Mathematical finance | Interest rates

Shadow rate

The shadow rate is an interest rate in some financial models. It is used to measure the economy when nominal interest rates come close to the zero lower bound. It was created by Fischer Black in his final paper, "Interest Rates as Options". The shadow rate derives from Fischer Black's insight that currency is an option. If someone has money, they can either (1) spend it today or (2) not spend it and have money tomorrow. Thus, when loans would return less money than was initially loaned out, investors will choose to "exercise the option" and not loan their money. Thus, the nominal short-term interest rate is always greater than or equal to zero. In Black's model, the shadow nominal short-term rate is what the nominal short-term rate would be if it was allowed to go below the zero lower bound. When the shadow nominal short-term rate is positive, the nominal short-term rate is equal to the shadow rate. But when the shadow short-term rate is negative - such as during deflation or a bad recession with low inflation - the nominal short-term rate will diverge and stay above zero. In Black's model, even when nominal short-term interest rates stay close to zero, the long-term nominal interest rates can be well above zero. This is because nominal interest rates behave like options and there is some chance that the shadow short-term rate becomes positive in the future. There is also a shadow real rate. The shadow real short-term rate is equal to the shadow nominal short-term rate minus expected inflation. Fischer Black published his paper in 1995 and mentioned that the most recent time that the USA had experienced the zero lower bound was the Great Depression. Shadow rate models got renewed interest with the Financial Crisis of 2007–2008 when interest rates plunged to near zero and, even, below zero in some instances. (Wikipedia).

Video thumbnail

Solve related rates two cars traveling

👉 Learn how to take the derivative of a function. Learn how to find the derivative of a function using the chain rule. The derivative of a function, y = f(x), is the measure of the rate of change of the function, y, with respect to the variable x. The process of finding the derivative of a

From playlist Solve Related Rates Problems

Video thumbnail

How to evaluate the limit of a function by observing its graph

👉 Learn how to evaluate the limit of an absolute value function. The limit of a function as the input variable of the function tends to a number/value is the number/value which the function approaches at that time. The absolute value function is a function which only takes the positive val

From playlist Evaluate Limits of Absolute Value

Video thumbnail

Using parent graphs to understand the left and right hand limits

👉 Learn how to evaluate the limit of an absolute value function. The limit of a function as the input variable of the function tends to a number/value is the number/value which the function approaches at that time. The absolute value function is a function which only takes the positive val

From playlist Evaluate Limits of Absolute Value

Video thumbnail

Easy way to find the derivative given the definition of derivative

👉 Learn how to evaluate the limit of a function using the difference quotient formula. The difference quotient is a measure of the average rate of change of the function over an interval, h. The limit of the difference quotient gives the derivative of the function. The difference quotient

From playlist Evaluate the Limit (PC)

Video thumbnail

Learn how to use the definition of a derivative to evaluate the limit

👉 Learn how to evaluate the limit of a function using the difference quotient formula. The difference quotient is a measure of the average rate of change of the function over an interval, h. The limit of the difference quotient gives the derivative of the function. The difference quotient

From playlist Evaluate the Limit (PC)

Video thumbnail

Intervals of increasing and decreasing function from a graph

👉 Learn how to determine increasing/decreasing intervals. There are many ways in which we can determine whether a function is increasing or decreasing but we will focus on determining increasing/decreasing from the graph of the function. A function is increasing when the graph of the funct

From playlist When is the Function Increasing Decreasing or Neither

Video thumbnail

Related Rates Similar Triangles 4K

I work through a 2 part example of Related Rates in Calculus which involve similar triangles. Part 1 0:55 Rate at which tip of shadow moves Part 2 13:16 Rate of change for a shadow length Related Rates Introduction Area & Volume https://youtu.be/DQ5T3ht5ahs Related Rates involving Right

From playlist Calculus (New)

Video thumbnail

Evaluate the limit for a value of a function

👉 Learn how to evaluate the limit of an absolute value function. The limit of a function as the input variable of the function tends to a number/value is the number/value which the function approaches at that time. The absolute value function is a function which only takes the positive val

From playlist Evaluate Limits of Absolute Value

Video thumbnail

How Fast is the Tip of the Shadow and Length of the Shadow Changing Related Rates Problem

In this video I do a related rates problem. The question is to determine How Fast is the Tip of the Shadow and Length of the Shadow Changing.

