Consider the following four debt securities which are the identical in every way characteristic except as noted.

- W. A corporate bond rated AAA
- X. A corporate bond rated BBB
- Y A corporate bond rated AAA with a shorter time to maturity than bond W and X
- Z A corporate bond rated AAA with the same time to maturity as bond Y that trades in a more liquid market than bonds W, X or Y. List the bonds in the most likely order of the interest rate (yield to maturity) of the bonds from highest to lowest.

From the above set of debt securities, the corporate bond X has the highest degree of risk. However, there is a high probability of the bond yielding a higher interest rate compared to the other bonds. Bond X which is rated BBBB can be categorised amongst the lower-medium grade bonds. However, the bond is above the non-investment categories of bonds. Bond W which is rated AAA is considered to have the highest degree of risk.

Under normal circumstances, bonds rated AAA are considered to be of high quality in that they have a higher return but a high risk. This is due to the fact that they have a relatively long yield to maturity. Decision to rank this bond at this point arises from the fact that its definite time of maturity is not known.Therefore, bond Y which is rated AAA and has a shorter duration to maturity compared to bond X and W has a relatively low degree of risk.

By having a relatively less duration to maturity, the probability of the bond paying off is high. This arises from the fact that the likelihood of the value of the bond being diminished by market changes is low. Bonds with long duration before attaining maturity are more likely to be diminished by market changes. As a result, investors can lose a significant amount of their investment in such bonds.

From the above set of securities, the corporate bond rated AAA but with same duration of maturity is the least risky. This arises from the fact that the bond trades in a relatively more liquid market compared to other bonds W, X and Y. By trading in a liquid market, it means that investors can easily dispose their investment in the bond to other investors in the market hence minimizing the chances of loss. Therefore, the bond that is likely to have the lowest interest rate the corporate bond Z rated AAA due to its high liquidity characteristic. The next bond with a lower interest rate is bond Y rated AAA but with a short duration to maturity.

The low rate of interest for this bond arises from the fact that the investors’ capital is not retained for a long duration. Corporate bond W rated AAA will have a relatively high interest rate. However, the bond with the highest interest rate is the corporate bond X that is rated BBB.

## Explain how an economist could use the slope of the yield curve to analyze the probability that a recession will occur and why the spread may matter

The yield curve is one of the most famous forecasting tools used by economics. Over the past few years, the yield curve has become popular with regard to predicting a recession. The yield curve is obtained by plotting the yield of a particular bond against the time to its maturity. The curve is usually upward sloping. The upward sloping nature of the curve shows that long-term bonds have relatively higher returns compared to short-term. However, long-term bonds are characterized by a high degree of risk hence the need to have higher returns. The difference between the long term rate and the short term rate is referred to as the spread and is usually positive.

There are three main shapes that the yield curve can take. These include normal shape, flat shape and an inverted shape. Past studies have shown that the shape of yield curve can be used to determine the probability of a recession occurring. According to these studies, there is a high probability of a recession occurring during a yield-curve inversion. The yield-curve inversion occurs when the yield (return) on short-term bonds is above the yield on long-term bonds.

Economists can use the slope of the yield curve to determine the probability of a recession occurring by determining the statistical relationship between the expected future economic growth and the future interest rate. The spread matters in that it gives a clue on the future interest rate. Economists can then evaluate the direction of the yield curve. However, to achieve this, economists must identify a particular spread which will be used in the prediction as the yield’s curve slope.In the United States, the most commonly used spread is the one of a 3 months treasury bill and that of a 10 year treasury bond.

*One year ago, you bought a bond for $10,000. You received interest of $400 at the end of the year, as well as your $10,000 principal. If the inflation rate over the last year was five percent, calculate the real return. Show your work*

Money loses value over time. One of the factors that contribute toward money losing its value is inflation. This means that investors are faced with a risk of losing their investment due to depreciation in the value of the currency over the time of the investment. In this case, the investor purchases a bond worth $10, 000. The interest received at the end of the year is equal to $400. This means that the value of the investment at the end of the year is equal to $10,000, that is, $10,000 + $400. From this the nominal rate of return can be calculated by dividing the interest with the principal amount and multiplying the result with 100.

