# The Mathematics of Fixed Interest securities pinpoints the risks

Fixed interest securities are priced using a standard readily available formula. The formula can be used to assess interest rate scenarios. If you buy a bond for \$100 paying a fixed coupon of 3.5% p.a. with a promise to repay \$100 in 5 years what happens to the price of \$100 if secondary market rates rise by 1% p.a. or fall by 1% p.a. on the day of purchase? What happens if we change the maturity to 1 year or 20 years?
By · 27 Jan 2015
By ·
27 Jan 2015
Fixed interest securities are priced using a standard readily available formula.  The formula can be used to assess interest rate scenarios.  If you buy a bond for \$100 paying a fixed coupon of 3.5% p.a. with a promise to repay \$100 in 5 years what happens to the price of \$100 if secondary market rates rise by 1% p.a. or fall by 1% p.a. on the day of purchase?  What happens if we change the maturity to 1 year or 20 years?

Working through each of these:
- if you buy a bond for \$100 paying a 3.5% coupon p.a. for 5 years with a promise to repay \$100 at the end of 5 years what is the price today if rates immediately fall to 2.5% p.a.?  \$104.70 according to the fixed rate bond formula.  What if rates rise to 4.5% p.a.?  \$95.50.
- if I change the maturity to 20 years and again rates fall to 2.5% p.a immediately, then the price rises from \$100 to \$115.70, a lot more than the rise for a 5 year bond.  If yields rise by 1% p.a. to 4.5% p.a. the price falls to \$86.90 again a lot more than for the 5 year bond.
- if I change the maturity to 1 year the price becomes \$101 if rates fall to 2.5% p.a. and \$99 if rates rise to 4.5% p.a..  This is much less volatile and why a low risk fixed interest portfolio needs to contain securities with short term maturities.

The larger the rate movements and the longer the term the greater the price volatility.

Of course, if you buy and hold your 5 year bond having paid \$100, earning 3.5% p.a. for 5 years and regaining your \$100 after 5 years, the secondary market movements pre maturity become irrelevant. Your return will be 3.5% p.a. on the \$100 over 5 years, provided the issuer doesn't default of course.

In the global financial crisis, rising interest rates caused by widening credit spreads saw many long term securities lose a lot of value.  Ultimately many regained their value as time went by and repaid their principal at maturity.  It is good to be a buyer in a crisis but that is only possible when you understand the mathematics.  Many companies and fund managers were unfortunately forced sellers as prices fell.  They didn't understand the maths.

Floating rate notes have a different pricing formula and therefore different scenarios.

 1 year 2 year 5 year 10 year 20 year Coupon 3.5% 3.5% 3.5% 3.5% 3.5% Yield 1.5% 102 104 109.6 118.5 134.5 2.5% 101 102 104.7 108.8 115.7 3.5% 100 100 100 100 100 4.5% 99 98 95.5 92 86.9 6.5% 97 94.4 87.3 78 66.5 10.5% 93 87.5 73 57 41.7