From playlist Related Rates

Video thumbnail

Calculus 1 - Derivatives and Related Rates (10 of 24) The Lamppost and the Shadow

Visit http://ilectureonline.com for more math and science lectures! I this video I will calculate how fast the length of the shadow is changing (dL/dt=?) and how fast the tip of the shadow is moving (dx(s)/dt=?). Next video in this series can be seen at: https://youtu.be/zh3MuGT-7eE

From playlist CALCULUS 1 CH 6.5 RELATED RATES

Video thumbnail

Related Rates - The Shadow Problem

This calculus video tutorial explains how to solve the shadow problem in related rates. A 6ft man walks away from a street light that is 21 feet above the ground at a rate of 3ft/s. At what rate is the length of the shadow changing? At what rate is the tip of the shadow moving? You nee

From playlist New Calculus Video Playlist

Video thumbnail

Find intervals that a function is increasing and decreasing

👉 Learn how to determine increasing/decreasing intervals. There are many ways in which we can determine whether a function is increasing or decreasing but we will focus on determining increasing/decreasing from the graph of the function. A function is increasing when the graph of the funct

From playlist When is the Function Increasing Decreasing or Neither

Video thumbnail

Shadow Man Related Rates Word Problem

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Shadow Man Related Rates Word Problem

From playlist Calculus 1 Exam 2 Playlist

Video thumbnail

Related Rates of Change - Shifting Shadow (1 of 2: Understanding the problem)

More resources available at www.misterwootube.com

From playlist Rates of Change

Video thumbnail

Calculus 1: Lecture 2.6 Related Rates

This is a real Calculus 1 classroom lecture. In this class I covered Related Rates. We did a few problems. - a warmup problem with x and y being functions of time - a related rates problem with a sphere - a related rates problem with a cube - a related rates problem with a triangle(house

From playlist Calculus 1 Full Lectures

Video thumbnail

Twisted Patterson-Sullivan Measure and Applications to Growth Problems (Lecture-2) by Remi Coulon

PROGRAM: PROBABILISTIC METHODS IN NEGATIVE CURVATURE (ONLINE) ORGANIZERS: Riddhipratim Basu (ICTS - TIFR, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Mahan M J (TIFR, Mumbai) DATE & TIME: 01 March 2021 to 12 March 2021 VENUE: Online Due to the ongoing COVID pandemic, the meeting will

From playlist Probabilistic Methods in Negative Curvature (Online)

Video thumbnail

How to use the definition of a derivative to evaluate the limit

👉 Learn how to evaluate the limit of a function using the difference quotient formula. The difference quotient is a measure of the average rate of change of the function over an interval, h. The limit of the difference quotient gives the derivative of the function. The difference quotient

From playlist Evaluate the Limit (PC)

Video thumbnail

Ex 2: Related Rates Problem -- Rate of Change of a Shadow from a Light Pole

This video explains how to determine the rate of change of a shadow from a person walking away from a street light. Site: http://mathispower4u.com

From playlist Related Rates

Video thumbnail

Determining when a function is increasing decreasing or constant

👉 Learn how to determine increasing/decreasing intervals. There are many ways in which we can determine whether a function is increasing or decreasing but we will focus on determining increasing/decreasing from the graph of the function. A function is increasing when the graph of the funct

From playlist When is the Function Increasing Decreasing or Neither

Related pages

Zero lower bound | Real interest rate | Yield curve | Nominal interest rate | Short-rate model | Interest rate