Finance Questions And Solutions

Interest Rates

Q1 Solutions

Risks may be classified as financial and non-financial risks (Actuaries, 2018). Examples of financial risks include:-

1. Credit risk- This is the risk where third parties such as borrowers or counterparty defaults on their payments.
2. Liquidity risk- This is a risk where an entity do not have enough financial resources to meet their obligations. Banks have a higher liquidity risk in comparison to other institutions, because they lend depositors’ funds and funds raised from money markets to other organizations, for longer periods than they offer to the providers of the funds.
3. Market Risk- Movements in investment market values, interest rates and inflation rates may lead to market risk.
4. Business risk- These are specific to the business undertaken as a bank, investing in a business or project that fails to be successful.
5. Systemic risk -Systemic risk cannot be diversified away. It is usually associated with succession failures such that failure of a big firm may cause the failure to other related entities. e.g. global financial crisis (Gangreddiwar, 2015). Furthermore, it tends to affects the entire industry.

Examples of non-financial risks include:-

1. Operational risk – This is an internal risk. It arises from inadequate or failed internal processes, systems and people.
2. External risk – This risk is beyond the company’s control as it arises from external events e.g. attacks, fire, tsunamis, regulation changes

References

Actuaries, I. a. (2018). Actuarial Risk Management. London: IFOA.

Gangreddiwar, A. (2015, September 29). 8 Risks in the Banking Industry Faced by Every Bank. Retrieved from Medici: https://gomedici.com/8-risks-in-the-banking-industry-faced-by-every-bank/ 

Q2 Solutions

Solve i  using Formula

i = (1 + )ⁿ -1 

1) 5.52% payable annually

i = (1 + )¹ -1

=5.52% 

2) 5.50% payable semi annually

 i = (1 + )²-1

=5.576% 

3) 5.48% payable quarterly

 i = (1 + )⁴ -1

=5.594% 

4) 5.45% payable monthly

 i = (1 + )¹² -1

=5.588% 

Answer: 5.52% payable annually provides the lowest cost of finance. 

Q3. Solutions

1. EMV decision rule

EMV (Fixed deposits) =0.3*5.5% + 0.5*5.5% +0.2*5.5%= 5.5%

EMV (Stock mutual fund) =0.3*12% + 0.5*9% +0.2*-2%= 7.7%

EMV (Bond) =0.3*10% + 0.5*8.7% +0.2*3%= 7.95% 

Answer: Select bonds because it has the highest payoff 

1. EOL (expected opportunity loss)

EOL (Fixed deposits) =0.3*(12%-5.5%) + 0.5*(9%-5.5%) +0.2*(5.5%-5.5%)=  3.70%

EOL (Stock mutual fund) =0.3*(12%-12%) + 0.5*(9%-9%) +0.2*(5.5%-(-2%))= 1.50%

EOL (Bond) =0.3*(12%-10%) + 0.5*(9%-8.7%) +0.2*(5.5%-3%)= 1.25% 

Answer: Select stock mutual funds because it has the lowest expected opportunity loss 

3.

=Maximum payoff – Expected payoff)*$100,000

Maximum payoff = 0.3*12% + 0.5*9% +0.2*5.5%= 9.20%

Expected Payoff = 7.95%

=(9.20% – 7.95%)*100,000

=1,250

Answer:  XYZ company should be willing to pay $1,250

4 Decision TreeQ4. Solutions

Dividend Growth Model

V = D/(K-g)

Where D= expected annual dividend next year, k is required return and g is the growth rateV= 0.32*(1+3%) +{(1+12%)^-1*0.32*(1+3%)²*(1+(1+2%)/(12%-2%))}

=0.32*1.03 +{1.12^-1*0.32*1.03²)*(1+1.02/0.1)}

=3.72 

Q5. Solutions

Formula 

C =  SN(d1) – Ke^-rT N( d2) 

d1 =  

d2 = d1 – σ 

We are given that  S = 42,  K = 40, σ = , r = 0.03, T = 3/12 = 0.25.

d1 =  

=0.357282 

d2 = d1 – σ 

=0.122762 

Therefore value of call option is:-

C =  42N(d1) – 40e^-0.03*0.25 N( d2) 

=42*0.63956-40* e^-0.03*0.25 *0.548852

=5.071 

Answer: Value $5.07 

Q6: Solutions

K =43, S₀= 42, S₂^u =45,  S₂^d =38, r = 0.04 

If S₂ = S₂^u =45, then the call option will be worth c₂^u =45-43=2

If S₂ = S₂^d =38, then the call option will be worth c₂^d = 0

V₂  =                      S₂^u  – c₂^u  = 45N -2 , If S2 = S₂^u =45

                              S₂^d – c₂^d = 38N,         If S2 = S₂^d =38 

For the risk-free portfolio that is used to value the stock option, these two values must be equal i.e. 

