QuestionFind the Macaulay duration of a RM1,000 bond with a $5\%$ coupon, maturing in 4 years, priced at RM965.35 and YTM $6.0\%$ .

Studdy Solution

STEP 1

Assumptions1. The par value of the bond is RM1,000. The annual coupon rate is5%
3. The bond matures in4 years4. The bond is currently priced at RM965.355. The yield to maturity (YTM) is6.0%
6. The bond pays coupons annually

STEP 2

First, we need to calculate the cash flows for each year. The annual coupon payment can be calculated by multiplying the par value by the coupon rate.
$Annual\, coupon\, payment = Par\, value \\times Coupon\, rate$

STEP 3

Now, plug in the given values for the par value and coupon rate to calculate the annual coupon payment.
$Annual\, coupon\, payment = RM1,000 \\times5\%$

STEP 4

Convert the percentage to a decimal value.
$\% =0.05$ $Annual\, coupon\, payment = RM1,000 \\times0.05$

STEP 5

Calculate the annual coupon payment.
$Annual\, coupon\, payment = RM1,000 \\times0.05 = RM50$

STEP 6

The cash flows for the first three years are the coupon payments and for the fourth year, it is the coupon payment plus the par value.

STEP 7

The present value of each cash flow can be calculated using the formula $PV = \frac{C}{(1 + r)^t}$ where- $C$ is the cash flow- $r$ is the yield to maturity (YTM) - $t$ is the time period

STEP 8

Calculate the present value of each cash flow. Convert the YTM to a decimal.
$6\% =0.06$

STEP 9

Calculate the present value of the cash flows for each year.
PV = \frac{RM50}{( +.06)^} $PV2 = \frac{RM50}{( +.06)^2}$ $PV3 = \frac{RM50}{( +.06)^3}$ $PV4 = \frac{RM,050}{( +.06)^4}$

STEP 10

Calculate the weighted time of each cash flow by multiplying the time period by the present value of the cash flow and dividing by the bond price.
$Weighted\, time = \frac{t \\times PV}{Price}$

STEP 11

Calculate the weighted time for each year.
$Weighted\, time = \frac{ \\times PV}{RM965.35}$ $Weighted\, time = \frac{ \\times PV}{RM965.35}$ $Weighted\, time3 = \frac{3 \\times PV3}{RM965.35}$ $Weighted\, time4 = \frac{4 \\times PV4}{RM965.35}$

STEP 12

The Macaulay duration is the sum of the weighted times.
$Macaulay\, duration = \sum Weighted\, times$

STEP 13

Calculate the Macaulay duration.
$Macaulay\, duration = Weighted\, time + Weighted\, time2 + Weighted\, time3 + Weighted\, time$ This will give the Macaulay duration of the bond in years.

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