## Finance Assignment: Understanding Portfolio Investment Analysis

**Question**

**Task: **
Assessment purpose: To allow students to demonstrate an understanding of various portfolio investment analysis calculation techniques, applicable to real world situations. Facility with these techniques will assist students to construct an investment portfolio, analyse the expected and actual performance of selected securities and to review and change - if deemed desirable - the content of a portfolio. The assessment will reflect the analysis which would be expected of students if, after graduating, they are working in a modern accounting practice or a fund manager’s office.

**Topic: Portfolio Investment Management Calculations and their Application to Decision-making. **

Task Details: This finance assignment consists of five questions (some with multiple parts) and students should answer all questions. Please note that you must solve each problem using the appropriate formula/e, which must be shown. All calculations and workings must also be shown.

**QUESTION 1.**

a. At 15 October, 2020, the share prices of Coal Ltd and Wood Ltd were $30 and $105 respectively. One year later, the respective share prices were $35 and $110.

i. Calculate the price-weighted percentage average return for these two stocks over the year to 15 October, 2021.

ii. Suppose instead, at 15 October, 2021, the final price of Coal Ltd was $35 (as above), but the price of Wood Ltd had fallen to $95. Calculate the revised price-weighted percentage average return for these two stocks over the year to 15 October, 2021.

b. What are both the payoff and the profit or loss per share for an investor in the following two situations?

i. Jean buys the June, 2022 expiration Paypal call option for $6.40 with an exercise price of $120, if the Paypal stock price at the expiration date is $132?

ii. Joan buys a Paypal put option for $4.50 with the same expiration date and exercise price as Jean’s call option, and the Paypal stock price is also $132 at the expiration date?

c. A large investor resident in your country seeks your advice on global investments.

i. State briefly two reasons why he/she should include international equities in his or her investment portfolio.

ii. Identify two risks which apply to the investor if he/she invests in international equities.

d. Two corporate bonds, issued respectively by F Ltd and G Ltd, have the same face value of $10,000 and the same term to maturity of

7 years. F Ltd’s bonds have a coupon rate of 8% per annum, payable half-yearly, and G Ltd’s bonds have a coupon rate of 7.8% per annum, payable bi-monthly (that is, every 2 months). Calculate the effective annual return (EAR) on each bond. [Show each answer as a percentage, correct to 2 decimal places.]

e. Asif is a fund manager with a share portfolio currently valued at $1 billion under management. He considers that the share market is much over-priced and fears a sharp downturn of 20% in the market by June, 2022, which will badly affect his share portfolio’s value and performance, which he wishes to protect. He seeks your advice as to whether he should take a short position in futures or buy a put option, each with an exercise price of $1 billion (the current value of his share portfolio). Explain each of the two strategies, and state your recommendation which Asif should follow, with reasons.

**QUESTION 2.**

a. The expected return of the market index over 2022 is 10%. The standard deviation of returns of the market index is expected to remain at its long-term average of 18%. The risk-free rate is 4%. Calculate:

i. the degree of risk aversion (commonly denoted by ‘A’) for an investor in the market index. ii. the Sharpe ratio of the market index portfolio.

b. The expected return of a risky portfolio in New Zealand over 2022 is 15%, while the risk-free rate is 7%. Terry wishes to set up a complete portfolio, with y (the proportion invested in the risky portfolio) = 0.75.

**REQUIRED:**

i. Define a “complete portfolio”.

ii. Describe the mix (or asset allocation) of Terry’s complete portfolio, including the percentages of each asset held.

iii. What is the expected return of Terry’s complete portfolio?

iv. What is the standard deviation of returns for Terry’s complete portfolio?

v. What is the Sharpe ratio for Terry’s complete portfolio?

c. Mabel is more risk averse than Terry, and her degree of risk aversion, A, is 4.0. Using the data supplied at the beginning of part

b. above, calculate the percentages of each asset class you would recommend she should hold in her optimal complete portfolio. [Show percentages correct to 2 decimal places.]

**QUESTION 3.**

a. Historical data for the All Ordinaries Index indicates that:

- the standard deviation of returns from the Index has been 17%; and - the degree of risk aversion (A) of an investor in the Index is 3.6.

**REQUIRED:**

i. What market risk premium is consistent with the above historical standard deviation?

ii. If the market risk premium is 12%, what would be the historical standard deviation?

b. The expected return of the market in Iceland is 15%. Stock H has a beta of 1.3 and the risk-free rate is 5%.

