CPD Spotlight Quiz Investing in Bonds Question 1 Risk of rates changing the basics All debt instruments have a market value that should be the sum of the present values of the component cash flows. In other words, the market value of a coupon bond s hould equal the present value of all of the coupons plus the present value of the redemption amount at maturity. So the value of the bond is dependent on the discount rate used or the yield and the cashflows associated with the bond along with their timing. Given that, once issued the bond maturity is fixed and cannot vary, the cashflows are clearly specified and cannot vary, any change in value can only be due to a change in yield. Yield change might come about because overall rates change or because the perceived credit risk of the issuer has changed. Either possibility will have a similar effect on the market value of the bond. If you invest in a fixed coupon bond with 5-year maturity and a redemption value of 100, which of the following would you be most concerned about? (a) yields rising (b) yields falling (c) yields staying the same (d) the yield curve changing from normal to inverse. (e) Don t know The right answer is (a) yields rising You would be concerned that rates/yields might rise because of the inverse relationship between bond values and rates/yields. In other words, rates rise and market values fall; rates fall and market values rise. This inverse effect is due simply to the greater effect of discounting future cashflows at a higher rate, giving a lower present value. Investors want a higher return for investing now in the hope of an expected cashflow in the future. That might be because investors see a higher risk associated with this bond s cashflows or their other possible investments offer higher returns rates generally have risen. The extreme case of this is rates being zero, when future cashflows are valued at their future value. When this happens, longer bonds become more valuable because their distant redemption is valued at 100%, as are all of the coupon payments. As rates rise, the discounting effect has most impact on distant cashflows, i.e. the redemption amount of a long bond so values fall faster on longer bonds than shorter bonds, assuming similar levels of coupons. Question 2 Value sensitivity
Market Value If the relationship between the value of a bond and its yield is plotted, a curve like the one below is produced. Market value of coupon bond v Yield 140.00 70.00 0.00 0% 20% Yield The slope of the line representing the relationship is indicative of the price sensitivity of the bond at that yield. Which of the following risk measures most nearly gives the slope of the curve in the diagram above? (a) (b) (c) (d) (e) duration modified duration convexity modified convexity don t know The right answer is (b) modified duration Duration is possibly the best known measure of risk in bond investment. It measures the weighted average timing of cashflows giving a result in years. So a bond with a duration of 5 years is likely to behave, in terms of price variability, in the same way as a zero coupon bond of that same maturity. In the case of a zero coupon bond there is only a single cashflow because there are no intermediate coupons. So the duration of a zero coupon bond is the same as its maturity. Modified duration is derived from duration dividing the duration result by (1+yield). This gives a measure from which we can make a first approximation of price change for a given yield change. This measure of price change is very close to being the slope of the line
above. But, importantly it is not actually the slope of the line. We calculate the price change by multiplying out original price modified duration yield change. This change can then be added to the original price if rates are falling or deducted from the original price if rates are rising. So the actual slope of the curve is ( original price modified duration ). Convexity is the second derivative of price with respect to yield, where duration is the first derivative. Duration and modified duration calculate straight line changes price changes are linear as yield changes. So duration and modified duration measures are useful only over relatively short yield ranges. Convexity tries to measure the curviness of the relationship or how much the true value deviates from the straight line estimate. Again, modified convexity is the measure used to calculate price deviation from the straight line. Question 3 Modified Duration We have a bond investment and use modified duration to estimate the price change we expect from a small change in yield. Which of the following is true of our estimate of the new price after a yield change? we will always overestimate the new price we will always underestimate the new price we will over-estimate the new price when yields fall but under-estimate when yields rise we will under-estimate the new price when yields fall but over-estimate when yields rise (a) don t know The right answer is (b) we will always underestimate the ne w price. Given the shape of the curve discussed before the straight line estimate, being the tangent to the curve, will always be below the curve, never above the curve, as shown below. The dotted line is the modified duration estimate while the curve is the fully recalculated value of the same bond.
Market Value Market value of coupon bond v Yield 210.00 140.00 70.00 0.00 0% 20% Yield This can cause some confusion because the implication of the estimate of the ne w price always being too low is that when yields are falling moving to the left in the diagram the price change estimate is too low. But when yields are rising, the price change estimate is too high. This is partly just different ways of saying the same thing but there is an important issue underlying the potential confusion. Value rises when rates fall are greater than value falls when rates rise. The differences are quite small though! Question 4
Market Values) Market Values of annual coupon bonds v Yield 250.0 200.0 150.0 100.0 50.0 0.0 0% 5% 10% 15% 20% Yield The chart above shows the relationship between market value and yield for two bonds with the same annual coupons but with different maturities. Which of the following is true? The dotted line represents a 10 year bond with a 10% coupon; the solid line represents a 5 year bond with a 5% coupon The dotted line represents a 5 year bond with a 5% coupon; the solid line represents a 10 year bond with a 5% coupon The dotted line represents a 20 year bond with a 10% coupon; the solid line represents a 10 year bond with a 5% coupon The dotted line represents a 20 year bond with a 5% coupon; the solid line represents a 5 year bond with a 5% coupon Don t know The right answer is (d) The dotted line represents a 20 year bond with a 5% coupon; the solid line represents a 5 year bond with a 5% coupon From the diagram both lines appear to take a value of 100 at 5% - so they represent bonds trading at par at 5% - in other words their coupon must be 5%, the par yield. At 0% yield the steeper (dotted) curve takes a value of 200. That is the redemption (100) plus a further 100 when future values are equal to present values. If the annual coupon is 5%, then future cash flows will be 100 if the maturity of the bond is 20 years. Using similar logic for the solid curve, the value looks to be 125 at 0% yield i.e. 100 redemption plus 25 or 5% annual coupon for 5 years.
