 2-year spot computed asr(3) = ^(l + q )(l + r2)(l + r3) 1. LO 12.2 The volatility of expected rates creates convexity, which lowers future spot rates. LO 12.3 The convexity effect can be measured by using Jensens inequality: E 1 (l + r) > 1 E[l + LO 12.4 Convexity lowers bond yields due to volatility. This reduction in yields is equal to the value of convexity. Thus, we can say that the value of convexity increases with volatility. The value of convexity will also increase with maturity, because the price-yield relationship will become more convex over time. LO 12.3 Risk-averse investors will price bonds with a risk premium to compensate them for taking on interest rate risk. 2018 Kaplan, Inc. Page 159 Topic 12 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 8 C o n c e p t C h e c k e r s 1. An investor expects the current 1-year rate for a zero-coupon bond to remain at 6%, the 1-year rate next year to be 8%, and the 1-year rate in two years to be 10%. What is the 3-year spot rate for a zero-coupon bond with a face value of$ 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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Topic 12 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 8
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.
2018 Kaplan, Inc.
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Topic 12 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 8
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 Page 162 2018 Kaplan, Inc. Topic 12 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 8 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.