# Equivalent Rates - Valuation Mathematics

• MHB
• logicandtruth
In summary, the conversation provides an example of valuing the right to receive £100,000 annually in advance in perpetuity, using a discount rate of 10%. The value is £1,100,000. The second question involves calculating the value of receiving £100,000 per annum quarterly in advance in perpetuity with an annual nominal rate of 10%. The correct quarterly rate is ~2.41%, resulting in a value of £1,062,344. The conversation also includes a formula for converting the annual rate to a quarterly rate.
logicandtruth
Hi all

I am currently working on questions focusing on valuation mathematics. A question on equivalent rates is perplexing me. The first question is straightforward, but I get stuck on the second question.

Q1. What is the value of the right to receive £100,000 annually in advance in perpetuity assuming a discount rate of 10%?

A1. £1,100,000

The formula below is for a level annuity that is received in perpetuity and in advance. Here in the UK typically commercial property leases are structured so tenants pay rents four times spread evenly over a year.

View attachment 8408

Q2. What is the value of the right to receive £100,000 per annum quarterly in advance in perpetuity assuming an annual nominal rate of 10%?

A. £1,062,344

I tried various iterations of the formula, but can't get the above answer. Any help would be much appreciated

#### Attachments

• Formula.PNG
1.5 KB · Views: 71
logicandtruth said:
Q2. What is the value of the right to receive £100,000 per annum quarterly
in advance in perpetuity assuming an annual nominal rate of 10%?

A. £1,062,344
Rate needs to be converted to the quarterly rate
that results in 10% annual; that rate is ~2.41%:
1.0241^4 = 1.10, so 10% effective.

Bank tatement will look like:
Code:
QUARTER PAYMENT INTEREST  BALANCE
0                   1,062,344
0  -25,000       0  1,037,344
1  -25,000  25,000  1,037,344 : 1037344*.0241= 25000
2  -25,000  25,000  1,037,344
...and so on till death do you part!
Sooooo...using formula:
PMT/r + PMT = 25000/.0241 + 25000 = 1,062,344

HOKAY?

I see where I went wrong, I was incorrectly converting the interest to a quarter rate:

Quarterly rate = (1 + annual rate )(1/4) – 1

Thank you, understood.

## 1. What is an equivalent rate?

An equivalent rate is a rate that yields the same amount of interest or return as another rate, but may have a different compounding frequency. For example, a 6% annual rate is equivalent to a 3% semi-annual rate, since both rates would result in a 6% return over a year.

## 2. How do you calculate equivalent rates?

To calculate equivalent rates, you need to know the initial rate and the compounding frequency. Then, use the following formula: equivalent rate = (1 + r/n)^n - 1, where r is the initial rate and n is the number of compounding periods per year. For example, if the initial rate is 6% and it compounds quarterly (4 times per year), the equivalent rate would be (1 + 0.06/4)^4 - 1 = 0.0614 or 6.14%.

## 3. Why are equivalent rates important in valuation mathematics?

Equivalent rates are important in valuation mathematics because they allow for accurate comparison and evaluation of different rates with varying compounding frequencies. This is especially useful when comparing investments or loans with different interest rates and compounding periods.

## 4. What is the difference between nominal and effective equivalent rates?

The nominal equivalent rate does not take into account the effects of compounding, while the effective equivalent rate does. The effective equivalent rate is a more accurate representation of the true return on an investment or the cost of a loan.

## 5. Can you provide an example of how to use equivalent rates in real life?

Sure, let's say you are comparing two savings accounts. One account offers a 5% annual interest rate, while the other offers a 4.8% semi-annual interest rate. To determine which account would provide a higher return, you would need to calculate the equivalent annual rate for the semi-annual option. Using the formula from question 2, we get (1 + 0.048/2)^2 - 1 = 0.0496 or 4.96%. This means that the annual rate of 4.8% compounded semi-annually is equivalent to a 4.96% annual rate. In this case, the first account with a 5% annual rate would provide a higher return.

• General Math
Replies
2
Views
8K
• General Math
Replies
1
Views
3K
• General Math
Replies
9
Views
2K
• Precalculus Mathematics Homework Help
Replies
10
Views
6K
• Precalculus Mathematics Homework Help
Replies
1
Views
832
• General Math
Replies
4
Views
4K
• Programming and Computer Science
Replies
29
Views
3K
• Other Physics Topics
Replies
6
Views
2K
• MATLAB, Maple, Mathematica, LaTeX
Replies
20
Views
4K
• Cosmology
Replies
1
Views
3K