###### EV Multiples

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Understanding the effects of assumptions on the estimated pension obligation and periodic pension costs helps in the** interpretation of a company’s financial statements**. Moreover, it aids in the **evaluation** of whether the assumptions are relatively **conservative or aggressive****.**

The amount of future pension liabilities under a DB plan requires assumptions such as:

- Employee turnover.
- Length of service.
- Rate of increase in compensation levels.
- Retirement age.
- Mortality.

Furthermore, the present value of these future liabilities requires assumptions on the **appropriate discount rate.**

Changes in actuarial assumptions are the **gains** and **losses**. An actuarial gain decreases the benefit obligation, and an actuarial loss increases the obligation. Apart from gains and losses, the estimates of a company’s pension liability also affect several components of periodic service costs, specifically the service costs and interest expense component.

The assumptions about the expected return on plan assets have a significant impact on the pension expense, which the company reports on the P&L under US GAAP.

To demonstrate this, let’s look at the following example:

ABC Company has a DB pension plan. An employee who has a salary of £90,000 in the coming year receives benefits for 30 years after retirement. He is also given credit for 5 years of prior service with immediate vesting. He has 10 years of service before retiring. The assumed discount rate is 10%, and the assumed compensation increase is 6% per annum.

We demonstrate how the pension obligation and periodic pension costs build up over the first 3 years.

Making the below further assumptions:

- No changes in actuarial assumptions.
- The company awards its employees all compensation increases on the first day of the service year.
- No additional adjustments are made to reflect the possibility that the employee may leave the company at an earlier date.

The information given can be broken down to:

$$\begin{array}{l|c} \textbf{Current salary }& \text{ £90,000}\\ \text{Years to retirement}&\hline10\\ \text{Annual compensation increase}&\hline6\%\\ \text{Discount rate}&\hline10\%\\ \text{Years for receiving the benefit}&\hline30\\ \hline\text{Prior years of service given credit}&\hline5\\ \end{array}$$

$$\begin{align*}\text{Estimated Final Salary}&=\text{Current year’s salary}\times[(1+\text{Annual compensation increase})^{\text{Years until retirement}}]\\&=\text{£152,053.11}\end{align*}$$

$$\begin{align*}\text{Estimated annual payment for each of the 30 years}&=\text{Estimated final salary}\times \text{Benefit Formula}\times\text{Years of service}\\ &=(152,053\times 0.04) \times (5+10)\\&=\text{91,231.86}\end{align*}$$

Value at the end of Year 10 (retirement date) of the estimated future payments = PV of £91,231.86 for 30 years at 10% discount rate

$$\begin{align*}&=\text{£91,231.86}\times\frac{1-1.10^{-30}}{0.10}\\&=\text{£860,034.94}\end{align*}$$

$$\begin{align*}\text{Annual unit credit}&=\frac{\text{Value at retirement date}}{\text{Years of service}}\\&=\frac{\text{£860,034.94}}{15}\\&=\text{ £57,335.66}\end{align*}$$

$$\small{\begin{array}{l|r|r|r} \textbf{Year}&1&2&3\\ \hline\text{Benefit attributed}&\\ \text{to:}&{}&{}&{}\\ \hline\text{Prior years}&\text{£286,678.33}&\text{£344,014.00}&\text{£401,349.67}\\ \hline\text{Current year}&\text{£57,335.66}&\text{£57,335.66}&\text{£57,335.66}\\ \hline\text{Total benefits earned}&\text{£344,013.99}&\text{£401,349.66}&\text{£458,685.33}\\ \hline\text{Opening obligation}&\text{£110,526.91}&\text{£145,895.52}&\text{£187,232.58}\\ \hline\text{Interest @10%}&\text{£11,052.691}&\text{£14,589.55}&\text{£18,723.26}\\ \hline\text{Current service costs}&\text{£24,315.92}&\text{£26,747.51}&\text{£29,422.26}\\ \hline\text{Closing obligation}&\text{£145,895.52}&\text{£187,232.58}&\text{£235,378.10}\\ \end{array}}$$

* Note:* The formulas for computing the values in the above table are as given in the previous learning objective.

To check if the value of the closing obligation is correct:

$$\begin{align*}\text{Closing obligation}&=[\text{Opening obligation}+\text{Interest@10%}+\text{Current service costs}]\\&=\frac{\text{Total benefits earned}}{(1+\text{Discount rate})^{\text{Years until retirement}}}\end{align*}$$

$$\text{Pension costs}=\text{Interest@10%}+\text{Current service costs}$$

Based on the example above, we can establish:

The effect on the Year 1 closing pension obligation and the pension cost if the assumed discount rate increases from 10% to 12%.

