## Estimating future retirement needs

Planning for future retirement needs should be done carefully. This can be attributed to the fact that retirement is an equally important aspect of the financial well-being of an individual just like investments and purchase of other assets. A challenge that most people face is coming up with clear plans for retirement. In most cases, people take too long to come up with these plans because of the competing needs of the limited finances that people have at their disposal. Some of the other challenges that people face are planning for the family, the effect of inflation, and the pension that will be received after retirement among others. These uncertainties often slow down the process of planning for retirement. Therefore, an individual needs to be aware of the risks they are exposed to and mitigate them. One of the possible ways of mitigating risks is by incorporating possible eventualities in the plans. This section seeks to estimate future retirement needs for Joshua. Specifically, the paper will focus on estimating the annual savings that the client needs to make to achieve the retirement plan.

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The calculation of the required annual savings can be broken down into four parts. The first part will focus on estimating the amount of household expenditure that is required during retirement. This is often expressed as a percentage of the level of household expenditure before retirement. In this example, the value is set at 80% of the current household expenditure. The second part of the calculation will estimate the total amount of annual income that an individual expects to earn during retirement. This section also estimates the annual shortfall. The shortfall is the amount by which the amount of annual household expenditure during retirement exceeds the expected annual income after retirement. The amount of annual savings will be calculated based on the estimated value of the shortfall. The third part of the solution will focus on adjusting the value of the estimated shortfall for the effects of inflation. The anticipated annual rate of inflation before retirement will be used. Thus, the annual shortfall will be multiplied by the inflation factor to arrive at the value of the annual shortfall adjusted for inflation. Failure to adjust the value for inflation will give misleading results. The final step yields the required annual savings. First, the total amount of funds that is required for retirement will be calculated. The inflation-adjusted value of the annual shortfall will be divided by the required rate of return after retirement to arrive at this value. Secondly, the compound interest factor will be obtained from the tables. In this case, the expected annual rate of return on investment before retirement will be used. The value of annual savings that is required will be arrived at through the division of the total amount of retirement fund required by the compound interest factor. The calculations are summarized in the table below.

From part 1, the estimated value of household expenditure during retirement is $67,200. The total annual income expected during retirement is estimated to be $53,000. The resulting annual shortfall amounts to $14,200. From part 3, the annual shortfall adjusted for inflation amounts to $61,344. From part 4, the resulting value of the annual end of year savings that is required for investment during the period until retirement amounts to $5,415.11.

## Price of preferred stock of Big Oil and number of shares

Preference shares are simpler to value than common stock. This can be attributed to the fact that they pay a constant amount of dividend. Besides, this category of shares does not have a maturity date. They are held on the books until a company liquidates. These features make them appear as a blend of debt and common stock. Investment in preferred stock offers a promise to pay dividend annually until perpetuity. However, payment of dividend on preferred stock highly depends on the profit earned by a company. Therefore, if a company makes profit in a specific financial year, then the dividends will be paid. However, during periods when a company does not make profits, it becomes impossible to pay dividends on preferred stock. In such cases, most companies often suspend the payment of dividend. When such companies return to a state of profitability, dividends on preferred stock are paid before dividends on common shares. Besides, the dividends in arrears are also paid full. Such companies always strive to make up for the dividends on preferred stock that were not paid. However, it is worth mentioning that when a company skips payment of dividend, the value of preferred stock declines. Therefore, this makes it important for a company to maintain the value of the preferred stock so that the shareholders do not make losses when the value of the preferred shares drops. This can be achieved by ensuring that one of the major goals of a company is profit maximization. If the required rate of return is not provided, then the capital asset pricing model will be used to estimate the discount rate. Calculation of the price of preferred stock is presented below.

- Annual dividend payments on preferred stock of Big Oil = $4
- Required rate of return = 8%
- Year 1dividend = 0
- Year 2 dividend = 0
- Year 3 dividend = $12
- Year 4 dividend = $4 onwards.

The solution for this problem will be approached in two steps. The first part of the solution will focus on the $12 dividend that is expected in three years. The second part of the solution will focus on the $4 that will be paid after the third year. Therefore, the present value of each part will be calculated separately and the two will be combined to obtain the price value of the shares.

## First part

Present value

= $12 / (1 + 0.08)^{3}

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= $9.53

## Second part

The second part will focus on estimating the present value of perpetuity from the third year.

- Present value (year 3)
- = $4 / 0.08
- =$50

- Present value of perpetuity
- = $50 / (1 + 0.08)3
- = $39.69

- Price of preferred shares = value of $12 dividend + value of $4 perpetuity
- = $9.53 + $39.69
- = $49.22

If the company had paid dividends consistently without skipping, then the value of the preferred stock will be $50. Even though the company eventually made the dividend payment for the periods that were skipped, the value of the preferred stock dropped slightly to $49.22. In this case, shareholders will lose $0.88 per share. The price of preferred stock for Big Oil Company is $49.22.

