The purpose of this page is to explain the usage and results that can be generated by the spreadsheet (Excel or OpenOffice) that automates most all the issues discussed here. Many bond investors will already understand the concept of rolling down the yield curve. Those who don’t should work through this page before making any changes to the default variables of the spreadsheet. The default variables are used to produce the graphs below. Understanding the concept of what should happen is not much use in the real world unless you can quantify your expectations. This spreadsheet allows you to do that. Its accuracy depends on the accuracy of your predictions about future interest rates.

#### Yield Curves

First you must understand the yield curve. Above is the graph of historical yields for various maturities of Government of Canada bonds. The relationship between the rates of different maturities at any specific date can be graphed with the different maturities along the bottom axis. When the rate differences are large the lines in the chart above splay apart. The yield curve at that time would look steeply rising. When the rates are similar across all maturities the yield curve would look flatter. It can even become inverted (e.g. 1990) with the shortest rates higher than the long rates. There are four factors determining what rate the market demands for different maturities.

- The liquidity preference reflects that loans repaid next year are less risky than those repaid later.
- The return required longer terms reflects the expected market rates that will be in effect at those later dates.
- Supply and demand for different debt terms may vary leading to different pricing.
- Investors have a maturity preference that will only change with sufficient pricing differences.

The market rates for the benchmark maturities that form the yield curve below are the variable inputs in the spreadsheet.

#### Price Curves

it is important to add the dimension of time to bond analysis. If you buy a 10-year bond, then next year you will own a 9-year bond, and the year after that an 8-year bond, etc. Each year your bond moves incrementally closer to maturity, resulting in lower volatility and shorter duration and demanding a lower interest rate. Since falling rates create increasing prices, the value of a bond initially will rise as the lower rates of the shorter maturity become its new market rate. Because a bond is always anchored by its final maturity, the price at some point must change direction and fall to par value at redemption.

Using the spreadsheet, a bond’s market value at different times in its life can be calculated. The chart above shows a 30-year bond issued at par. In one case the yield curve is steep and the bond is predicted to have a large capital gain before later falling in price. When the yield curve is flatter, the capital gain is predicted to be much less. Notice the word ‘predicted’. This chart assumes that the market rates that determine the yield curve do not change over the 30 years of the bond’s life.

The bond’s increasing and then declining value has a few implications.

1) Advice to retired investors is often to hold a bond ladder of different maturities. Each bond is allowed to mature. The proceeds fund your living costs. But if there is an existing capital gain in the bond, it may be better to sell it BEFORE maturity. Remember transaction costs.

2) Bond ETFs have mandates to invest in bonds of specific maturities. Try to figure out whether their holding period will allow for the realization of capital gains from rolling down the yield curve. Stay away from those where the allowed maturities are within a very tight range. A Canadian ETF that holds to maturity a ladder of 5 year bonds seems to capture all the worst aspects of the price curve.

3) An ongoing bond portfolio can ride each bond down the yield curve and roll over to a new long-term bond at the optimal point to benefit from the capital gain. Remember that if you reinvest the larger capital in another long-term bond you also assume the risks of that longer maturity. The spreadsheet has a box to calculate your expected bump up in yield from this action.

#### Total Returns

The optimal time to exit a bond is not when its valuation is the highest. You probably think “I’ll get the same coupon $$ every year anyway, so I should try to maximize the capital gain”. The error in that argument is that a stable coupon $$ becomes a smaller % yield as the bond’s value increases. You are trading off yield for capital gains. To properly evaluate your situation you need to calculate the yearly ‘total return’ – the sum of the coupon yield plus the capital gain from the changing valuation. You compare this total return to your alternate investment possibilities.

The chart above shows that the yearly total-returns are quite different from the chart of the capital gains only. When the yield curve is flat there is little variability in the returns over time. But when the yield curve is steep returns are much higher to start and much lower approaching maturity. For reference, this bond was issued at par with a 4.4% coupon and yield-to-maturity.

#### Rising Rates

When the media talks about rising interest rates, too often investors presume that rates all along the yield curve will rise by the same amount – the curve will move up in parallel. Because longer-term bonds have a larger duration that rise in rates will cause a larger capital loss for them, than for short-term bonds. But rates do NOT rise equally along all the maturities. Almost always, the long maturity’s rate will change much less. When rates rise by flattening the yield curve, the greater change in rates at the short end will offset to some extent the advantage provided by the shorter bond’s lower duration. So staying in short maturities may not provide the protection from rising rates that you expect.

Looking at the chart above, a flattening of the yield curve from a steep curve will change your expected yearly total returns from the blue line to the red line. The rise in rates causes a capital loss but if you hold to maturity to recover the loss with the higher yearly returns of the red line.

For this discussion’s purpose pretend that rising interest rates change from the steep yield curve to the flat one – all in one year. The example used here (and the default variables in the spreadsheet) has the 30-year bond’s rate the same in both the steep and the flat yield curve situations. Everyone will have capital losses except for those with 30-years remaining, because the market rates for that maturity have not changed. The size of the loss varies. In this example the bond with only 4 years (50 months) remaining shows the largest capital loss, not the longer term bonds with the larger duration. Notice that the chart above shows the one year’s return experienced ONLY ONCE by the bond depending on its term-to-maturity when the rate change happened. The bottom axis does NOT represent time.

But the capital loss is not the only thing happening to your returns. A better evaluation of the situation measures each year’s ‘total return’ – the combination of the coupon’s yield, the gain from rolling down the yield curve plus the capital loss. In this example you see a total-return loss of only 4% – pretty manageable compared to the 8% capital loss.