Interest rate risk, spot and forward rates

Interest rate risk is the risk that the rate on money a company has borrowed, or has invested, moves against it. It is not a remote possibility — a business carrying floating-rate debt is exposed every time the Bank of Zambia moves its policy rate.

The exposure is the gain or loss the movement produces. A company with ZMW40 million of floating-rate borrowing facing a two-point rise in rates is ZMW800,000 a year worse off, and that lands entirely on profit before tax.

Which is why a small movement in rates makes a large difference to reported profit. Put that ZMW800,000 against a company earning ZMW3 million before interest and the rate rise has taken a quarter of it. The interest cover working in Lesson 2 showed the same mechanism from the other side: fixed interest amplifies whatever happens, and a rise in rates is a rise in the amount being amplified.

A spot interest rate is the rate today on money lent from today until a single point in the future. A forward interest rate is the rate today on money that will be lent starting at some future date, for a specific period after that.

• A two-year spot rate is what you earn by lending now and being repaid in two years.
• A one-year forward rate beginning in year 1 is what you would earn by lending for the twelve months between year 1 and year 2 — agreed now, but not starting yet.

The notation is compact once you know it reads left to right as a period. The leading subscript is when the lending starts, the trailing one is when it ends.

Spot rates are not one number. There is a different one for every maturity, and each cash flow ought properly to be discounted at the spot rate for the period in which it arrives rather than at a single average rate for the whole security.

Spot rates are not published as a list. They are extracted from the prices of government zero coupon bonds, because a zero coupon bond has exactly one cash flow — which makes its yield the pure spot rate for that maturity, with nothing else mixed in.

One-year, two-year, three-year and four-year government zero coupon bonds are priced at ZMW86.96, ZMW77.64, ZMW71.89 and ZMW65.35.

Read the column. Rates are highest at one year and fall as maturity lengthens — the market expects short-term rates to come down. That downward shape is a signal in itself, and the next step is about what it means.

The common mistake is dividing rather than taking the root. 100 ÷ 65.35 = 1.53, and treating that as 53% over four years and dividing by four gives 13.3% instead of 11.2% — the difference is compounding, which simple division ignores.

Spot rates and forward rates are locked together by one idea: lending for two years must earn the same as lending for one year and then relending for the second. If it did not, an investor could borrow one way and lend the other and make money for nothing, and the rates would move until the gap closed.

Rearrange for whichever rate is missing. With a two-year spot rate of 6.25% and a one-year spot rate of 5%:

The same construction extends to any two maturities, and there is usually more than one route to the same answer.

Take the spot rates from the last section and find the two-year forward rate starting in two years — the rate for years 3 and 4 taken together. Four years must equal two years followed by that rate for two more.

And the one-year forward rate beginning in year 3 — four years being three years plus one more:

Two rules keep this straight. Raise each bracket to the number of years that rate covers, and take a root at the end equal to the length of the forward period you are solving for — a two-year forward needs a square root, a one-year forward needs none.

Once you have a spot rate for each maturity, each of a bond’s cash flows can be discounted at the rate that belongs to its own year rather than at one rate applied to all of them. That is the more accurate valuation, and it is what the market actually does.

MBS has 10% ZMW1,000 four-year bonds redeemable at nominal value, with interest paid annually. The spot rates are 4% for one year, 4.5% for two, 5% for three and 6% for four.

Each factor is 1 ÷ (1 + that year’s spot rate) raised to that year’s number — 1 ÷ 1.045² for year 2, not 1 ÷ 1.045 and not 1 ÷ 1.04². The rate changes by row and so does the exponent.

The bond trades well above its ZMW1,000 nominal value because its 10% coupon is far above the 4% to 6% the market is paying. That is the same relationship the valuation lesson established, now stated one year at a time.

Notice how lopsided the present values are. The year 4 line is ZMW871 of a ZMW1,145 price — 76% of the bond’s value sits in its final cash flow. Which is why a change in the four-year rate moves this bond’s price far more than a change in the one-year rate does, and why measuring that sensitivity needs a weighted measure rather than just the maturity.