5.

In general, models that can handle embedded options have the following five characteristics:

Begin with the fundamental model to derive estimates of treasury spot rates.

Estimate the volatility (degree of likely changes) in interest rates.

Develop an interest rate tree, based upon the volatility estimate, of future interest rates.

Model probabilities are set such that the model correctly predicts the current treasury bond price(s).

Develop rules for the exercise of the embedded options.

1.B: Yield Measures, Spot Rates, and Forward Rates

a: Explain the sources of return from investing in a bond (coupon interest payments, capital gain/loss, and reinvestment income).

Coupon interest payments. The series of cash payments made at fixed intervals as specified in the bond indenture.

Capital gain or loss. The difference between the purchase price and the sales price (or maturity value if the bond is held until maturity).

Reinvestment income. The interest income that accrues on the reinvestment of any cash flows that occur during the specified holding period.

b: Compute the traditional yield measures for fixed-rate bonds (current yields, yield to maturity, yield to first call, yield to first par call date, yield to put, yield to worst, and cash flow yield).

The current yield is concerned only with coupon cash flow, but does not consider capital gains/losses or reinvestment income. Suppose that we have a 3-year, $1,000 par value, 6% semiannual coupon bond. The cash coupon payment per year = cash coupon payment per year = maturity value * stated coupon rate = $1,000 * 0.06 = $60. If the bond price is $852.480 (@YTM = 12%, then the current yield = $60/$852.48 = 7.038%.

Yield to maturity (YTM) measures the internal rate of return to a bond. It is the most popular of all yield measures used in the marketplace. Continuing with the example from above, YTM = 12% (see calculation below).

PV = -852.48, N = 6, FV = 1,000, PMT = 30; CPT I/Y = 6.00 * 2 = 12%.

Yield to call (YTC): Some bonds may be called (repurchased prior to maturity) at the option of the issuer. Investors are typically interested in knowing what the yield will be if the bond is called by the issuer at the first possible date. This is called yield to first call or yield to call (YTC). There are two modifications to our YTM formula necessary to determine yield to first call, 1) maturity date is shortened to the first call date, and 2) maturity value is changed to call price.

Continuing from the previous example, assume the cash value of each coupon is $30, and the first call price is $1,060 in 2 years, N = 4, FV = 1,060, PMT = 30, PV = -852.48; CPT I/Y = 8.85 * 2 = 17.707%. Note that it isn’t very likely that the issuer would call a 6% bond when we can see that the current required rate is much greater than 6%.

Yield to first par call is calculated in exactly the same way, except that the number of years is to first par call, and FCP becomes par value.

Yield to put: Some bonds may be put (sold back to the issuer prior to maturity) at the option of the holder. Investors are typically interested in knowing what the yield will be if the bond is put to the issuer at the first possible date. This is called yield to put (YTP). There are two modifications to our YTM formula necessary to determine yield to put, 1) maturity date is shortened to the first put date, and 2) maturity value is changed to put price.

Suppose that we have a 3-year, $1,000 par value, 6% semiannual coupon bond. We observe that the value of the bond is $852.480. The cash value of each coupon is $30, and the first put price is $1,000 in 2 years. Therefore:

N = 4, FV = 1,000, PMT = 30, PV = -852.48; CPT I/Y = 7.39 * 2 = 14.788%.

Yield to worst (YTW) involves the calculation of yield to call (YTC) and YTP for every possible call or put date, and determining which of these results in the lowest expected return. Some “fixed income” securities have payment structures that are amortizing – the payments include both interest and principal. In many cases the amount of the principal repayment can be greater than the amount required to amortize the loan over its original life. In these cases, it is said that prepayment is occurring.

Cash flow yield (CFY) incorporates a projection as to how these prepayments are likely to occur. Once we have this in hand, we can calculate CFY via an internal rate of return measure, similar to the YTM.

c: Explain the assumptions underlying tradational yield measures and the limitations of the traditional yield measures.

YTM and present-value based yield measure assumptions:

All coupons will automatically be reinvested to maturity at a rate of return that equals the bond's YTM.