Market Values) Question 5 Market Values of annual coupon bonds v Yield 250.0 200.0 150.0 100.0 50.0 0.0 0% 5% 10% 15% 20% Yield Using the same diagram, it appears as though the longer bond, represented by the dotted line, changes value by much more than the shorter bond at low yields. But at higher yields, to the right of the diagram it looks as though the change in value, represented by the steepness of the curve, is similar to the shorter bond. Imagine that current yields are very high by today s standards and that you are considering investing in one of the two bonds: one a 20-year bond with a 5% annual coupon, the other a 5-year bond with a 5% annual coupon. Which of the following would fairly represent the potential for the risk of your investment if yields fell from 18% to 17%? Both bonds would change in value by about the same proportion In money terms the longer bond will rise in value by more than the shorter bond The longer bond would rise in value by a much higher proportion than the shorter bond The longer bond would fall in value by a higher proportion than the shorter bond Don t know The right answer is (c) The longer bond would rise in value by a much higher proportion than the shorter bond When rates are 18%, the longer bond will be valued at: PV of redemption + coupon annuity factor, i.e. 20 20 1 (1 18 %) 30. 41 100 (1 18 %) 5 When rates fall to 17% the value will become:
20 20 1 (1 17 %) 32. 47 100 (1 17 %) 5 This is an increase of 2.06 or 6.77% The shorter bond will be valued in exactly the same way the numbers being: 5 5 100 (1 18 %) 5 1 (1 18 %) 59. at 18% 5 35 5 1 (1 17 %) 61. 61 100 (1 17 %) 5 This is a rise in value of 2.26 or 3.81% at 17% So the longer bond has increased in value by only 2.06 whereas the shorter bond has risen in value by 2.26 shorter bond rises by more! So answer (b) is not right. In percentage terms the longer bond has increased by 6.77% whereas the shorter bond has risen by a mere 3.81%. An investment when yields are 18% would therefore benefit more from investing in the longer bond, if rates where to fall to 17%. Question 6 Investing with a fixed time horizon Investment by non-financial companies in financial instruments is fundamentally different from investment for financial companies. For non-financial companies the broad thrust of investment is in real assets rather than financial assets. When investment in financial assets does arise, it is often as a short term investment; finding a profitable home for funds until they are required. This fixed timescale means that it is rare to be able to invest in a bond until maturity so fixing a longer term return. Instead, the investor must sell at market yields in the future and be exposed to a form of interest rate risk rates may change so that the investment yields insufficient funds to make the payment. Fortunately, we can immunise our investment to a certain extent by ensuring that the duration of the investment is equal to the duration of the payment, i.e. the timing of the payment if it is a single outflow. This is fine if we can find a single investment with the appropriate duration. But often we need to split the total investment to achieve the desired duration. We can then vary the proportions in each component to achieve the portfolio duration that we need. We need to make a payment in 2 years time. We have the funds now and wish to invest them until the time of the payment. We have a choice of investments with duration of either 1 year (Investment A) or 5 years (Investment B). In what proportions should we split the investment to try to ensure that the proceeds of the investment portfolio will cover the payment? 20% in Investment A, 80% in investment B 25% in Investment A, 75% in Investment B 75% in Investment A, 25% in Investment B 80% in Investment A, 20% in Investment B (a) Don t know The right answer is (c) 75% in Investment A, 25% in Investment B
We need to create a portfolio with a duration of 2 years, so we need to mix the two investments to create that. So, 0.75 1year + 0.25 5years = 2.0 years. Other mixes result in duration not equal to 2 years, the timing of the payment, and therefore more exposed to interest rate risk. It may be easier to see this as a diagram: T 0 1 year Duration of Investment A 2 years Payment Required 5 years Duration of Investment B We are trying to centre the investment on 2 years to coincide with the payment timing. Investment A is 1 year before while Investment B is 3 years after, so the split should reverse those timing differences Investment A having ¾ of the portfolio and Investment B having ¼ of the portfolio. All we are doing here is to create a weighted average. The logic underlying this is that the initial investment required is the present value of the payment. This initial amount should be split between the potential investments as above. When Investment A matures the proceeds are reinvested at then-ruling rates until required, i.e. the time of the payment. At the time of the payment the investment in Investment B is sold at market value that value being determined by then-ruling rates. The proportions above are those that will provide the expected (or hoped-for) value over a range of future rates. The flaw in the logic is, of course, that the then-ruling rates covers a full year in the example above. The argument allows for only two potential rates present rates and future rates whereas we all know that future rates are rarely steady over such a long time period.