**REQUIRED:**

i. What is the expected return of Stock H, according to the CAPM?

ii. What is the alpha of a stock? (Definition or explanation required.)

iii. What is the alpha of Stock H, if Iceland Stockbrokers, investors in - and researchers of - the stock, believe that Stock H will provide a return this year of:

I. 20%; or alternatively, if they consider the return this year will be:

II. 14%?

c. Based on your answers to part b. iii. above, is Stock H over-priced, underpriced or fairly priced in each of the situations I. and II.? Would you recommend that Iceland Brokers buy more of – or sell – or just hold Stock H in each of these situations?

d. Jackie, an analyst with Betta Brokers, uses a two-factor (F1 and F2) CAPM index method to evaluate the expected return of stock in Z Ltd. The model uses the following data: E(R) of F1 = 12%; E(R) of F2 = 8%; ? (beta) of F1 = 1.3; ? (beta) of F2 = 0.4; and Rf (risk-free rate) = 5%.

**What is the expected return of a share in Z Ltd?
QUESTION 4.
A. The yield curve for Government-guaranteed zero-coupon bonds is based as follows:
REQUIRED: **

i. What are the implied one-year forward rates for years 1, 2 and 3 respectively?

ii. If the expectations hypothesis of the term structure of interest rates is correct, in one year’s time, what will be the yield to maturity on a one-year zero-coupon bond?

iii. Based on the same hypothesis as in ii. above, in one year’s time, what will be the yield to maturity on a two-year zero-coupon bond?

B. On 15 January, 2021, you bought a Government bond, with a face value of $1,000; a term to maturity of 5 years; a coupon rate of 6% per annum payable yearly, and a yield to maturity of 5% per annum. You paid the market price of $1,043.76 for the bond.

On 15 January, 2022, you sold the bond to Jill, providing her with a yield to maturity of 4% per annum. [NOTE: You bought and sold the bond immediately after payment of the interest coupon due on 15 January each year – that is, the interest payments due on 15 January in 2021 and 2022 are not included in the bond prices.]

**REQUIRED:**

i. What price would Jill have paid for the bond? [Show answer correct to the nearer cent.]

ii. What is your holding period return for holding the bond for one year, receiving the January, 2022 interest coupon, then selling the bond? [Show answer as a percentage, correct to 2 decimal places.]

C. With the aid of hypothetical illustrative examples, briefly explain each of the Expectations and the Liquidity preference hypotheses relating to the term structure of interest rates. Which of the two hypotheses do you consider to be the more relevant? Why?

**QUESTION 5.**

A. Briefly explain the following concepts relating to bond portfolio management.

i. Duration.

ii. Convexity.

iii. Immunisation.

B. Illustrate your answer to A. above with the calculation of the duration and convexity of a bond with a face value of $1,000, term to maturity of 3 years, a coupon rate of 6% per annum, payable yearly, and a yield to maturity of 4% per annum.

[NOTE: As a by-product of these calculations, you should calculate the current market price of the bond, which price should be used as a base or starting point to your answers required in C. i. and C. ii. below.]

C. Calculate the expected price of the bond described in B. above, if the yield to maturity fell immediately to 3% per annum, by each of the following 3 methods.

i. The duration adjustment method.

ii. The duration-with-convexity adjustment method.

iii. The present value of future cash flows method.

D. Which of the methods listed in C. above is most accurate? Why?

E. Explain how a pension fund can use zero-coupon bonds to immunize its obligation to pay out $10 million a year in pensions in perpetuity, if the forecast long-term interest / discount rate is 5% a year forever.

**Answer**

**Finance Assignment Part 1
a. i. Price-weighted percentage average return**

Given,

The share price of Coal Ltd (Beginning of the year) = $30

Share price of Wood Ltd (Beginning of the year) = $105

Sum of the share price of Coal Ltd and Wood Ltd (Beginning of the year) = $(30 + 105) = $135

Share price of Coal Ltd (End of the year) = $35

Share price of Wood Ltd (End of the year) = $110

Sum of the share price of Coal Ltd and Wood Ltd (End of the year) = $(35 + 110) = $145

Price weighted percentage average return of the two stocks = [(145 – 135) / 135] * 100%

