Special Topics in Finance (FIN-253)

15. (a)

Given:

• Average daily assets = Rs 1,800 million
• Securities sold during the year = Rs 600 million
• Securities purchased during the year = Rs 500 million

i) Portfolio turnover rate

Portfolio turnover rate is normally based on the lesser of purchases and sales:

[ \text{Turnover rate}=\frac{\min(\text{Purchases, Sales})}{\text{Average daily assets}}\times 100 ]

[ =\frac{500}{1800}\times 100 = 27.78% ]

Portfolio turnover rate = 27.78%.

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ii) Total fees paid to investment managers and other administrative expenses

Total expenses of the fund:

[ \text{Total expenses}=1.2%\times 1800=\text{Rs }21.6\text{ million} ]

Management fee paid to investment managers:

[ \text{Management fee}=0.8%\times 1800=\text{Rs }14.4\text{ million} ]

Other administrative expenses:

[ \text{Other administrative expenses}=21.6-14.4=\text{Rs }7.2\text{ million} ]

Total fees paid to investment managers = Rs 14.4 million.
Other administrative expenses = Rs 7.2 million.

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15. (b)

Given:

• Average return on portfolio P, (R_p = 35%)
• Average return on market, (R_m = 28%)
• Beta of portfolio P, (\beta_p = 1.2)
• Beta of market, (\beta_m = 1.0)
• Standard deviation of portfolio P, (\sigma_p = 42%)
• Standard deviation of market, (\sigma_m = 30%)
• T-bill rate, (R_f = 6%)

i) Treynor and Sharpe measures

Treynor measure:

[ T=\frac{R_p-R_f}{\beta} ]

For portfolio P:

[ T_p=\frac{35-6}{1.2}=\frac{29}{1.2}=24.17% ]

For the market:

[ T_m=\frac{28-6}{1.0}=22% ]

Sharpe measure:

[ S=\frac{R_p-R_f}{\sigma} ]

For portfolio P:

[ S_p=\frac{35-6}{42}=\frac{29}{42}=0.6905 ]

For the market:

[ S_m=\frac{28-6}{30}=\frac{22}{30}=0.7333 ]

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ii) Performance interpretation and conflicting results

• According to the Treynor measure, portfolio P outperformed the market because (T_p = 24.17% > T_m = 22%).
• According to the Sharpe measure, portfolio P underperformed the market because (S_p = 0.6905 < S_m = 0.7333).

These two measures may give conflicting results because they use different definitions of risk:

• Treynor measure uses beta, which measures only systematic or market risk. It assumes the portfolio is well diversified, so unsystematic risk is ignored.
• Sharpe measure uses standard deviation, which measures total risk, including both systematic and unsystematic risk.

Here, portfolio P has a higher beta than the market, so its excess return per unit of systematic risk is higher. But its total standard deviation is also much higher than the market’s. In fact, if P were perfectly diversified, its standard deviation would be:

[ 1.2\times 30%=36% ]

But its actual standard deviation is (42%), so it carries substantial unsystematic risk. The Sharpe measure penalizes this extra total risk, while the Treynor measure does not.

Also, using CAPM, the required return on P is:

[ R_p = 6 + 1.2(28-6)=32.4% ]

Since the actual return is (35%), portfolio P has a positive alpha of (2.6%). Thus, on a systematic-risk-adjusted basis, P outperformed the market; but on a total-risk-adjusted basis, it underperformed. If the portfolio were fully diversified, the two measures would generally not conflict.

Given:

• Risk-free rate, (R_f = 6%)
• Market return, (R_m = 12%)
• Beta of ABC stock, (\beta = 1.5)
• Current dividend, (D_0 = \text{Rs }20)
• Supernormal growth, (g_1 = 20%) for 2 years
• Normal growth after year 2, (g_2 = 5%) indefinitely
• Current market price = Rs 260

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a. Required rate of return on ABC stock

Using CAPM:

[ K_e = R_f + \beta (R_m - R_f) ]

[ K_e = 6% + 1.5(12% - 6%) ]

[ K_e = 6% + 9% = 15% ]

Required rate of return = 15%.