In general, models that can handle embedded options have the following five characteristics:

Begin with the fundamental model to derive estimates of treasury spot rates.

Estimate the volatility (degree of likely changes) in interest rates.

Develop an interest rate tree, based upon the volatility estimate, of future interest rates.

Model probabilities are set such that the model correctly predicts the current treasury bond price(s).

Develop rules for the exercise of the embedded options.

1.B: Yield Measures, Spot Rates, and Forward Rates

a: Explain the sources of return from investing in a bond (coupon interest payments, capital gain/loss, and reinvestment income).

Coupon interest payments. The series of cash payments made at fixed intervals as specified in the bond indenture.

Capital gain or loss. The difference between the purchase price and the sales price (or maturity value if the bond is held until maturity).

Reinvestment income. The interest income that accrues on the reinvestment of any cash flows that occur during the specified holding period.

b: Compute the traditional yield measures for fixed-rate bonds (current yields, yield to maturity, yield to first call, yield to first par call date, yield to put, yield to worst, and cash flow yield).

The current yield is concerned only with coupon cash flow, but does not consider capital gains/losses or reinvestment income. Suppose that we have a 3-year, $1,000 par value, 6% semiannual coupon bond. The cash coupon payment per year = cash coupon payment per year = maturity value * stated coupon rate = $1,000 * 0.06 = $60. If the bond price is $852.480 (@YTM = 12%, then the current yield = $60/$852.48 = 7.038%.

Yield to maturity (YTM) measures the internal rate of return to a bond. It is the most popular of all yield measures used in the marketplace. Continuing with the example from above, YTM = 12% (see calculation below).

PV = -852.48, N = 6, FV = 1,000, PMT = 30; CPT I/Y = 6.00 * 2 = 12%.

Yield to call (YTC): Some bonds may be called (repurchased prior to maturity) at the option of the issuer. Investors are typically interested in knowing what the yield will be if the bond is called by the issuer at the first possible date. This is called yield to first call or yield to call (YTC). There are two modifications to our YTM formula necessary to determine yield to first call, 1) maturity date is shortened to the first call date, and 2) maturity value is changed to call price.

Continuing from the previous example, assume the cash value of each coupon is $30, and the first call price is $1,060 in 2 years, N = 4, FV = 1,060, PMT = 30, PV = -852.48; CPT I/Y = 8.85 * 2 = 17.707%. Note that it isn’t very likely that the issuer would call a 6% bond when we can see that the current required rate is much greater than 6%.

Yield to first par call is calculated in exactly the same way, except that the number of years is to first par call, and FCP becomes par value.

Yield to put: Some bonds may be put (sold back to the issuer prior to maturity) at the option of the holder. Investors are typically interested in knowing what the yield will be if the bond is put to the issuer at the first possible date. This is called yield to put (YTP). There are two modifications to our YTM formula necessary to determine yield to put, 1) maturity date is shortened to the first put date, and 2) maturity value is changed to put price.

Suppose that we have a 3-year, $1,000 par value, 6% semiannual coupon bond. We observe that the value of the bond is $852.480. The cash value of each coupon is $30, and the first put price is $1,000 in 2 years. Therefore:

N = 4, FV = 1,000, PMT = 30, PV = -852.48; CPT I/Y = 7.39 * 2 = 14.788%.

Yield to worst (YTW) involves the calculation of yield to call (YTC) and YTP for every possible call or put date, and determining which of these results in the lowest expected return. Some “fixed income” securities have payment structures that are amortizing – the payments include both interest and principal. In many cases the amount of the principal repayment can be greater than the amount required to amortize the loan over its original life. In these cases, it is said that prepayment is occurring.

Cash flow yield (CFY) incorporates a projection as to how these prepayments are likely to occur. Once we have this in hand, we can calculate CFY via an internal rate of return measure, similar to the YTM.

c: Explain the assumptions underlying tradational yield measures and the limitations of the traditional yield measures.

YTM and present-value based yield measure assumptions:

All coupons will automatically be reinvested to maturity at a rate of return that equals the bond's YTM.