fixed-income

One of the harder ideas in fixed income is risk-neutral probabilities. In this video, I'd like to specifically illustrate, and define, what we mean by risk-neutral probabilities. I will do this in three steps. The first one is just a simple example of a coin toss, where my objective is to...

The key rate shift technique overcomes the key limitation of duration and DV01 which is that they must assume a parallel shift in the yield curve because they are single-factor risk measures. The key rate shift technique, on the other hand, is multi-factor: the term structure is carved into a...

In this playlist, David has already recorded at least ten videos on duration and convexity which are the two most common measures of single-factor interest rate risk. So, in this video, we wrap it up in one simple explanation that tries to illustrate both duration and convexity and how we apply...

In this video, David shows you how to calculate modified convexity by matching the modified convexity that Tuckman shows in Table 4.6 in Chapter 4 of his book, Fixed Income Securities. You can find Tuckman's Fixed Income Securities book here: https://amzn.to/2SOMGzv

The bullet portfolio invests in a single medium-term bond. The corresponding barbell portfolio invests the same amount of capital and achieves the same duration, but invests in a mix of the short-term plus long-term bond. But the barbell portfolio will have greater convexity. Tuckman explains...

The previous videos in this playlist have illustrated how we calculate the two most popular measures of single factor interest rate sensitivity, that is duration and dv01, also called price value of the basis point. Now, knowing how these calculations work we will apply them to understand some...

Duration plus a convexity adjustment is a good estimate (approximation) of the bond's price change. We can express this change in percentage terms(%) as given by ΔP/P = -D*Δy + 0.5*C*(Δy)^2; or we can express this in dollar terms ($) as given by ΔP =∂P/∂y*Δy + 0.5*∂^2P/∂y^2*(Δy)^2.

Effective convexity approximates modified convexity (just as effective duration approximates modified duration). Mathematically, convexity is a function of the bond's second derivative with respect to yield: convexity = 1/P*∂^2P/∂y^2. Convexity is illustrated by the curvature (i.e., non-linear)...

Macaulay duration is the bond's weighted average maturity (where the weights are each cash flow's present value as a percent of the bond's price; in this example, the bond's Macaulay duration is 2.8543 years. Modified duration is the true (best) measure of interest rate risk; in this example...

Using my rebuild of Bruce Tuckman's Table 4-6, this video illustrates the calculation of Macaulay and modified duration. Macaulay duration is the bond's weighted average maturity. Modified duration is the best measure of the bond's interest rate risk.

Effective duration approximates modified duration. Both express interest rate sensitivity: an effective (or modified) duration of 6.2 years tells us to expect a 0.620% price change if the yield changes by 10 basis points; i.e., 0.10% ∆y * 6.2 years = 0.620% ∆P. Effective duration is given by...

The DV01 is dollar change in the position for a one basis point (1 bps) decline in the interest rate (typically, yield). The DV01 is expressed per $100 face amount; for example, $0.035 implies that when rates drop by one basis point, the bond will increase in value by $0.035 per $100 face...

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 4, One-factor Risk Metrics and Hedges). The DV01 stands for "dollar value of an .01% (one basis point)." It is also called the Price Value of a Basis Point (PVBP). It is the bond's or fixed...

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 3, Returns, Spreads and Yields). The Carry-Roll-Down is the price change in the bond due exclusively to the passage of time. It is only one component of a bond's total profit and loss (P&L). The...

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 3, Returns, Spreads and Yields). The three basic term structure scenarios are: 1. Realized Forwards; 2. Unchanged Term Structure, and 3. Unchanged Yields. Realized Forwards implicitly assumes...

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 3, Returns, Spreads and Yields). Yield to maturity (aka, yield) is the single rate that discounts a bond's cash flows to a present value that matches the bond's traded (observed) price.

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 3, Returns, Spreads and Yields). The bond spread is the rate that, when added to the term structure, equates the discounted cash flow to the bond's observed market price. Bond spread is a key...

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 3, Returns, Spreads and Yields). The Gross Realized Return is the holding period return (HPR), so it includes the bond's price change and any coupon income. The Net Realized Return subtracts...

This follows Tuckman's example in Chapter 2. When the yield is unchanged, a bond pulls to par. HOWEVER, the assumption of unchanged yield is unrealistic. Here we assume the term structure (of spot and forward rates) in unchanged. Specifically, this is a 2.5-year swap (or bond) where the fixed...

Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 2, Spot, Forward and Par Rates). Given the swap rate curve, we can infer the discount function (i.e., set of discount factors), spot rate curve, forward rate curve and par yield curve.