= 7.40%

a. ii. Revise price-weighted percentage average return

Given,

Price of Wood Ltd = $95

Sum of the share price of Coal Ltd and Wood Ltd (End of the year) = $35 + $95 = $130

Sum of the share price of Coal Ltd and Wood Ltd (Beginning of the year) = $(30 + 105) = $135

Price weighted percentage average return of the two stocks

= [(130 - 135) / 135] * 100%

= -3.70%

b. i. Payoff and profit or loss per share for an investor in expiration PayPal call option

Given,

Call option price = $6.40

Exercise price = $120

Expiration date price = $132

Pay-off:

= Stock price at expiration – strike price

= $132 - $120

= $12

Profit per share:

= Pay-off per share – premium

= $12 - $6.40

= $5.60

b. ii. Payoff and profit or loss per share for an investor in expiration PayPal put option

Given,

Put option price = $4.50

Expiration date price = $132

Pay-off = It will be zero because the strike price is lesser than the stock price.

Loss per share:

= pay-off per share – premium

= $0-$4.50

= -$4.50

c. i. Two reasons for including international equities in his or her investment portfolio
The two reasons for investing in international equities are discussed below:

1. Diversification: Domestic index funds will assist in the limited exposure to international stocks, and many agree that these funds will not allow the investor to get complete diversification (Torrente and Uberti, 2021). In order to get the real exposure of the international stocks and make the portfolio more and more diversified, the investor is required to invest in international equities. The international stock markets represent over 40% of the world's equity investment.

2. Geographic advantages: The international equities will help the investor boost returns by exposing dollars to the faster-growing economy. Different nations have different agendas, such as government leadership, taxation, access to natural resources and many (Belderbos, Tong and Wu, 2020). These factors impact the stocks' performance, resulting in rapid growth.

c. ii. Two risks applied to the investor investing in international equities

The two risks applied to the investors investing in international equities are discussed below:

1. Currency risk: If the dollar weakens then the international holdings will provide a hedge against the movement of the currency (Wiriadinata, 2018). However, if the dollar becomes strong, then the international stock's performance will become weaker.

2. Geopolitical risk: In the case of any political unrest, the nation can also suffer from an economic downturn (Smales, 2021). On such an occasion, the international stocks might fall, resulting in overall returns.

d. Effective annual return on each bond

Given,

Face value of bond issued by F Ltd = $10000

Time to maturity = 7 years

Coupon rate = 8% p.a. (half-yearly)

Effective annual return of bond issued by F Ltd:

= (1 + coupon rate/compounding periods) ^compounding periods – 1

= (1 + 8%/2) ^2 – 1

= 8.16%

Given,

Face value of bond issued by G Ltd = $10000

Time to maturity = 7 years

Coupon rate = 7.8% p.a. (bi-monthly)

Effective annual return of bond issued by G Ltd:

= (1 + coupon rate/compounding periods) ^compounding periods – 1

= (1 + 7.8%/6) ^2 – 1

= 8.06%

**e. Explaining the two strategies and recommending Asif**

The two strategies and the explanation has been discussed below:

1. Short position in futures: A short hedge is taking a short position in future contracts. When the assets are being expected to be sold then the future contracts are being enforced. In the case of a sharp downturn, the prices will fall since Asif has short sell the futures; therefore, when it is sold, Asif will gain (Wang et al., 2021). However, if the prices rise, then Asif will suffer losses.

2. Buy a put option: Buying a put option will limit the loss that Asif might incur. If there is an economic downturn, Asif buying off the put option will reduce the effective loss (Wang et al., 2021). However, Asif will still suffer a loss.

**Recommendation:**

It is advised to Asif to a short position in the future. Then, if there is an economic downturn, the Asif will sell the futures and gain the return, which will effectively nullify the loss that the portfolio might have experienced after the economic downturn. On the other, if the future price increases, Asif will suffer loss; however, the increase in the portfolio return will nullify the loss.

