Realized Return

• A bond’s realized return is calculated by comparing the initial investment’s value with its final value. To take a simple example, suppose a bond is bought for USD 98 immediately after a coupon payment date. It earns a coupon of USD 1.75 in six months and is worth 98.5 at that time. The capital gain from the increase in the bond’s price is  98.5 – 98 = 0.5. The realized return from the bond over six months is therefore: (98.5 – 98 +1.75)/98 = 2.296%.
• This is (2 x 2.296)% = 4.592% per annum with semi-annual compounding.
• If the return over a longer period (e.g., one year) is to be determined, the investment of the coupon received at the six-month point must be considered as well. Suppose the invested proceeds from the coupon earn 1.1% for the following six months and that the bond is worth USD 98.7 after one year. The realized return on the bond over the one-year period is (98.7 + 1.75 +1.75 x 1.011 – 98)/98 = 4.305% or 4.305% per annum with annual compounding These returns are referred to as gross returns because the cost of financing the bond purchase is not considered.
• The net return is the return after financing costs have been subtracted. Suppose that the position in the bond has been financed at 3% per annum (semi-annually compounded). The cost of financing the purchase of the bond for six months in our first example 3/2 of 98 = 1.47. The net realized return over six months is therefore: (98.5 + 1.75 – 98 -1.47)/98 = 0.796%.
• This is 2 x 0.796 = 1.592% per annum with semi-annual compounding. The cost of financing the bond for one year is (1.015 x 1.015 – 1) x 98 = 2.962. The net realized return over one year is therefore: (98.5 + 1.75 + 1.75 x 1.011 – 98 – 2.962)/98 = 0.01283 or  1.283%(with annual compounding).

Spreads

• An investor buying a Treasury security may be interested in calculating the excess return earned over the return provided by other Treasury securities. This can be done by asking the following question:
  • What spread do we have to add to the prevailing Treasury forward rates so that the present value of the cash flows on the security equals the price paid for the security?
• Suppose Treasury forward rates (expressed with semi-annual compounding) are as presented in this Table. Spot rates can be calculated by compounding forward rates. Discount factors can be calculated in a direct way from spot rates.
• The six-month discount factor is

• The 12-month discount factor is

• The 18-month discount factor is

Finally, the 24-month discount factor is

These discount factors are summarized in the following Table

Period (months)	Forward Rate (%)
0 – 6	0.7
6 – 12	1.2
12 – 18	1.6
18 – 24	2.0

Maturity (Years)	Discount Factor
0.5	0.996512
1.0	0.990569
1.5	0.982707
2.0	0.972977

• Now suppose that an investor buys a two-year Treasury bond with a coupon of 2.5% for USD 101.5. The first thing the investor might do is value the bond. The value of the bond (per USD 100 of face value) is 102.226

The investor has managed to buy the bond for USD 101.5 when the theoretical value is 102.226. This is a gain of 102.226 – 101.5 = 0.726. To convert this gain into a spread, the amount by which the forward rates would have to be increased for the theoretical price to be USD 101.5 needs to be considered. Suppose a spread of s is added to each forward rate. Thus, the value of the required spread is determined by solving

• Another type of calculation of spread sometimes compares one market to another market.
• For example, the spread between the AA-rated corporate bond market and the Treasury market can be of interest. The spread can be calculated, which when added to Treasury forward rates, gives the market price of an AA-rated bond. In general, the spread calculated will depend on maturity. For example, three-year AA-rated bonds might provide a spread over Treasuries of 50 basis points per year, whereas five-year AA-rated bonds might provide a spread over Treasuries of 80 basis points per year.

Yield To Maturity

• A bond’s yield to maturity is the single discount rate, which if applied to all the bond’s cash flows, would make the cash flows’ present value equal to the bond’s market price. For example, suppose a two-year bond with a coupon of 2.5% sells for USD 102. If the yield is y (expressed with semi-annual compounding), then

• The yield to maturity is a convenient measure because the price of a bond can be unambiguously converted into its yield to maturity (and vice versa).

• Formally, the yield to maturity (expressed with semi-annual compounding) for a bond with price P, lasting T years, and paying a coupon of c, is the value of y that solves

This equation assumes the price is observed immediately after a coupon payment date (i.e., that T is an integral number of half years). If the bond is being observed between coupon payment dates, and P is the quoted price, the cash (dirty) price must be calculated first. This is the quoted price plus accrued interest. It is the price that would be paid by the purchaser (and received by the seller).

Annuities

• An annuity is a financial product that pays out a fixed stream of payments to an individual, primarily used as an income stream for retirees.

• The value of an annuity paying c/2 every six months is

where y is the yield.

Perpetuity

• Perpetuity is perpetual annuity i.e. an annuity that lasts forever.

If in a perpetuity, payments are received semi-annually at the rate of c per year, then the perpetuity’s value is c/y

Properties Of Yield To Maturity

• Some straightforward properties of the yield to maturity are as follows:
  • When the yield to maturity is equal to the coupon rate, the bond sells for its face value.
  • When the yield to maturity is less than the coupon rate, the bond sells for more than its face value. If time passes with no change to the yield, the price of the bond declines.
  • When the yield to maturity is greater than the coupon rate, the bond sells for less than its face value. If time passes with no change to the yield to maturity, the price of the bond increases.
  • If the term structure is flat with all rates equal to R, the yield to maturity is equal to R for all maturities.