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b. Expected dividends per share for next two years

[ D_1 = D_0(1+g_1) = 20(1.20) = \text{Rs }24 ]

[ D_2 = D_1(1+g_1) = 24(1.20) = \text{Rs }28.80 ]

Expected dividends:

• Year 1: Rs 24
• Year 2: Rs 28.80

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c. Expected selling price at the end of year 2

First find dividend for year 3:

[ D_3 = D_2(1+g_2) = 28.80(1.05) = \text{Rs }30.24 ]

Using constant growth model:

[ P_2 = \frac{D_3}{K_e - g_2} ]

[ P_2 = \frac{30.24}{0.15 - 0.05} = \frac{30.24}{0.10} = \text{Rs }302.40 ]

Expected price at end of year 2 = Rs 302.40.

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d. Intrinsic value of the stock today

[ P_0 = \frac{D_1}{(1+K_e)^1} + \frac{D_2}{(1+K_e)^2} + \frac{P_2}{(1+K_e)^2} ]

[ P_0 = \frac{24}{1.15} + \frac{28.80}{(1.15)^2} + \frac{302.40}{(1.15)^2} ]

[ P_0 = 20.87 + 21.78 + 228.66 ]

[ P_0 = \text{Rs }271.30 ]

Intrinsic value today = Rs 271.30.

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e. Is the stock underpriced or overpriced? Would you buy it?

• Intrinsic value = Rs 271.30
• Market price = Rs 260

Since:

[ \text{Intrinsic value} > \text{Market price} ]

[ 271.30 > 260 ]

The stock is underpriced by:

[ 271.30 - 260 = \text{Rs }11.30 ]

Conclusion: The stock is underpriced. Since it is selling below its intrinsic value, it is expected to provide a return higher than the required 15%. Therefore, I would prefer to buy the stock.

17. Fixed Income Security

Meaning [3 marks]

A fixed income security is a financial instrument that promises the investor a fixed or determinable stream of periodic payments (interest or coupon) at stated intervals, plus repayment of the principal amount on a specified maturity date. Because the return is contractually fixed at the time of issue, the investor knows in advance how much and when he will receive. Examples: government bonds, corporate debentures, treasury bills. The main risks borne by the holder are interest rate risk, default (credit) risk and inflation risk.

Types of Fixed Income Securities [12 marks]

1. Government / Sovereign Bonds Issued by the central government (in Nepal, through Nepal Rastra Bank) to borrow long-term funds. They carry almost no default risk because the government can tax and borrow to repay. Interest is usually paid semi-annually and principal at maturity. Example: Nepal Government Development Bonds. Their yield serves as the benchmark risk-free rate for the market.

2. Treasury Bills Short-term money market instruments issued by the central bank on behalf of the government, maturing within one year (91, 182, 364 days in Nepal). They are sold at a discount and redeemed at face value; the difference is the investor's return. They are highly liquid and virtually risk-free.

3. Corporate Bonds / Debentures Long-term debt securities issued by companies to raise capital. The issuer pays a fixed coupon and repays principal at maturity. They carry higher default risk than government bonds, so they offer a higher yield. They may be secured (backed by specific assets/mortgage) or unsecured (debentures), and may be convertible (convertible into equity shares) or non-convertible.

4. Municipal / Local Government Bonds Issued by local bodies or municipalities to finance infrastructure such as roads, water supply and schools. They offer a yield higher than government bonds but lower than corporate bonds, and their credit quality depends on the fiscal strength of the issuing local body.

5. Money Market Instruments (Short-term) Besides treasury bills, these include commercial papers (unsecured short-term promissory notes issued by large, creditworthy firms), certificates of deposit (negotiable time deposits issued by banks) and banker's acceptances. They are highly liquid and traded in large denominations.

6. Preferred Stock (Fixed-income characteristic) Although equity in form, preferred shares pay a fixed dividend and rank before common shareholders for dividend and liquidation proceeds. They behave like perpetual fixed income securities, though the dividend is not a legal obligation.

7. Asset-Backed / Mortgage-Backed Securities Securities created by pooling loans (home loans, auto loans, credit card receivables) and issuing claims against the pooled cash flows. The holder receives periodic payments from the underlying loan instalments. They are fixed income in nature but carry prepayment risk.

8. Zero-Coupon Bonds Bonds issued at a deep discount and redeemed at face value, with no intermediate interest payment. The return is the difference between purchase price and maturity value. They are useful for investors who want a known lump sum at a future date.

9. Inflation-Linked / Indexed Bonds Bonds whose coupon or principal is adjusted for inflation (e.g., Treasury Inflation-Protected Securities). They protect the real purchasing power of the investor's return.