**Part 2
a. i. Degree of risk aversion**

Given,

Expected return in 2022 = 10%

Standard deviation = 18%

Risk-free rate = 4%

**Degree of risk aversion:**

= (Expect return – risk free rate) / (standard deviation) ^2

= (10% - 4%)/18%^2

= 1.85

a., ii. Sharpe ratio

Sharpe ratio:

= (Portfolio return – risk-free rate) / standard deviation

= (10% - 4%) / 18%

= 33.33%

b. i. Complete portfolio

The complete portfolio is a combination of risky assets, with return Rp and Rf being the return of risk-free assets (Kostadinova et al., 2021). The portfolio’s expected return will be E(Rc) = Wp * E(Rp) + (1 – Wp) Rf.

b. ii. A mix of Terry's complete portfolio

Terry's portfolio can be categorized into two kinds of assets that are risky and non-risky. The allocated percentage of risky asset in the portfolio is 75%, and the percentage of assets that qualifies to be non-risky is 25%. Hence, Terry prefers taking risky allocation of assets to the portfolio.

**b. iii. Expected return of Terry’s portfolio**

Expected return:

= Weight of the portfolio * expected return of risky portfolio + (1 - weight of the portfolio) * risk-free rate

= 0.75 * 15% + (1 – 0.75) * 7%

= 13%

b. iv. Standard deviation of return

Standard deviation of return:

= [(75% - 13%) ^2 + (25% - 13%) ^2] ^1/2

= 20%

b. v. Sharpe ratio

Sharpe ratio:

= (Portfolio return – risk-free rate) / standard deviation

= (13% - 7%) / 20%

= 30%

c. Percentages of each asset class to be recommended to Mabel

Given,

A = 4

Standard deviation = 24.5%

Expected return – risk free rate = A * (standard deviation) ^2 = 4 * (0.245) ^2 = 0.24 = 24%

Expected return = 24% -risk free rate = 24% - 7% = 17%

Since, for generating 15% return the portfolio had 75% risky assets therefore, to generate 17% return the proportion of risky asset will increase to 85%. Therefore, Mabel will have to invest in 15% of non-risky assets.

Finance Assignment Part 3

a. i. Market risk premium

Return expected – risk free rate = A * (standard deviation) ^2

Also, market risk premium = Expected return rate – risk free rate

Therefore, market risk premium = A * (standard deviation) ^2 = 3.6 * (17%) ^2 = 10.40%

a. ii. Historical standard deviation

Market risk premium = 12%

Standard deviation:

= (Market risk premium / risk coefficient) ^1/2

= (12%/3.6) ^1/2

= 1.66%

b. i. Expected return of stock H

Beta = 1.3

Risk-free rate = 5%

Expected market return = 15%

Expected return of stock H:

= Risk free rate + (Beta * market return)

= 5% + (1.3 * 15%)

= 24.50%

**b. ii. Alpha of stock**

Alpha is an index used to determine the highest possible return concerning the least amount of the risk.
Alpha of stock H:

= Return of portfolio – risk free rate – beta * (Return of market – risk free rate)

= 24.50% - 5% - 1.3 * (15% - 5%)

= -6.50%

b. iii. Alternatively, alpha of stock H

I. Return = 20%

Alpha = 20% - 5% - 1.3 * (15% - 5%) = -11%

II. Return = 14%

Alpha = 14% - 5% - 1.3 * (15% - 5%) = -17%

**c. Recommending the stock H to Iceland Brokers**

I. When return is 20%, the alpha is -11% therefore, it can be concluded that the stock is way too risky. Hence, selling the stock will be good option.

II. When the return is 14%, the alpha is -17%; therefore, it can be concluded that the stock is way too risky as well. Hence, the stocks must be sold.

d. Expected return of a share in Z Ltd

E(R) of F1 = 12%

Beta of F1 = 1.3

Risk-free rate = 5%

Expected return of F1:

= 5% + (1.3 * 12%)

= 20.60%

E(R) of F2 = 8%

Beta of F2 = 0.4

Expected return of F2:

= 5% + (0.4 * 8%)

= 8.20%

Expected return = (20.60% + 8.20%) / 2 = 28.80% / 2 = 14.40%

Part 4

A. i. Implied one-year forward rate for 1, 2, and 3

Formula for implied rate:

= (1+YTMn) ^n/(1+YTMn-1) ^n-1

Years YTM Implied rates

1 8%

2 9% [(1+9%) ^2/ (1+8%) ^1]-1=10.01%

3 10% [(1+10%) ^3/ (1+9%) ^1]-2=12.03%

A. ii. YTM on a one-year zero-coupon bond

10.01% is the YTM of one-year zero coupon bond.

A. iii. YTM on a two-year zero-coupon bond

12.03% is the YTM two-year zero-coupon bond.

