## How negative interest rates mean redoing finance textbooks..

Prof JR Varma has a nice post on the topic.

By assuming non negative interest rates, finance textbooks arrive at many results that are false in a negative rates world. Finance theory does not rule out negative rates – theory requires only bond prices to be non negative, and this only prevents interest rates from dropping below −100%. In practice also, early 2015 saw interest rates go negative in many countries. The BIS 2015 Annual Report (Graph II.6, page 32) shows negative ten-year yields in Switzerland, and negative five year yields in Germany, France, Denmark and Sweden in April 2015.

Let us take a look at how many textbook results are no longer valid in this world:

• The formula for the present value of a perpetuity PV=1/r yields the absurd result that the present value is negative when r is negative. In fact, the present value is infinite (the geometric series diverges for negative r).
• Interestingly, the formula for a growing perpetuity PV=1/(r−g) is still valid under the text book assumption that r>g. But this requires negative g in a negative rates world. That is why the 1/r formula for the zero growth case fails.
• It is no longer true as the textbooks claim that an American call option on a non dividend paying stock would never be exercised prematurely and is therefore the same as a European call. If the call is sufficiently deeply in the money, the holder would want to pay the exercise price as early as possible to avoid the tax (of negative rates) on cash holdings.
• The opposite text book claim about puts is now false. The textbook result is that a deep out of the money put could be exercised early to realize the cash flow early. In a negative rates world, we want to postpone the realization of cash (and avoid paying negative rates on that cash). Consequently, in a negative rates world, American puts would never be exercised early. Even the non dividend paying assumption is not needed for this result.
• It is no longer true that the modified duration of a bond is slightly less than the duration; with negative rates, the modified duration of a bond is slightly more than the duration. Modified duration is given by MD=D/(1+r); if r is negative, the denominator is less than unity and the ratio is therefore more than the numerator.
• Negative rates have not so far generally translated into negative coupons. For example, the Swiss Governmentand German Government have sold bonds with non negative coupons at a premium to par to achieve negative yields. If this trend continues, then in a negative rates world, there will be no par bonds and no discount bonds, and the concept of a par bond yield curve becomes problematic.
• Over a period of time, probably negative coupon bonds will emerge. Warren Buffet’s Berkshire Hathaway sold a convertible bond with a negative coupon way back in 2002. With negative coupons, it is no longer true that the duration of a bond cannot exceed its maturity. It is also not true that for the same maturity, the zero coupon bond has the longest duration. For example, a simple calculation shows that a ten year par bond with a −1% coupon and a −1% yield has a duration of 10.47 years.

How do you explain these ideas?