Bond yield curves April 2006 Paper 1 Q7(iii)

In the CA1 webinar on 6th August 2008, I was asked about the Examiners' report solution to the tricky exam question (CA1 paper 1, April 2006, Q7(iii)) on the bond yield curve theories. Here are some examples to help explain the solution. The solution is in bold and my comments are in italics.

To cost the guarantee, it is necessary to estimate future 1 year yields (using the above theories and the yield curve) and compare these rates to the possible guaranteed rate.

The examiners report is talking about estimating future 1-year spot rates from the yields on the yield curve, eg:

10-year yield on yield curve is 5.5%
11-year yield on yield curve is 5.7%
12-year yield on yield curve is 5.9%

We could estimate the future 1-year spot rates as follows:

1-year spot rate (years 10 to 11) = i(10)

Solve (1.05)^10 * (1 + i(10)) = 1.057 ^ 11 so i(10) = 7.72%

1-year spot rate (years 11 to 12) = i(11)

Solve (1.057) ^ 11 * (1 + i(11)) = 1.059 ^ 12 so i(11) = 8.13%

The yield curve is upward sloping so based on the expectations theory the market expectations for future short-term (1-year) interest rates is that the one-year rates will progressively increase.

The figures above show this.

Expectations theory says that the yield curve is based on expectations of future short term interest rates. I have used this assumption in my calculations above.

Part of the upward slope of the yield curve relates to liquidity rather than the expectation of increases to short-term rates. Thus based on the liquidity preference theory using the yield curve to estimate future short-term rates risks overestimating future rates therefore understating the cost of the investment guarantee.

Liquidity preference theory "uplifts" the yield curve as investors require a risk premium for liquidity risk. The uplift is greater for longer-term, less liquid bonds.

Without the liquidity preference uplift, we might find that the yields on the yield curve would be (deducting the uplift):

10-year yield on yield curve is 5.45%
11-year yield on yield curve is 5.6%
12-year yield on yield curve is 5.75%

I have assumed uplifts of 0.5% (10 year), 1% (11 year) and 1.5% (12 year), ie they increase with term.

The corresponding 1-year spot rates would be:

i(10) = 7.11%
i(11) = 7.41%

Since these 1-year spot rates are lower than before, when we included the liquidity preference uplift, we overestimated the 1-year spot rates.

If we overestimate the 1-year spot rates, it is more likely that the guaranteed fixed rate will NOT bite. Hence we will have underestimated the cost of the guarantee.

Market segmentation theory says that the yields on long and short bonds may move somewhat independently, so the yield curve does not necessarily predict the future short-rates so care should be exercised in using the yield curve to anticipate the future cost of investment guarantees.

This is saying that in order to estimate the 1-year spot rates, we should also strip out the effects of market segmentation theory. Really, we need to be left with the effects of expectations theory alone.

Using the initial yield curve to anticipate the cost of investment guarantees risks mis-stating the anticipated cost, however, there may be no better objective information available.

We have seen this with the above numerical example.

If there is other market information indicating that future short-term yields will be different from those predicted from the initial yield curve then this suggests an arbitrage opportunity exists.

Suppose that the yield curve is the same as before but the 1-year spot rates are different to what we had estimated before, suppose instead that they are:

i(10) = 7.08%
i(11) = 7.21%

One investor could buy a 10 year bond yielding 5.45% and then reinvest the proceeds at the end of 10 years in a 1-year zero-coupon bond yielding 7.08% and then reinvest the proceeds at the end of that year in another 1-year zero-coupon bond yielding 7.21%. The average return on this investment is expected to be [1.0545 ^ 10 * 1.0708 * 1.0721 ] ^ (1/12) - 1 = 5.73%.

This is less that the average return of 5.75% that could be achieved by investing in a 12-year bond from the start. Hence an arbitrage opportunity exists.

Initial market yield information should therefore be used carefully and with judgement when estimating the cost of investment guarantees.

The yield curve will vary over time. Uncertainty over predicting future yield curves will increase the further in the future the prediction is required there is an expanding funnel of doubt. Therefore the actual cost of the investment guarantee may be considerably different from that anticipated at outset.

To estimate the cost of the guarantee, assumptions as to future yields must be estimated. The further into the future we estimate, the more likely we are to be wrong and hence for the actual experience.

All crystal clear I expect now!!!

Anna

 

Hi Frances

We need to be a little bit careful here as we don't actually "know" what the future one-year spot rates will be, we can only estimate them from the current yield curve. What is important for this question is that we are trying to estimate future short term (one-year) yields using the current yield curve, in order to price the investment guarantee (since that guarantee is based on one-year bond yields).

The reason why we need to try to strip out the effects of liquidity preference and market segmentation theories is because those two theories state that the yield curve depends partly on factors that affect long term bond yields, and not just factors that affect short term bond yields. So before we can use the yield curve to estimate future short term bond yields, we would need to somehow remove any part of the yield curve that reflects aspects relating to long term bond yields - because these are not relevant to what we are doing. Such aspects would include how liquid long term bonds are (liquidity preference) and issues relating to supply/demand of long term bonds (market segmentation).

It is not so much a case therefore of us deciding to "use pure expectations theory alone", but more that we cannot use the yield curve in order to determine future short term bond yields without removing anything that has caused the yield curve to be the shape it is which relates to long term bond yields rather than to short term bond yields. [These influences will of course be seen towards the right hand side of the curve, at longer terms to redemption, so most of this "stripping out" will need to be done at these longer terms.]

Does that make a little more sense? Do let us know if you need further explanation.

 

Great - but do of course post again if you need further clarification. As Anna says, this was a tricky question!