$ 400/ $10,000*100= 4%

However, considering the fact that rate of inflation over the past 5 years was 5%, the value of investment was affected. Real rate refers to the nominal rate that is adjusted to take into account the effect of the inflation. In this case the real return is equal to

-1% that is, 4%-5%=1%. Despite earning an interest of $400, there was a loss in the value of the money since the real return is -1%.

*Suppose that the price of a stock is $50 at the beginning of a year and $53 at the end of the year, and it pays a dividend of $2 during the year. Calculate the stock’s current yield, capital-gains yield, and the return. Show your work for three separate calculations.*

According to Croushore (2007, p.189), stocks total returns are comprised of two main elements which include the capital gains yield and the current yield. Capital gains yield refers to the percentage increment in the value of a stock. On the other hand, the stocks current yield refers to the proportion of the stocks paid to the investors as dividends.

## Current Yield

This is calculated by dividing the amount of the dividend paid by price of the security at the beginning of a particular financial year. The value obtained is multiplied by 100. Therefore, in this case the stock’s current yield is equal to 4%.

=$2/$50*100 =4%.

## Capital gains yield

This is calculated by finding the difference between the price of the stock at the end of the year and its price at the beginning of the year. The difference is dividend by the value of the stock at the start of the year. The result is multiplied by 100. In this case, the price of the stock at the end of the year is $ 53 while its price at the beginning of the year was $50 which means that the stock gained $3, that is $ 53-$ 50= $3. Therefore, the stocks capital gain yield is obtained as follows;

$ 3/$ 50*100= 6%

## Rate of return

This is obtained by summing up the stocks capital gains yield and the current yields. In this case, the rate of return is therefore equal to 4%+6% = 10%.

*Use the capital-asset pricing model to predict the returns next year of the following stocks, if you expect the return to holding stocks to be 12 percent on average, and the interest rate on three-month T-bills will be two percent. Calculate a stock with a beta of -0.3, 0.7, and 1.6. Show your work for three separate calculations.*

The Capital Asset Pricing Model (CAPM) shows the relationship between the risk and return of a particular security. The CAPM postulates that the price of a particular stock is dependent on two main variables. These include the stock’s risk factor and the time value of money. The CAPM integrates beta which measures the risk of a particular stock relative to the existing market risk in determining the expected rate of return. The following formula is used in calculating the expected rate of return.

R_{J}= R_{f} +B (R_{m} +R_{f})

Where

Rj= Expected return on the security

R_{f}= the rate of return on a particular risk-free stock for example a treasury Bill.

R_{m}=the expected rate of return on a certain market portfolio. Rm can be represented by the average interest rate on all assets. For example it can be represented by the Dow Jones 30 Industrials or the Standards & Poor’s 500 Stock Composite Index

B=Beta, represents the sensitivity of a security to market movement.

In this case, the expected rate of return on holding the stock is 12% which represents the R_{m}. On the other hand, the risk free rate (R_{f}) of holding a 3 month treasury bond is 2%. Therefore, the annual risk free rate of the Treasury bond is 8%.

### Calculation of R_{J} when beta =-0.3

Rj = 2% + -0.3(12%-8%)

=2%+-.3(4%)

=2%+-1.2%

=0.8%

### Calculation of expected return when beta =0.7

Expected return =R_{J}= R_{f} +B (R_{m} +R_{f})

Rj = 2% + 0.7(12%-8%)

= 2%+0.7(4%)

=2%+2.8%

=4.8%

### Calculation of expected return when beta=1.6

Expected return =R_{J}= R_{f} +B (R_{m} +R_{f})

=2%+1.6 (12%-8%)

= 2%+1.6(4%)

=2%+6.4%

=8.4%

## Reference List

Croushore, D. (2007). *Money and banking; a policy oriented approach*. Boston, MA; Houghton Mifflin.