10. Callable and Putable Bonds Callable bonds give the issuer the right to redeem before maturity when interest rates fall; putable bonds give the holder the right to sell back to the issuer when rates rise. These embedded options affect the bond's price and yield.

Conclusion

Thus, fixed income securities range from risk-free treasury bills to higher-yielding corporate and asset-backed instruments. They suit investors seeking regular, predictable income and capital safety, and they form the backbone of the debt market by channelling savings into government and corporate borrowing.

Solution:

Let returns be in percent. Number of years (n=5).

a. Average return and standard deviation for each asset

Formula:

[ \bar R=\frac{\sum R}{n},\qquad \sigma=\sqrt{\frac{\sum (R-\bar R)^2}{n}} ]

Asset A

Returns: (10,12,14,16,18)

[ \bar R_A=\frac{10+12+14+16+18}{5}=14% ]

[ \sum (R_A-\bar R_A)^2=(-4)^2+(-2)^2+0^2+2^2+4^2=40 ]

[ \sigma_A=\sqrt{\frac{40}{5}}=\sqrt{8}=2.83% ]

Asset B

Returns: (10,16,14,12,10)

[ \bar R_B=\frac{10+16+14+12+10}{5}=12.4% ]

[ \sum (R_B-\bar R_B)^2=(-2.4)^2+(3.6)^2+(1.6)^2+(-0.4)^2+(-2.4)^2=27.2 ]

[ \sigma_B=\sqrt{\frac{27.2}{5}}=\sqrt{5.44}=2.33% ]

Asset C

Returns: (10,12,14,16,16)

[ \bar R_C=\frac{10+12+14+16+16}{5}=13.6% ]

[ \sum (R_C-\bar R_C)^2=(-3.6)^2+(-1.6)^2+(0.4)^2+(2.4)^2+(2.4)^2=27.2 ]

[ \sigma_C=\sqrt{\frac{27.2}{5}}=\sqrt{5.44}=2.33% ]

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b. Covariance between A and B, and A and C

Formula:

[ Cov(A,B)=\frac{\sum (R_A-\bar R_A)(R_B-\bar R_B)}{n} ]

[ Cov(A,C)=\frac{\sum (R_A-\bar R_A)(R_C-\bar R_C)}{n} ]

[ Cov(A,B)=\frac{-8}{5}=-1.6%^2 ]

[ Cov(A,C)=\frac{32}{5}=6.4%^2 ]

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c. Return and standard deviation of the portfolios

Equal weights are used, so:

[ w_A=w_B=0.5,\qquad w_A=w_C=0.5 ]

Formula:

[ R_p=w_A R_A+w_B R_B ]

[ \sigma_p^2=w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_A w_B Cov(A,B) ]

Portfolio AB

[ R_{AB}=0.5(14)+0.5(12.4)=13.2% ]

[ \sigma_{AB}^2=0.5^2(8)+0.5^2(5.44)+2(0.5)(0.5)(-1.6) ]

[ \sigma_{AB}^2=2+1.36-0.8=2.56 ]

[ \sigma_{AB}=\sqrt{2.56}=1.6% ]

Portfolio AC

[ R_{AC}=0.5(14)+0.5(13.6)=13.8% ]

[ \sigma_{AC}^2=0.5^2(8)+0.5^2(5.44)+2(0.5)(0.5)(6.4) ]

[ \sigma_{AC}^2=2+1.36+3.2=6.56 ]

[ \sigma_{AC}=\sqrt{6.56}=2.56% ]

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d. Which portfolio is recommended and why?

Portfolio AB is recommended.

Reasons:

1. Portfolio AB has much lower risk: standard deviation (1.6%) compared with (2.56%) for AC.
2. Its return is only slightly lower: (13.2%) compared with (13.8%) for AC.
3. Assets A and B have negative covariance ((-1.6)), so their returns partly offset each other. This provides a real diversification benefit.
4. Assets A and C have positive covariance ((6.4)), so their returns move together. There is little risk reduction from combining them.
5. Coefficient of variation: [ CV_{AB}=\frac{1.6}{13.2}=0.121 ] [ CV_{AC}=\frac{2.56}{13.8}=0.186 ] AB gives better return per unit of risk.

Therefore, for a normal risk-averse investor, Portfolio AB is better. Portfolio AC may be chosen only by an investor who wants a slightly higher return and can tolerate higher risk.

