# Test 4 2015 Question 6

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(i) A government issues two bonds, BondShort which is a 5 year bond with a coupon of 6% p.a. and BondLong, a 20 year bond with a coupon of 3% p.a. Today, the market interest rates increased drastically at all terms. State which bond will experience more of a fluctuation in price, and give reasons for your answer. (2)

(ii) Calculate the price of an inflation-linked bond which pays a coupon of 6% per annum, semi-annually in arrear, and which is redeemable at 120% for a nominal amount of R150 000 after 10 years. You may assume a nominal yield to maturity of 7% p.a. compounded semi-annually, and an annual rate of inflation of 4% effective. (4)

(iii) In the well developed economy of Perfidious Albion there are two bonds, A and B, with identical characteristics. Bond A has a better credit rating than Bond B. State which of these bonds you expect to earn a higher yield and explain why? (2)

answered Nov 2, 2016 by (200 points)
selected Mar 1, 2017 by Rowan

(i)

There are two elements to each bond which will influence the bond’s price:

Term (5 years versus 20 years)
BondLong’s longer term means that it has many coupons which are received far into the future.  An increase in interest rates will cause these later coupons to be discounted at a higher rate (due to more years of discounting), causing the coupons to have a low present value when compared to BondShort’s coupons.

Coupon (6% p.a. versus 3% p.a.)
BondLong has smaller coupon amounts compared to BondShort.  This means that a larger percentage of the bond’s value is realised at redemption.  An increase in interest rates causes the redemption value of BondLong to be worth proportionally less than BondShort’s redemption due to the aforementioned higher discounting.

Both of these effects contribute to BondLong experiencing more of a fluctuation in price than BondShort.  Note that I have included a more thorough description than what is required for 2 marks.

(The course reader more thoroughly explains the effect that changing interest rates have on a bond’s price in terms of its term and coupon value)

(ii)

Drawing a timeline is hugely beneficial to answering a question like this.  I cannot draw one here, but I highly recommend drawing one to help answer this question.

The price of the bond is calculated by discounting all future cashflows.  This is split into coupons and the redemption amount.

An important thing to note in this question is that the bond is inflation-linked.  This means that the cash flows cannot simply be discounted at the rate of 7%.  We need to calculate a discount rate which accounts for the fact that the cash flows will increase in line with inflation.

Let $$i^{(2)}$$ = 7% (interest rate)
and j = 4% (inflation)

If k is the new interest rate, we have:

1 + k = (1 + $$i^{(2)}$$)/(1 + j)1/2

If this looks confusing, remember that the interest rate is nominal compounded semi-annually, while the inflation rate is effective.

So k = 1.49%

We can now calculate the price of the bond:

(Note that the coupon is based on the nominal value of R150 000)

Price = (150 000 x 6% x 0.5)  $$a_\bar{20|}$$$$_k$$
+ (120% x 150 000)$$(1 + k)^{-20}$$
= R 211 239.98

(A final observation is that the redemption also increases in line with inflation, meaning it should also be discounted at the rate k.)

(iii)

Bond B should provide a higher yield than Bond A.  Bond B’s lower credit rating means that it is riskier than Bond A (an example of this risk is that there would be a higher probability that the issuer of Bond B will default on its payments).  This means that an investor would require a higher yield to hold Bond B opposed to Bond A.