## Calculation of number of shares

The number of shares will be arrived at through the division of the total amount that is available for investment by the estimated price per share.

= $9,500 / $49.22

= 193 shares

Thus, with a total of $9,500 Joshua can purchase 193 shares of preferred stock for Big Oil.

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## Price of Bonds of Bank Corp.

A bond, as a debt instrument, pays the principal amount at maturity. In addition, there are periodic interest payments. Further, the cash flow on bonds is fairly certain. The price of a bond is estimated by adding the present value of the future stream of expected cash flow. In other words, if an investor receives periodic interest payments, then the present value of that investment can be estimated using these payments. The total amount that will be received by the investor is the sum of the present value of periodic interest payments and the maturity value of the bond. The details of the bond are presented in the table below.

From the information provided, the period to maturity of the bond is 10 years. That is between 1^{st} January 2016 and 31^{st} December 2025. The periodic interest payment will be calculated at 7% of the par value. The first step is to estimate the periodic interest rate. The coupon rate is 7% and interest is paid semi-annually. The calculation of the amount of interest is presented below.

- = ½ * 7% * $100
- = $3.5

Since the interest is paid semi-annually, the required rate of return will be halved to 4% (8% *1/2) and the time to maturity will be doubled to 20 periods (10 years * 2).

- = $3.5 * [1 – (1 + 0.04)
^{-20}] / 0.04 + $100 (1 + 0.04)^{20} - = $3.5 * [1 – (1.04)
^{-20}] / 0.04 + $100 (1 + 0.04)^{20} - = $3.5 * [0.543613053] / 0.04 + $100/ (1 + 0.04)
^{20} - = $3.5 * 13.5903 + 45.6387
- = $47.5661 + 45.6387
- = $93.20

Thus, a bond that is settling on 1^{st} January 2016 that has a par value of $100 and a maturity date of 31^{st} December 2025 with a coupon rate of 7% and a required rate of return of 8% will be priced at $93.20. This is based on the assumption that the redemption value of the bond is equal to the par value of $100. The result above has a number of implications to a bondholder. All the possible factors that can affect the price of a bond are included in the calculations. The result above can be interpreted in a number of ways. The difference between the settlement and maturity date will be determined by the duration that the investor will hold the bond. If this duration is longer, then the price of bond will be lower. Also, the required rate of return can be compared to the coupon rate. If the difference between these rates is large, then the price of the bond will be low. Finally, the difference between the redemption value and par value affects the price of the bond. If the difference in these two prices is large, then the price of the bond is likely to be lower. Therefore, before investing in bonds, an investor needs to look at these factors and make decisions that will maximize returns.

## Number of bonds

Calculation of the number of bonds will be arrived at through the division of the total amount that is available for investment and the price of the bond that is estimated above. The calculations are presented below.

- Number of bonds = cash available for the investment / price of the bond
- = $9,500 / $93.20
- = 101.93 bonds

When rounded off to the nearest whole number, then the investor should buy 102 bonds.

## Calculating the price of the stock and number of shares

The estimation of the price of the stock is multifaceted. This can be attributed to the fact that the future stream of cash flow from investments made in shares cannot be predicted with certainty. The stream of cash flow that is expected from common stock highly depends on the financial performance of an entity. The first step in the calculation of the price of the stock is the estimating the required rate of return. In this case, the capital asset pricing model (CAPM) will be used. Thus, since the amount of dividend and the future share price is not known with certainty, then it is an indication that there are risks associated with investing in common stock. Therefore, it is important to estimate the value required rate of return that incorporates the risk that are associated with investing in common stock. The CAPM model is stated below.

- Required rate of return = risk free rate of return + beta * (expected return on the market – risk free rate of return)
- Risk premium = expected return on market – risk free rate of return
- Risk free rate of return = 3%
- Beta = 1.2
- Risk premium = 6%
- Required rate of return (discount rate)
- = 3% + 1.2 * 6%
- = 9.6%

This discount rate of 9.6% will be used to estimate the price of common stock. It can be noted that the dividend payment is expected to grow at different rates. Therefore, a two stage dividend discount model will be used to estimate the price of the shares. The dividend is expected to grow at the rate of 8% per annum over a period of 3 years and at 4% per year onwards. The most recent divided that is paid by the company amounts to $2 per share. The price of the shares will be calculated in two steps. The first step is to calculate the present value of the future stream of dividend payment during the period of high growth rate (8%). The second step is to calculate the future stream of payment that will be received after the period of high growth rate. The calculations are presented in the table below.