Given:

• Current market interest rate, r₀ = 9%
• Expected interest rate after one year, r₁ = 7%
• Holding period = 1 year
• Bond A: 25-year zero coupon bond, par Rs. 1,000
• Bond B: 20-year, 5.5% coupon bond, par Rs. 1,000
• Both bonds carry the same agency rating

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a. Which bond maximizes capital gain income?

Bond A: 25-year Zero Coupon Bond

Purchase price today (at r = 9%, n = 25):

$P_0 = \frac{1000}{(1.09)^{25}} = \frac{1000}{8.6231} = \text{Rs. }115.97$

Selling price after one year (remaining maturity = 24 years, r = 7%):

$P_1 = \frac{1000}{(1.07)^{24}} = \frac{1000}{5.0724} = \text{Rs. }197.15$

Capital gain = P₁ − P₀ = 197.15 − 115.97 = Rs. 81.18

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Bond B: 20-year, 5.5% Coupon Bond

Annual coupon = 5.5% × 1000 = Rs. 55

Purchase price today (at r = 9%, n = 20):

$P_0 = 55 \times PVIFA_{9\%,20} + 1000 \times PVIF_{9\%,20}$

$P_0 = 55 \times 9.1285 + 1000 \times 0.1784$

$P_0 = 502.07 + 178.43 = \text{Rs. }680.50$

Selling price after one year (remaining maturity = 19 years, r = 7%):

$P_1 = 55 \times PVIFA_{7\%,19} + 1000 \times PVIF_{7\%,19}$

$P_1 = 55 \times 10.3356 + 1000 \times 0.2765$

$P_1 = 568.46 + 276.51 = \text{Rs. }844.97$

Capital gain = P₁ − P₀ = 844.97 − 680.50 = Rs. 164.47

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Comparison

Decision: Bond B provides a higher capital gain of Rs. 164.47 compared to Rs. 81.18 from Bond A. Therefore, to maximize capital gain income, buy Bond B (20-year, 5.5% coupon bond).

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b. Which bond maximizes total return?

Total return = Capital gain + Coupon interest (if any)

Bond A (Zero Coupon)

• Capital gain = Rs. 81.18
• Coupon interest = Rs. 0
• Total return = Rs. 81.18

Bond B (5.5% Coupon)

• Capital gain = Rs. 164.47
• Coupon interest = Rs. 55
• Total return = 164.47 + 55 = Rs. 219.47

Comparison of Total Return (in %)

Decision: Although Bond B gives a higher absolute total return (Rs. 219.47), the percentage total return is much higher for Bond A (70.0% vs 32.25%). Since the zero coupon bond was purchased at a deep discount (Rs. 115.97), the same absolute gain represents a far greater return on the amount invested.

Therefore, to maximize total return from the investment, buy Bond A (25-year zero coupon bond).

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c. Why did one bond provide better capital gains than the other?

The difference arises from bond price sensitivity to interest rate changes, which depends mainly on maturity and coupon rate:

1. Longer maturity → greater price sensitivity: Bond A has a longer maturity (25 years vs 20 years), so its price changes more (in percentage terms) when interest rates fall. A longer time horizon means the discounting effect of the lower rate compounds over more periods, producing a larger percentage price increase.

2. Lower coupon → greater price sensitivity: Bond A pays no coupons (zero coupon), so its entire value comes from the single payment at maturity. Bond B pays 5.5% coupons annually, which are received early and are less affected by the interest rate change. The bulk of Bond B's value is recovered through coupons before maturity, reducing its exposure to the rate change.

3. Duration effect: The zero coupon bond's duration equals its maturity (25 years), while the coupon bond's duration is shorter than its maturity (because coupons are received earlier). Higher duration means greater percentage price change for a given change in yield.

However, despite Bond A having higher duration (and thus higher percentage price sensitivity), Bond B gave a larger absolute capital gain. This is because Bond B's base price (Rs. 680.50) was much larger than Bond A's (Rs. 115.97). Even though Bond A's percentage price increase was larger (about 70% vs about 24%), the absolute gain was smaller because it was computed on a much smaller base.

Conclusion: Bond B gave a larger absolute capital gain because it had a much higher base price to start with. Bond A gave a larger percentage capital gain because its longer maturity and zero coupon feature made it far more sensitive to the fall in interest rates from 9% to 7%.