Perpetuity, Sinking Funds and Bonds for CUET Applied Maths

AnswerC · ₹75,000

1. A perpetuity pays for ever; its present value is $\frac{R}{i}$, the sum that earns $R$ a year in interest.

2. $\frac{6{,}000}{0.08} = 75{,}000$ rupees.

AnswerA · Only I

1. I: a sum of $\frac{R}{i}$ earns exactly $R$ a period in interest, for ever. True.

2. II: far-off payments are worth less and less today, so the total stays finite, $\frac{R}{i}$. False.

AnswerA · ₹30,000

1. A perpetuity's payment is the yearly interest on its value: $R = PV \times i$.

2. $5{,}00{,}000 \times 0.06 = 30{,}000$ rupees a year.

AnswerD · ₹1,20,000

1. $PV = \frac{R}{i} = \frac{9{,}000}{0.075}$.

2. $= 1{,}20{,}000$ rupees.

AnswerD · ₹2,50,000

1. Present value of a perpetuity $= \frac{R}{i}$.

2. $\frac{15{,}000}{0.06} = 2{,}50{,}000$ rupees.

AnswerB · ₹21,600

1. A perpetuity pays the interest and keeps the fund whole.

2. $3{,}60{,}000 \times 0.06 = 21{,}600$ rupees a year.

AnswerC · ₹50,000

1. Payments are quarterly, so use the quarterly rate: $i = \frac{12\%}{4} = 3\% = 0.03$.

2. $\frac{1{,}500}{0.03} = 50{,}000$ rupees.

AnswerB · 6%

1. $PV = \frac{R}{i}$, so $i = \frac{R}{PV} = \frac{2{,}400}{40{,}000}$.

2. $i = 0.06$, that is 6% a year.

AnswerD · ₹40,000

1. Payments are half-yearly, so use the half-yearly rate: $i = \frac{10\%}{2} = 0.05$.

2. $\frac{2{,}000}{0.05} = 40{,}000$ rupees.

AnswerA · ₹10,000

1. At 5%: $\frac{3{,}000}{0.05} = 60{,}000$. At 6%: $\frac{3{,}000}{0.06} = 50{,}000$.

2. It falls by $60{,}000 - 50{,}000 = 10{,}000$ rupees.

AnswerC · Both I and II

1. I: $\frac{R}{i}$ gets smaller as $i$ grows. True.

2. II: $\frac{2R}{i} = 2 \times \frac{R}{i}$. True.

AnswerC · ₹2,65,000

1. Today's payment is made at once; the rest is an ordinary perpetuity.

2. $15{,}000 + 2{,}50{,}000 = 2{,}65{,}000$ rupees.

AnswerC · ₹3,00,000

1. $\frac{15{,}000}{0.05}$.

2. ₹3,00,000: a lower rate needs a bigger fund to pay the same amount.

AnswerC · ₹2,10,000

1. Today's payment is worth ₹10,000 as it stands; the rest is an ordinary perpetuity worth $\frac{10{,}000}{0.05} = 2{,}00{,}000$.

2. In all, $10{,}000 + 2{,}00{,}000 = 2{,}10{,}000$ rupees.

AnswerB · ₹8,264

1. Two years from now it is an ordinary perpetuity, worth $\frac{1{,}000}{0.1} = 10{,}000$ at that time.

2. Discount it back 2 years: $10{,}000 \times 0.8264 = 8{,}264$ rupees.

AnswerD · ₹54,000

1. Today's payment counts in full, ₹4,000; the rest is an ordinary perpetuity worth $\frac{4{,}000}{0.08} = 50{,}000$.

2. In all, $54{,}000$ rupees.

AnswerA · Only I

1. I: replacing a machine or repaying a bond are typical targets of a sinking fund. True.

2. II: a sinking fund is set aside for its purpose until the date; a savings account can be drawn on freely. False.

AnswerB · ₹1,525

1. The deposits add up to $3 \times 10{,}000 = 30{,}000$.

2. Interest $= 31{,}525 - 30{,}000 = 1{,}525$ rupees.

AnswerD · ₹4,43,840

1. Four deposits of ₹1,10,960.

2. $4 \times 1{,}10{,}960 = 4{,}43{,}840$ rupees.

AnswerB · ₹56,160

1. The deposits total ₹4,43,840.

2. Interest $= 5{,}00{,}000 - 4{,}43{,}840 = 56{,}160$ rupees.

AnswerB · ₹16,380

1. The deposits $R$ must grow to $A$: $R \times \frac{(1 + i)^{n} - 1}{i} = A$, so $R = \frac{A\,i}{(1 + i)^{n} - 1}$.

2. $R = \frac{1{,}00{,}000 \times 0.1}{0.61051} \approx 16{,}380$ rupees.