B. i. Price paid for the bond

Face value = $1000

Term to maturity = 5 years

Coupon rate = 6%

YTM = 4% p.a.

Market price = $1043.76

Price paid by Jill for the bond:

= Periodic coupon payment * 1-(1+YTM) ^-number of periods till maturity/YTM + face value/(1+YTM) ^ number of periods

= 1*1-(1+4%) ^-5/5%+1000/ (1+4%) ^5

= $826.37

B. ii. Holding period return

Holding period return:

= (826.37-787.85)/787.85 *100%

= 4.89%

**C. Expectations and liquidity preference hypothesis**

The Expectations hypothesis helps in predicting the interest rate in the short-term in the future based on the present long-term rates of interest (Caldeira and Smaniotto, 2019). According to this hypothesis, an investor will be earning the same amount of interest by investing into two consecutive bond investments that are one-year bond and two-year bond.
In the case of the liquidity preference hypothesis, it is suggested that an investor should demanding an interest rate that is higher or premium on the stocks with the long-term maturities that will have a high risk for considering all the factors to be same, the investors will be preferring cash or assets that qualify as high liquid (Xie, Wang and Meng, 2019).

**Part 5
A. i. Duration**

Duration means measuring the bond price sensitivity or other debt instruments to a fluctuation in the rate of interest (Di Asih and Abdurakhman, 2021). The bond duration can be confused as the time to maturity as duration measurements are also calculated in years. However, the fact is that the bond term is linear, whereas the duration is a non-linear concept.

**A. ii. Convexity**

Convexity is the measurement of the curvature of the degree of the curve in the relationship between the bond’s price and the bond’s yield (Homaifar and Michello, 2019). Convexity is a tool for measuring and managing the exposure of portfolio to a market risk. Convexity helps in establishing the duration of a bond changes as the change in the rate of interest. When the duration of bond increases with the increase in yields, the bond is said to have a negative convexity.

**A. iii. Immunisation**

Immunization is a risk mitigation strategy that matches the duration of the liabilities and assets to reduce the impact of rate of interest on the net worth (Lapshin, 2019). It is a strategy to mitigate the risk that will match the assets and the liability duration so that the portfolio values are protected against the changes in the rate of interest.

**B. Calculation of duration and convexity of bond**

Face value = $1000

Term to maturity = 3 years

Coupon rate = 6% p.a. payable yearly

YTM = 4% p.a.

Duration of the bond:

term cash flow present value time weighted pv(pv * term)

1 60 60/(1+4%)^1=57.69230769 57.69230769

2 60 60/(1+4%)^2=55.47337278 110.9467456

3 1060 1060/(1+4%)^3=942.3361402 2827.008421

Sum=1055.501821 Sum= 2995.647474

Price of the bond = $1055.50

Duration = Time weighted pv (sum) / price of the bond = 2995.64 / 1055.50 = 2.83

term cash flow present value time weighted pv (pv * term) convexity time weighted pv(pv*term*(term+0.5)

1 60 57.69230769 60/(1+4%)^1=57.69230769 86.53846154

2 60 55.47337278 60/(1+4%)^2=55.47337278 277.3668639

3 1060 942.3361402 1060/(1+4%)^3=942.3361402 9894.529472

1055.501821 Sum=2995.647474 Sum=10258.4348

Convexity = sum of convexity time weighted pv / (present value (1+YTM) ^2) = 10258.43 / (1055.50*(1+4%) ^2) = 8.98

C. i. Duration adjustment method

Modified duration = Duration/(1+YTM) = 2.83 / (1+3%) = 2.74

Percentage change in price = -2.74 * (-25%) = 69%

Change in price of the bond = $1055.50 * 0.69 = +$727.95

Price of the bond = $1055.50 + $727.95 = $1783.45

C. ii. Duration with convexity adjustment method

Convexity adjustment = Convexity of the bond * 100 * (change in YTM) ^2 = 8.98 * 100 * (4%-3%) ^2 = 0.089 = 8.9%

Change in the price of bond = Duration * yield change + convexity adjustment = 2.83 * -0.25 + 8.9% = -61.85%

Bond price = $1055.50* -65% = $369.425

C. iii. Present value of the future cash flow method

YTM 3%

0 1 2 3

57.69/(1+3%)^1=56.00970874 57.69

55.47/(1+3%)^2=52.28579508 55.47

942.33/(1+3%)^3=862.3654399 942.33

Total sum of the price of bond = $970.66

**D. Most accurate method**

Duration with convexity adjustment is the most accurate because it considers the convexity of the bond, which, when omitted, create a large breadth of inaccurate measurement.