Carry Roll-Down

• The carry roll-down is designed to estimate the return achieved if there is no change to some aspect of the interest rate environment.
• The most common assumption when the carry roll-down is calculated is that forward rates are realized (i.e., the forward rate for a future period remains unchanged through time). When the beginning of the future period is reached, the spot rate for the period equals the forward rate. For example, suppose the term structure is flat at 4% (with semi-annual compounding) and that an investor owns a five-year bond paying a 4% coupon with a face value of USD 100. The price of the bond is USD 100, and all forward rates are 4%. For the carry roll-down, it is  assumed that forward rates remain unchanged through time. This means that the term structure continues to be flat at 4%.
• Now consider a slightly more complicated example. The term structure is still flat at 4% (with semi-annual compounding) and an investor owns a five-year bond paying a 5% coupon.
• For a more complete example, consider again a Treasury bond with a coupon of 2.5%. The forward rates for this bond are shown in this Table and the bond is currently valued at USD 102.226.
• To calculate the carry roll-down, it is assumed that the forward rates are realized. This means that after six months, the forward rates for the 0–6, 6–12, and 12–18 month periods are the same as those observed today for the 6–12, 12–18, and 18–24 month periods.
• This assumption is shown in the second Table.

Period (Months)	Forward Rate (%) (Semi-Annually Compounded)
0 – 6	0.7
6 – 12	1.2
12 – 18	1.6
18 – 24	2.0

Period (Months)	Forward Rate (%) (Semi-Annually Compounded)
0 – 6	1.2
6 – 12	1.6
12 – 18	2.0

• The price of the bond at the end of six months under the carry roll-down assumption is

The calculations show that if forward rates are realized, we expect the price of the bond to decrease from USD 102.226 to 101.334 after six months.

During this time, a coupon of 1.25 will be received.

The carry roll-down per USD 100 face value is therefore

• A quicker way of calculating the carry roll-down (assuming forward rates are realized) is to assume the return earned on any bond over the next period is always the prevailing one-period rate.

In the previous example, the six-month rate is 0.7% semi-annually compounded. The return earned over the next six months is therefore 0.7/2 = 0.35%.

The current value of the bond is USD 102.226. This carry roll-down is therefore:

• Note that the bond portfolio we are considering here does not matter. If forward rates are realized over the next period, the portfolio will always earn the prevailing market rate for the next period. The carry roll-down for the portfolio is the prevailing rate for the next period applied to the current value of the portfolio. We can extend this result so that it applies to several periods. If forward rates are realized for several periods, the gross return realized will be the rate in the market applicable to those periods.

• This result has implications for trading strategies. Should long-maturity be bought or short-maturity bonds be bought ?
  • If forward rates are expected to be realized, the return will be same in both cases.
  • If realized rates are expected to be less than forward rates, however, long-maturity bonds will provide a better gross return.
  • If realized rates are expected to be greater than forward rates, the reverse is true, and a series of short maturity bonds will provide the best gross return.

• Alternative Carry Roll-Down Assumptions

An alternative to the “forward rates are realized” assumption in carry roll-down calculations is to assume the interest rate term structure stays unchanged. An unchanged term structure would mean that the forward rates after 6 months will become those in this Table

Period (Months)	Forward Rate (%)
0 – 6	0.7
6 – 12	1.2
12 – 18	1.6

The carry roll-down is then

per USD 100 of face value.

• An argument in favor of the unchanged term structure assumption is that an upward-sloping term structure reflects investor risk preferences. Investors demand an extra return to induce them to invest for long maturities. If investors’ risk preferences are not expected to change, the term structure can be reasonably expected to retain its shape.

• Another assumption sometimes made in carry roll-down calculations is that a bond’s yield to maturity will remain unchanged. The one-period gross return, assuming the yield remains unchanged, is the yield itself. If the coupons are invested at the yield over many periods, the gross return is also the yield.

• A criticism of this assumption is that normally, the yield on a bond cannot be expected to remain unchanged.
  • In an upward-sloping term structure environment, the yield of a coupon-bearing  bond is expected to increase as the bond’s maturity is approached.
  • Similarly, in a downward-sloping term structure environment, the yield of a coupon-bearing  bond is expected to decrease as the bond’s maturity is approached

• A final possibility is for an investor to make personal estimates of future rates and use these as the basis for calculating the carry roll-down.

P&L Components

• The profit and loss (P&L) from a fixed-income portfolio can be split into several components. The components are as follows:
  • The Carry Roll-Down: This has already been discussed.
  • Rate Changes: This is the return realized when realized rates differ from those assumed in the carry roll-down.
  • Spread Changes: This is the return realized when a bond’s spread relative to other bonds changes.

Period (Months)	Forward Rate (%)
0 – 6	1.0
6 – 12	1.4
12 – 18	1.8

Initial Price of Bond	101.5
Carry Roll-Down	0.54
Rate Changes	0.3
Spread Changes	0.1
Final Value of Bond	101.19
Cash-Carry	1.25