Interest rates and time

Time, credit and interests rates are inseparable.  Bear in mind from the first installment of this blog series that money has a time value: the dollar you hold in your hand today is worth more than that same dollar two, five or ten years from now.  How much more?  To explain this requires some mathematics.  Now, pondering mathematical formulas is undoubtedly far from the top of the favorite Internet activities of most people, and you are by all means invited to skip this next paragraph if the preponderance of numbers should appear less than appealing.  On the other hand if there is one formula in the whole pantheon of financial principia mathematica worth actually committing to memory it is probably the one having to do with the time value of money.

So here it is: PV = FV / (1+r)ⁿ where FV=future value, PV=present value, r is the annual rate of interest and n is the time period (generally expressed in years). Consider the $1,000 5-year bond with a 5% interest rate.  Using the time value of money formula we see that what the lender actually receives over the term of the bond, in principal and interest, is equal to $1,000 discounted by the lender’s own deemed value of that $1,000, expressed by the interest rate.

FV (principal paid back at maturity) = $1,000

PV (that principal expressed in present value terms) = $1,000 / (1+0.05)⁵ = $783.5262

So from the lender’s perspective the opportunity cost of foregoing a dollar today for a dollar five years from now is mathematically expressed as a discount on that dollar.  The compensation for that opportunity cost, of course, is the coupon interest payment.  Thus:

FV (annual coupon) = $50 each year for five years

PV (annual coupon) = $50 / (1+0.05)ⁿ where n = 1, 2, 3, 4, 5 (five equal payments at the end of each year the bond is outstanding).

PV (total coupons) = ($47.61905+45.35147+43.19188+41.13512+39.17631) = $216.4738

So PV (total) = $783.5262 + $216.4738 = $1000.00, Q.E.D.

As you delve into the world of credit instruments the mathematics can become fantastically complicated.  Just remember that most of it ultimately derives from this fundamental relationship between money today and money tomorrow.

Here’s a practical application of that relationship: how time affects the price of credit instruments in reaction to changes in interest rates.  Market interest rates change every day for a wide variety of reasons, some of which we will cover in the third installment of this series (including monetary policy, the business/economic cycle, inflation and the general price of risk).  The key point to retain (again, we will walk through some mathematics below, but if you want to skip that just remember this): All else being equal, the price of a credit instrument with a longer term to maturity is more sensitive to a given change in interest rates than is an instrument with a shorter maturity. You may hear a financial news report say something like this: “investors are moving into shorter term bonds amid fears that the Fed will raise interest rates”.  That sentiment is what this example is all about.

To illustrate, let’s assume here that we are evaluating zero-coupon bonds.  A zero coupon bond’s market value is simply the present value of the future par amount (e.g. $1,000) at the prevailing rate of interest.  For example the present value of a zero coupon bond with five years to maturity and a 5% interest rate is $783.53 – an investor pays that amount today and gets $1,000 back in five years with no coupon payments in between.  What about if the bond matures in one year?  Then the present value is $1,000 / (1+0.05)¹ = $952.38.

But what if the level of prevailing market interest rates rises from 5% to 6%?  The market price of the bond must also change – you couldn’t sell a bond yielding 5% if investors can get yields of 6% elsewhere for equivalent instruments.  How will your price change?  Just re-calculate that same time value formula with the “r” value changed from 5% to 6%.  The new market price for the five-year ZCB is thus $747.26 and that for the one-year ZCB is $943.40.  The thing to notice here is the magnitude of the change in price.  With a 1% change in rates the one-year ZCB price changed from $952.38 to $943.40, a decrease of 0.94%.  By contrast the change in price of the five-year bond was a 4.63% decline (from $783.53 to $747.26).  In other words, the longer-term bond was more sensitive than the shorter-term bond to the same level of change in interest rates.  Since investors know that bonds with longer maturities react more to interest rate changes than do short-term bonds, they will be more inclined to buy long-dated instruments if they think market interest rates are going down (i.e. bond prices are going up), and will prefer shorter maturities if they think rates are likely to go up.