**E. Pension fund using zero-coupon bonds to immunize its obligation**

The perfect immunization of pension funds using zero-coupon bonds is possible because it will guarantee that the movement in the interest rates will have no virtual impact on the value of the portfolios. It will be easier for the fund manager to reach the desired amount that will be paid to the pension holders. If the interest rates are fluctuating then the fund manager will not be able to pay out $10 million. Therefore, in order to pay out of $10 million a year in pensions immunization is the perfect solution.

**References**

Belderbos, R., Tong, T.W. and Wu, S., 2020. Portfolio configuration and foreign entry decisions: A juxtaposition of real options and risk diversification theories. Strategic Management Journal, 41(7), pp.1191-1209. https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/smj.3151

Caldeira, J.F. and Smaniotto, E.N., 2019. The expectations hypothesis of the term structure of interest rates: The Brazilian case revisited. Applied Economics Letters, 26(8), pp.633-637. https://www.researchgate.net/profile/Emanuelle-Smaniotto/publication/325837687_The_expectations_hypothesis_of_the_term_structure_of_
interest_rates_The_Brazilian_case_revisited/links/5e869938299bf13079746f8b/
The-expectations-hypothesis-of-the-term-structure-of-interest-rates-
The-Brazilian-case-revisited.pdf

Di Asih, I.M. and Abdurakhman, A., 2021. Delta-Normal Value at Risk Using Exponential Duration with Convexity for Measuring Government Bond Risk. DLSU Business & Economics Review, 31(1), pp.72-80. https://www.dlsu.edu.ph/wp-content/uploads/2021/08/DLSUBER.2021.July_.6maruddani-revised.pdf

Homaifar, G.A. and Michello, F.A., 2019. A generalized algorithm for duration and convexity of option embedded bonds. Applied Economics Letters, 26(10), pp.835-842. http://www.mtsu.edu/econfin/Convertiblesfinal.pdf

Kostadinova, V., Georgiev, I., Mihova, V. and Pavlov, V., 2021, February. An application of Markov chains in stock price prediction and risk portfolio optimization. In AIP Conference Proceedings (Vol. 2321, No. 1, p. 030018). AIP Publishing LLC. https://aip.scitation.org/doi/pdf/10.1063/5.0041119

Lapshin, V., 2019. A nonparametric approach to bond portfolio immunization. Mathematics, 7(11), p.1121. https://www.mdpi.com/2227-7390/7/11/1121/htm

Smales, L.A., 2021. Geopolitical risk and volatility spillovers in oil and stock markets. Finance assignment The Quarterly Review of Economics and Finance, 80, pp.358-366. https://www.researchgate.net/profile/Lee-Smales/publication/334386206_Geopolitical_Risk_and_Volatility_Spillovers_in_Oil_and_
Stock_Markets/links/6045ee08299bf1e07862c1fa/Geopolitical-
Risk-and-Volatility-Spillovers-in-Oil-and-Stock-Markets.pdf

Torrente, M.L. and Uberti, P., 2021. Geometric Diversification in Portfolio Theory. https://www.researchgate.net/profile/Maria-Laura-Torrente/publication/351691895_Geometric_Diversification_in_Portfolio_
Theory/links/60a4ff23299bf10613724784/Geometric-Diversification-in-Portfolio-Theory.pdf

Wang, L., Ahmad, F., Luo, G.L., Umar, M. and Kirikkaleli, D., 2021. Portfolio optimization of financial commodities with energy futures. Annals of Operations Research, pp.1-39. https://link.springer.com/article/10.1007/s10479-021-04283-x
Wiriadinata, U., 2018. External debt, currency risk, and international monetary policy transmission. The University of Chicago. https://knowledge.uchicago.edu/record/345/files/Wiriadinata_uchicago_0330D_14350.pdf

Xie, Y., Wang, Z. and Meng, B., 2019. Stability and bifurcation of a delayed time-fractional order business cycle model with a general liquidity preference function and investment function. Mathematics, 7(9), p.846. https://www.mdpi.com/2227-7390/7/9/846/htm