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## Calculations of dividends

The growth rate for the first 3 years is 8%. The resulting value of dividend at the end of the third year is $2.519424. However, in the second stage, the growth rate of dividend is 4%. Thus, dividend after the fourth year will be calculated as shown below.

= $2.519424 * 1.04

= $2.62020096

It will be assumed that dividend will remain constant for eternity. The present value of the perpetual dividend will be calculated using the formula presented below.

Value of stock = dividend per share that is paid at year end / (discount rate – growth rate)

This formula that will be used to calculate the present value at the end of the third year is shown below.

= $2.62020096 / (0.096 – 0.04)

= $2.62020096 / 0.056

= $46.7893

The final step is the calculation of the present values of the stream of benefits that the customer will earn from the investment is summarized in the table below.

The sum of all the present values ($41.37) is the price of the common stock. This price can aid Joshua in making a number of decisions. For instance, the investor can compare this value to the prevailing market price of the shares. If the prevailing market price is lower than the price calculated above, it can be an indication that the shares are undervalued. This can be interpreted that the growth rate that is used in the model is higher than what the market expects. In this case, it may be important to reduce the growth rates that are used in the model with an aim of ensuring that the outcome of the model is consistent with the perception of the market. From another point of view, an investor can assume that the prevailing market prices are lower than intrinsic value. Thus, the shares are trading at a cheaper price. The investor can gain by purchasing these shares because the prices are bound to rise. On the other hand, if the intrinsic value is lower than the market price, then it indicates that the shares are overvalued. In this case, the prices of the shares are likely to fall in the future and an investor is likely to make losses. Therefore, it can be an indication of a bad investment. Another interpretation of this scenario is that the market expects the share prices to grow at a rate that is higher than what is used in the model. In this case, it may be necessary to adjust the growth rates that are used in the model upwards.

## Calculating the number of shares

The number of shares will be calculated through the division of the total amount available for investment by the price per share. The calculations are presented below.

Number of shares = Total amount of money available for the investment / price per share

= $9,500 / $41.37

= 229.6 shares.

When the estimated value is truncated, then Joshua should buy 229 shares of Thunderbird.

## Expected return and standard deviation of the portfolio

### Correlation coefficient

## Expected return of the portfolio

= Weight of stock A * expected return of stock A + Weight of stock B * expected return of stock B + Weight of stock C * expected return of stock C

= (30% * 10%) + (50% * 12%) + (20% * 9%)

= 3% + 6% + 1.8%

= 10.8%

## Standard deviation of the portfolio

The first step is to calculate the variance of the portfolio. The standard deviation of the portfolio will be the square root of variance. The formula for calculating variance is presented below.

= (Weight of stock A)^{2} * (variance of stock A) + (weight of stock B)^{2} * (variance of stock B) + (weight of stock C)^{2} * (variance of stock C) + (2 * weight of stock A * weight of stock B * covariance_{AB}) + (2 * weight of stock A * weight of stock C * covariance_{AC}) + (2 * weight of stock B * weight of stock C * covariance_{BC})

## Calculation of covariance

It can be noted that the formula entails the use of covariance. However, only the values of the correlation coefficient between the assets are provided. There is a mathematical relationship between covariance and correlation. The values of covariance will be estimated using this relationship as presented below.

The correlation between two stocks = Covariance between the two stocks

Standard deviation of stock 1 * standard deviation of stock 2

Covariance = Correlation between two stock * products of standard deviation of the two stocks

Covariance_{AB} = correlation_{AB} * standard deviation A * standard deviation B

= 0.8 * 0.14 * 0.18

= 0.02016

Covariance_{AC} = correlation_{AC} * standard deviation A * standard deviation C

= 0.6 * 0.14 * 0.12

= 0.01008

Covariance_{BC} = correlation_{BC} * standard deviation B * standard deviation C

= -0.4 * 0.18 * 0.12

= -0.00864

### Calculation of portfolio variance

= (0.3^{2} * 0.14 * 0.14) + (0.5^{2} * 0.18 * 0.18) + (0.2^{2} * 0.12 * 0.12) + (2 * 0.3 * 0.5 * 0.02016) + (2 * 0.3 * 0.2 * 0.01008) + (2 * 0.5 * 0.2 * -0.00864)

= 0.001764 + 0.0081 + 0.000576 + 0.006048 + 0.0012096 – 0.001728

= 0.0159696

### Portfolio standard deviation

= √0.0159696

= 0.1264

=12.64%

Therefore, if Joshua wants to invest in the three stocks, then the portfolio will yield an expected return of 10.8%. The standard deviation (a measure of risk) of the portfolio will be 12.64%.