Note that the estimated final salary and the annual payments after retirement are the same, i.e., £152,053.1 and £91,231.86, respectively.

However, the value at the end of Year 10 (retirement date) of the estimated future payments = PV of £91,231.86 for 30 years at a 12% discount rate:

$$\begin{align*}&=\text{£91,231.86}\times\frac{1-1.12^{-30}}{0.12}\\&=\text{£734,889.42}\end{align*}$$

$$\begin{align*}\text{Annual unit credit}&=\frac{\text{Value at retirement date}}{\text{Year of service}}\\&=\frac{\text{£734,889.42}}{15}\\&=\text{£48,992.63}\end{align*}$$

$$\begin{array}{l|r} \textbf{Year}&1\\ \hline\text{Benefit attributed}&\\ \text{to:}&{}\\ \hline\text{Prior years}&\text{£244,963.15}\\ \hline\text{Current year}&\text{£48,992.63}\\ \hline\text{Total benefits earned}&\text{£293,955.78}\\ \hline\text{Opening obligation}&\text{£78,871.58}\\ \hline\text{Interest @ 12%}&\text{£9,464.59}\\ \hline\text{Current service costs}&\text{£17,667.23}\\ \hline\text{Closing obligation}&\text{£106,003.40}\\ \end{array}$$

If the discount rate increases from 10% to 12%, Year 1 **closing pension obligation decreases** by: \(\text{£145,895.52}- \text{£106,003.40} = \text{£39,892.12}.\)

The Year 1 **pension cost declines** from:\(\text{£35,368.61}=(\text{£11,052.691}+\text{£24,315.92})\) to \(\text{£27,131.82}=(\text{£9,464.59}+\text{£17,667.23})\).

The change in the interest component is a function of the **decline in the opening obligation** (which decreases the interest component) and the **increase in the discount rate** (which increases the interest component). In this case, the increase in the discount rate dominated, and the interest component increased. The **current service costs **and **the opening obligation** both declined because of the increase in the discount rate.

* Discount rate*: A higher discount rate decreases obligation on the balance sheet and lowers periodic pension costs.

* Compensation rate*: A higher rate of compensation increase leads to a higher obligation on the balance sheet and higher service costs.

* Expected rate of return*: A higher expected rate of return does not affect the balance sheet as the fair value of plan assets is used on the balance sheet. It lowers the periodic pension expense under US GAAP.

## Question

Based on the information given in the above example, what is

most likelyto be the impact of Year 1 closing pension obligation if the assumed annual compensation increases from 6% to 8%?A. The closing obligation increases by £30,989.

B. The closing obligation decreases by £60,729.

D. The closing obligation increases by £26,729.

## Solution

The correct answer is C.$$\begin{align*}\text{Estimated final salary}&=\text{£90,000}\times 1.08^9\\&=\text{£179,910.42}\end{align*}$$

$$\begin{align*}\text{Estimated annual payment for each of the 30 years}& = (\text{Estimated final salary}\times\text{Benefit formula}) \times \text{Years of service}\\&=\text{£179,910.42}\times0.04\times(5+10)=\text{£107,946.25}\end{align*}$$

Value at the end of Year 10 (retirement date) of the estimated future payments = PV of £107,946.25 for 30 years at 10% discount rate

$$\begin{align*}&=\text{£107,946.25}\times{\frac{1-1.10^{-30}}{0.10}}\\&=\text{£1,017,600.07}\end{align*}$$

$$\begin{align*}\text{Annual unit credit}&=\frac{\text{Value at retirement date}}{\text{Years of service}}\\&=\frac{\text{£1,017,600.07}}{15}\\&=\text{£67,840.00}\end{align*}$$

$$\begin{array}{l|r} \textbf{Year}&1\\ \hline\text{Benefit attributed}&\\ \text{to:}&{}\\ \hline\text{Prior years}&\text{£339,200.03}\\ \hline\text{Current year}&\text{£67,840.00}\\ \hline\text{Total benefits earned}&\text{£407,040.03}\\ \hline\text{Opening obligation}&\text{£130,776.30}\\ \hline\text{Interest @ 12%}&\text{£13,077.63}\\ \hline\text{Current service costs}&\text{£28,770.78}\\ \hline\text{Closing obligation}&\text{£172,624.71}\\ \end{array}$$

A 2% increase in the assumed compensation rate (6% to 8%) increases the pension obligation by: \(\text{£172,624.71} – \text{£145,895.52} = \text{£26,729.19}\).

Reading 12: Employment Compensation: Post-Employment and Share-Based

*LOS 12 (d) Explain and calculate the effect of a defined benefit plan’s assumptions on the defined benefit obligation and periodic pension cost.*