Investments

V₂  = 45N – 2 = 38 N . Therefore N = 0.2857

Hence ,

 V₂  =                     45*0.2857 -2  = 10.86

                              38 * 0.2857 = 10.86

When this is discounted to its present value, it must be equal to the value of the portfolio at time t =0

V₀ = S₀N – C ₀ = V₂e^-r*2/12

C _{0 =} S₀N – V₂e^-r/12 = (42*0.2857)-(10.86* e^-0.04*2/12) = 1.215 

The value of call option is 1.215 

Hedge Ratio:

Hedge Ratio = f u − f d S 0 − u − S 0 − d = 2 − 0 *42 – 45  − 42- 38 = 0.2857 

i.e There is a 28.57% changes in option values when per $1 dollar change in stock price. 

Q7: Solutions

Using calculator

Statistic	ABC Stock	XYZ Stock
Average Returns	1.34%	1.39%
Standard deviation	0.77%	3.30%
Geometric mean	1.34%	1.35%
Sharpe Ratio	1.74	0.41
VaR (parametric)	14,230	(809,604)
Correlation coefficient	0.078	0.078

 When selecting a portfolio, investors should look for assets that are less correlated, high risk adjusted performance, both absolute and relative to the market, and good inflation hedging properties. From above statistic, ABC has a higher Sharpe ratio and positive VaR suggesting that the stock have a good risk adjusted performance. Therefore, I would recommend that Mr John to invest more in ABC stock.

Part b- VaR Combined portfolio

portfolio weight of ABC                      50%    

portfolio weight of XYZ                       50%

return on ABC                                     1.34%

return on XYZ                                      1.39%

Standard deviation of ABC                 0.77%

Standard deviation of XYZ                  3.30%

expected return of the portfolio         50%*(1.34% +1.39%)=1.37%

standard deviation of the portfolio    1.69% 

VaR = [Expected Weighted Return of the Portfolio – (z-score 95% CI * standard deviation of the portfolio)] * portfolio value

=(1.37%-(1.65*1.69%))*20,000,000

=($285,209.18) 

VaR for combined portfolio is lower than the VaR for stock ABC and higher than VaR for stock XYZ.

optimal weights for A and B based on Minimum Risk approach

Wabc =

Wxyz= 1 – Wabc 

Q8: Solutions

Part A

Using calculator volatility of ABC is 0.00005937 and XYZ is 0.00108721

ABC

 l= 0.96

σ_n-1² = 0.00005937

 U²_{n-1 =} ((52.93-51.80)/51.80)^2 = 0.000476

EVMA : σ_n²= l σ_n-1² + (1-l) U²_n-1

σ_n² = 0.96*0.00005937 + (1-0.96)*0.000476

=0.0000760 

XYZ

 l= 0.96

σ_n-1² = 0.00108721

 U²_{n-1 =} ((67.35-67.14)/67.14)^2 = 0.0000098

EVMA  : σ_n²= l σ_n-1² + (1-l) U²_n-1

σ_n² = 0.96*0.00108721 + (1-0.96)*0.0000098

=0.0010441 

Part B

GARCH (1, 1) model

σ ² _t= ω + α *u²_t−1 +  β ×σ² _{t –1} 

ABC Shares

ω= 0.000003, α=0.04, and β=0.82

 g?ABC?= 1 – a- b= 1-0.04-0.82= 0.14

V (ABC) = w/ g= 0.000003/ 0.14= 0.00002143

Volatility = √ 0.00002143 = 0.004629 

XYZ Shares

 ω= 0.000002, α=0.05, and β=0.82

g?XYZ?= 1 – a- b= 1-0.05-0.82= 0.13

V (XYZ) = w/ g= 0.000002/ 0.13= 0.00001538

Volatility= √ 0.00001538 = 0.003922 

Part c

Garch (1,1) can be used to estimate the long run volatility

ABC Shares

ω= 0.000003, α=0.04, and β=0.82

 g?ABC?= 1 – a- b= 1-0.04-0.82= 0.14

V (ABC) = w/ g= 0.000003/ 0.14= 0.00002143

Volatility = √ 0.00002143 = 0.004629 

XYZ Shares

 ω= 0.000002, α=0.05, and β=0.82

g?XYZ?= 1 – a- b= 1-0.05-0.82= 0.13

V (XYZ) = w/ g= 0.000002/ 0.13= 0.00001538

Volatility= √ 0.00001538 = 0.003922 

Comment

The long-run average volatility of XYZ shares is less than ABC. Thus XYZ has a lower risk than ABC . Using the risk return relationship, expected returns for ABC are greater than XYZ.