The most comprehensive educational resources for finance

Key Risks Associated with Investing in Bonds

Bonds are a lucrative investment class for investors and portfolio managers. However, just like any other investment, investing in bonds also has many risks associated with it. This article lists the key risks: Interest Rate Risk: Bond prices are inversely related to interest rates. When interest rates rise, bond prices fall and vice versa. Call

Sensitivity Analysis (Duration and Convexity)

The sensitivity analysis of fixed-income instruments refers to how the price moves in relation to each of sensitivity estimates such as price, duration and convexity. The relationships are mainly represented by the following three equations: Equation 3 obviously implies that the effect on price is positive once convexity is positive. This normally holds true to

Sensitivity Measures for a Portfolio

In financial markets assets are managed at a portfolio level which in other words means a large number of securities are managed in a group. It would be very inconvenient to consider movements in each position that composes the portfolio individually. A more practical and efficient solution would be to measure and manage the sensitivities

Economic Interpretation of Sensitivity Measures

Duration can be interpreted as the average time receipt of each cash flow weighted by its present value.  This applied to all cash flows is called Macaulay duration. The duration of a coupon paying bond can be written as: This equation can also be re-written as follows: Each weight is the present value of each

Sensitivity Measures for Fixed Income Investments

The derivatives associated with fixed income instruments have special terminology for them. The negative of the first derivative is called the Dollar Duration (DD).  Where  is called the modified duration. Thus dollar duration is:  The price is the dirty price of the bond including any accrued interest. The dollar value of the basis point measures

Taylor Series Expansion

In financial markets participants would like to measure the effect of changes in the price of the bond due to changes in yield. This enables better risk management of financial assets as the impact of asset values is determinable. Recomputing the value of the bond using the changed yield comes across as an obvious solution.

Infinite Series and Its Applications

Bonds consist of fixed coupons and the pricing of the bond involves the use of combinations of infinite series.  A typical example would be the sum of terms that increase at an infinite rate. This formula can be proved by multiplying both sides of the equation by (1-a) and then cancelling common terms on both

Price Yield Relationship for Bonds

For a bond with a cash flow pattern the present value can be calculated using the following formula: Where the numerator denotes the coupon or principal in that period, t is the number of periods, T is the number of periods to final maturity and y is the discount factor per period. For a bond

Discounting of Cash Flows

A zero coupon bond is a fixed income instrument with only one cash flow i.e. the face value of the bond which is paid to the investor at the maturity date. It is a more appropriate instrument for exemplifying the discounting of cash flows. Present Value The price of the zero coupon bond is determined

Measuring Interest Rate Risk: Full Valuation Approach

We know that bond prices are sensitive to interest rate changes. A portfolio of bonds will suffer a loss if the interest rates rise and vice versa. Similarly, a short bond position will make losses when interest rates fall. What a portfolio manager is interested in is to know the exact losses his portfolio will