AnswerC · ₹22,531.25

1. The fund is the amount of an annuity: $A = R \times \frac{(1 + i)^{n} - 1}{i}$.

2. $A = 5{,}000 \times \frac{0.3605}{0.08} = 5{,}000 \times 4.50625 = 22{,}531.25$ rupees.

AnswerD · $R = \frac{A\,i}{(1 + i)^{n} - 1}$

1. The deposits form an annuity whose amount must be $A$: $R \times \frac{(1 + i)^{n} - 1}{i} = A$.

2. Solving for $R$: $R = \frac{A\,i}{(1 + i)^{n} - 1}$.

AnswerD · $R\,\frac{(1 + i)^{n} - 1}{i}$

1. Each deposit grows for the periods left after it: the first for $n - 1$, the last for none.

2. Adding the geometric series gives $R\,\frac{(1 + i)^{n} - 1}{i}$.

AnswerB · I, III and IV only

1. A sinking fund has a purpose, a date and a deposit plan; a savings account has none of these restrictions.

2. II is false: the money stays until the date it is meant for.

AnswerA · ₹45,317

1. $R = \frac{A\,i}{(1 + i)^{n} - 1} = \frac{1{,}50{,}000 \times 0.1}{0.331}$.

2. $R \approx 45{,}317$ rupees.

AnswerC · ₹31,525

1. $A = R \times \frac{(1 + i)^{n} - 1}{i} = 10{,}000 \times \frac{0.157625}{0.05}$.

2. $= 10{,}000 \times 3.1525 = 31{,}525$ rupees.

AnswerB · ₹1,10,960

1. Sinking-fund payment $= \frac{F \cdot i}{(1 + i)^{n} - 1} = \frac{5{,}00{,}000 \times 0.08}{0.36049}$.

2. $\frac{40{,}000}{0.36049} \approx 1{,}10{,}960$ rupees.

AnswerC · ₹3,31,625

1. Per half-year, $i = 0.04$ and there are $n = 6$ deposits.

2. $A = 50{,}000 \times \frac{0.2653}{0.04} = 50{,}000 \times 6.6325 = 3{,}31{,}625$ rupees.

AnswerC · ₹25,749

1. Per quarter $i = \frac{8\%}{4} = 0.02$, and there are $n = 8$ deposits.

2. $A = 3{,}000 \times \frac{0.171659}{0.02} \approx 25{,}749$ rupees.

AnswerA · ₹45,718

1. Per half-year $i = 0.06$, and $n = 4$ deposits.

2. $R = \frac{2{,}00{,}000 \times 0.06}{0.262477} \approx 45{,}718$ rupees.

AnswerD · ₹2,30,797

1. The first deposit has earned one year's interest; the second has just gone in.

2. $1{,}10{,}960 \times 1.08 + 1{,}10{,}960 = 1{,}10{,}960 \times 2.08 \approx 2{,}30{,}797$ rupees.

AnswerB · ₹80

1. The coupon is the coupon rate times the face value.

2. $0.08 \times 1{,}000 = 80$ rupees a year.

AnswerD · At a discount

1. A bond sells at par at its face value, at a premium above it and at a discount below it.

2. ₹950 is below ₹1,000, so it sells at a discount of ₹50.

AnswerB · 9%

1. A year's coupons are $2 \times 45 = 90$ rupees.

2. $\frac{90}{1{,}000} = 9\%$ a year.

AnswerC · 1 April 2029

1. The maturity date is when the face value is repaid, here 5 years after issue.

2. 1 April 2024 plus 5 years is 1 April 2029.

AnswerD · At a premium

1. A price above the face value is a premium; below it, a discount; equal to it, par.

2. ₹104 is above ₹100, so the bond sells at a premium.

AnswerC · ₹40

1. The coupon is worked out on the face value, not the price paid.

2. $8\% \times 500 = 40$ rupees.

AnswerA · 4%

1. The discount is $500 - 480 = 20$ rupees.

2. As a share of face value: $\frac{20}{500} \times 100 = 4\%$.

AnswerD · Below its face value, at a discount

1. The coupons are fixed. If buyers want a higher return than they pay, buyers will only pay less for the bond.

2. So the price falls below the face value: a discount.

AnswerA · A-II, B-IV, C-I, D-III

1. The face value is repaid on the maturity date; the coupon rate fixes the yearly interest on it.

2. The yield to maturity is the rate of return a buyer earns at today's price, held to the end.

AnswerA · Only I

1. I: the fixed coupons are now worth less against new bonds paying more, so the price falls. True.

2. II: the coupon is fixed by the coupon rate and face value when the bond is issued. False.

AnswerC · 10%

1. The coupon is $8\% \times 1{,}000 = 80$ rupees a year.

2. Current yield $= \frac{80}{800} = 10\%$, more than the coupon rate because the price is below par.

AnswerA · Only I

1. I: paying exactly the face value, the buyer earns exactly the coupon rate. True.

2. II: paying less than face value, the buyer earns the coupons plus the gain to face value, so the yield is above the coupon rate. False.

AnswerB · ₹225

1. The yearly coupon is $9\% \times 5{,}000 = 450$ rupees.

2. Paid in two halves: 225 rupees every half-year.

AnswerC · 8.33%

1. Current yield is the annual coupon as a percentage of the price paid.

2. $\frac{40}{480} \times 100 \approx 8.33\%$.

AnswerD · 12.5%

1. In the year the investor gets the ₹40 coupon and gains $500 - 480 = 20$ on redemption: ₹60 in all.

2. $\frac{60}{480} \times 100 = 12.5\%$.

AnswerD · The present value of its coupons plus that of its face value

1. A bond's owner receives the coupons and, at maturity, the face value.

2. Its value today is the present value of all of these at the required rate of return.

AnswerA · ₹890

1. Discount the repayment back 2 years.

2. $1{,}000 \times 0.8900 = 890$ rupees.

AnswerB · ₹90

1. The coupon is the coupon rate times the face value.

2. $9\% \times 1{,}000 = 90$ rupees.

AnswerD · At a discount, because the yield is above the 9% coupon rate

1. Buyers can earn 10% elsewhere, but the bond pays only 9%.

2. So they pay less than ₹1,000 for it: a discount.

AnswerB · ₹1,000

1. Its value is $\frac{100}{1.1} + \frac{1{,}100}{(1.1)^{2}}$, the present value of the coupons and the repayment.

2. When the required return equals the coupon rate this is exactly the face value, ₹1,000: the bond sells at par.

AnswerA · Both A and R are true, and R is the correct explanation of A.

1. Discounting the coupons and the face value at 7% gives more than the face value, so A is true.

2. R gives the reason, so R explains A.

AnswerC · ₹1,018.49

1. In a year the holder receives $1{,}000 + 100 = 1{,}100$ rupees.

2. Value $= 1{,}100 \times 0.9259 = 1{,}018.49$ rupees, a premium.

AnswerA · ₹7,513.15

1. Its only payment is the face value in 3 years.

2. Value $= \frac{10{,}000}{1.331} \approx 7{,}513.15$ rupees.

AnswerC · ₹136.20

1. The coupons are an annuity: PV $= R\,\frac{1 - (1 + i)^{-n}}{i} = 50 \times \frac{1 - 0.8638}{0.05}$.

2. $= 50 \times 2.724 = 136.20$ rupees.

AnswerA · Both A and R are true, and R is the correct explanation of A.

1. R: the coupons and face value do not change, but each is divided by a larger $(1 + i)^{t}$. True.

2. Smaller present values add to a lower price, so A is true because of R.

AnswerA · ₹982.64

1. Price $= \frac{90}{1.1} + \frac{1{,}090}{1.1^{2}} = 81.82 + 900.83$.

2. About ₹982.64.

AnswerC · ₹1,017.83

1. Price $= \frac{90}{1.08} + \frac{1{,}090}{1.08^{2}} = 83.33 + 934.50$.

2. About ₹1,017.83, a premium.

AnswerB · ₹965.24

1. Year 1 brings ₹80 and year 2 brings $80 + 1{,}000 = 1{,}080$.

2. Value $= 80 \times 0.9091 + 1{,}080 \times 0.8264 = 72.73 + 892.51 = 965.24$ rupees, below par because 8% is less than 10%.

AnswerC · ₹103.70

1. In a year the holder receives $100 + 12 = 112$ rupees.

2. Value $= 112 \times 0.9259 \approx 103.70$ rupees: a premium, as 12% beats 8%.

AnswerC · ₹1,034.66

1. Coupons: $70 \times 3.4651 = 242.557$. Face value: $1{,}000 \times 0.7921 = 792.10$.

2. Value $\approx 1{,}034.66$ rupees, a premium, since 7% beats 6%.

AnswerB · ₹964.29

1. Year 1: ₹60; year 2: $60 + 1{,}000 = 1{,}060$.

2. Value $= 60 \times 0.9259 + 1{,}060 \times 0.8573 = 55.554 + 908.738 = 964.292$, about ₹964.29, a discount.